Linear conjugation energy for

Proceeding as we did for the allyl radical, it is easily seen that the electron charge distribution is uniform (one n electron onto each carbon atom, alternant hydrocarbon) and the spin density is zero, as expected for a state with S = Ms = 0 since the two bonding MOs are fully occupied by electrons with opposite spin. The delocalization (or conjugation) energy for linear butadiene is ... 

Table III. Conjugation Energies for Linear Polyenes (kcal/mol)...

The Progression of Orbitals and Energies for Linear Conjugated Systems... 

Figure 12.7 A summary of the molecular orbitals and relative energies for linear conjugated pi systems of 1 to 6 p orbitals. The phase of the ends of a linear pi system molecular orbital alternates as the energy of the MO increases. The fundamentai (lowest MO) is always symmetric.

Figure 2. Results of calculations for conjugated polyenes. (Left) Results of full calculations, where the dashed line (la) gives the normalized r-energies for linear polyenes and the solid line (Ila) gives the energies for cyclic polyenes. The line at 2.14 eV represents the limit for an infinite chain. (Right) Results of calculations in which the antisymmetry principle was not included. The energies for the linear polyenes and the 4n + 2 cyclic polyenes are unchanged, but the An cyclic polyenes are calculated to be too stable. (Reprinted with permission from ref 124a. Copyright 1996 Elsevier Science.)...

Dewar et al.83 showed that the energy of linear polyenes is an additive quantity of bond energy increments. This enabled Dewar to quantify the thermochemical stability or instability of a cyclic molecule relative to a reference molecule. The reference molecule R for a conjugated cyclic molecule... 

For linear, conjugated polyenes, each molecular orbital has a unique energy. 

Pr and Pr are the radial momenta conjugate to R and r, respectively UC and uv are the translational and vibrational reduced masses t and j are the orbital and rotational angular momenta and V Is the full potential energy function V depends on the three variables R, r and Y where y Is the angle between R and r such that y 0 for linear COH geometries In the lOS treatment, the quantities t,j and y are taken to be parameters, with I chosen In the usual way (from 0 to < In the quantum calculations and 0 to imax classical calculations with... 

The energies of the molecular orbitals can also be deduced by the same device, used for linear conjugated systems, of inscribing the conjugated system inside a circle of radius 2/3. There is no need for dummy atoms, since the sine curves go right round the ring, and the picture is therefore that shown in Fig. 1.44. 

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