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SIGMATROPIC REARRANGEMENTS A general sigmatropic rearrangement is shown in Figure 122 A molecule with two conjugated systems of m and n p electrons rearranges in such a way that the s bond appears to migrate across each p system, forming a molecule with the same description but with the p bonds shifted by one carbon atom and the s bond in a new position Here m and n are even integers One or both may be zero A ``zero'' may indicate either that there is no active p system adjacent to that end of the s bond or that there is an active p system which consists of a single empty p orbital (a trivial p system with no electrons) In order that a sigmatropic rearrangement takes place, there must be at least one active p system Sigmatropic rearrangements are sometimes described by a pair of numbers iY j .

Fig. 1.9 A frequency-independent (wideband) antenna in (a) theoretical realisation and (b) practical realisation (bow-tie antenna).

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Equations (50a) (50d) contain terms that describe (a) the absorption of each beam (a), and (b) phase changes in each beam (ibs ikDn0) due to the mean intensityinduced change in the refractive index (Dn0), and nal terms on the right-hand side of the equations involve diffraction from the cosinusoidal modulation in the refractive index (Dn) that exchanges energy between pairs of beams (A1:A3 and A2:A4) that are Bragg-matched to the thermally induced refractive index grating. For general solution of Eq. (50), with arbitrary input strengths and temporal pulse shapes, it is convenient to use a numerical computational method. Analytical solutions can, however, be derived for some important special cases. One such case is when A1, A3 @ A2, A4 and the optical driving terms s and t can be considered functions of A1 and A3 only, and beam A2 as a weak probe beam. When beam A4 is initially zero at the input to the medium, this beam will be created in the medium by diffraction of beam A2 from the refractive index grating induced by writing beams A1 and A3. An important parameter of the interaction is the diffraction ef ciency of beam A2 which can de ned as h jA4/A2j2 I4/I2. For equal writing beam intensities (I1 I3) the diffraction ef ciency can be shown to be given by the simple expression

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These count the number of atoms over which the two ends of the s bond migrate, counting the atom at each end as ``1'' Thus the Cope rearrangement illustrated in Figure 123 is a [3,3] rearrangement since each end of the migrating s bond moves over three atoms In order that the formation of the new (rearranged) s bond may be feasible, the two ends of the active p systems must be able to approach each other in such a way as to enable the terminal p orbitals to overlap, or, in the case of a 1Y j rearrangement (ie, only one end of the s bond migrates), the nonmigrating end of the s bond must be able to approach the other end of the active p system closely enough so that new bond formation can be initiated.

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Stereochemical Considerations In a sigmatropic rearrangement, with maximum labeling, the number of elements of stereochemistry is preserved There are four: chirality at each end of the s bond and geometric isomerism at the far end of each active p system After the rearrangement, the roles are reversed A maximum of 16 2 4 stereoisomeric products may be formed Even if one end of the s bond does not migrate but is an asymmetric center, the stereochemistry of that center may change The stereochemical mode of reaction of a s bond (as a component) may be speci ed as suprafacial or antarafacial in a fashion parallel to the p components A sigma component of the reaction always has two electrons (a p component may have any even number, including zero) A s bond component is.

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