Picket-fence model

In this book we shall write the Hamiltonian as an (algebraic) operator using the appropriate Lie algebra. We intend to illustrate by many applications what we mean by this cryptic statement. It is important to emphasize that one way to represent such a Hamiltonian is as a matrix. In this connection we draw attention to one important area of spectroscopy, that of electronically excited states of larger molecules,4 which is traditionally discussed in terms of matrix Hamiltonians, the simplest of which is the so-called picket fence model (Bixon and Jortner, 1968). A central issue in this area of spectroscopy is the time evolution of an initially prepared nonstationary state. We defer a detailed discussion of such topics to a subsequent volume, which deals with the algebraic approach to dynamics. 

Figure 10.13 Perspective view of a picket fence model of the porphyrin ring system of hemoglobin. Note that the oxygen molecule is bound at an angle (there is a four-fold statistical distribution of the terminal O because of rotation about the Fe—O bond). [From J. P. Collman, R. R. Gagne, C. A. Reed, W. T. Robinson, and G. A. Rodley, Proc. Natl. Acad. Sci. USA 71,1326 (1974).]...

The X-ray crystal structures of the active sites of sperm whale oxyMb and an invertebrate oxyHb are shown in Figure 7. As anticipated from the picket-fence model, O2 binds to the active sites in a bent, end-on fashion trans to the proximal histidine. A comparison of the structures in Figures 5 and 7 shows that the torsional disorder of the Fe—O—O unit around the Fe—O bond axis in the picket-fence porphyrin O2 adduct is quenched in Mb/Hb, apparently by hydrogen bonding interactions with residues lining the 02-binding pocket—the so-called distal ... 

For the general case, the analytical form of cannot be further discussed from expression (170). However a closed expression may be obtained for the special case of the uniform or picket-fence model, where it is assumed that the nonradiant levels couple to the radiant state with the same matrix element v i = v. We also assume that the levels are equally spaced, Ssi — e,+i — El — D, and the widths are identical, ji=y. Hence,... 

Table 16.3 Electonic absorption specta of oxyhemoglobin and a picket fence model compound...

The picket fence model as carried out in Problem F requires that the coupling between the bright and dark states does not vary too much between one dark state and another nearby so that the actual coupling can be replaced by its mean... 

Figure A7.1 Tiers of increasingly dense manifolds of dark states. The simple picket fence model assumes only one tier of dark states. Only the zero-order bright state, ZOBS, is optically accessible. But it is spread by intramolecular coupling over the dark states. The shading of each dark state indicates how much it contributes to the mixing.

Figure 9.15 A one-dimensional view of the electronic states and their potentials for the electrocyclic ring opening of cyclohexadiene. Shown is an optical absorption to a bright electronic state, IB, with another dark electronic state, 2A, close by. A conical intersection allows a facile conversion from the bright to the dark state and a second conical intersection allows a return to the ground electronic state on which the cyclic and open ring are two isomers, as shown (estimates for the time of passage via the intersections are also shown, in femtoseconds). A delayed process leads to the linear trans isomer. Experimental and computational studies show that the one-dimensional view, while providing a summary of the essential features, misses some key details, for example the role of the twist in reaching the second conical intersection. Such a view, even when another coordinate is included, is complementary to the description using states that we employed in the picket fence model in Chapter 7.

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