The Circle Mnemonic

The circle mnemonic has also been adapted for use with linear systems. A linear polyene with m p orbitals (where tn is an integer 1) is transformed 

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The circle mnemonic was de vised by Arthur A Frost a theoretical chemist at North western University... 

The circle mnemonic was devised by Arthur A. Frost, a theoretical chemist at Northwestern University. 

FIGURE 5.3 The VB structures for singlet and triplet states of C3H3 +, along with the graphical representation of their interaction matrix elements. The spread of the states is easily predicted from the circle mnemonic used in simple Hiickel theory. The expressions for the VB structures (dropping normalization) are deduced from each other by circular permutations 1 , = ab — ab, 1 <1>2 = bc — bc, 3 = ca — cI = ab, 34>2 = bc, 33 = ca. ... 

In a pericyclic reaction the array of basis orbitals of the reacting molecule is cyclic halfway along the reaction coordinate. The array will either be of the Mobius or the Hiickel type. Since the circle mnemonics give the distribution of MO energies... 

The circle mnemonic leads directly to a familiar result of HMO theory. As illustrated in Figure 4.25, the molecular orbitals of monocyclic conjugated n... 

As is often the case with simple HMO theory, it is the energy levels of the Mobius systems and not the MOs themselves that are of primary interest. Zimmerman developed a circle mnemonic, analogous to the circle mnemonic used with HMO theory in Chapter 4, which provides a shortcut to finding tt e Mobius energy levels. For Mobius MOs, the polygon corresponding to the cyclic molecule is inscribed in the circle of radius 26 with one side down (not with a comer down as was done for Huckel systems). This procedtue is illustrated in Figure 11.104 for both (a) Huckel MOs and (b) Mobius MOs for cyclopropenyl, cyclobutadiene, and benzene. 

Figure 11.101 (page 765), calculate the energy levels of a Mobius cyclopro-penyl system and verify that they are identical with those given by the circle mnemonic (Figure 11.104, page 767). 

The circle mnemonic for determining the tt system molecular orbital diagram for cyclic hydrocarbons at the HOckel level. 

A useful mnemonic device for quickly setting down the HMOs for cyclic systems is Frosts circle.If a regular polygon of n sides is inscribed in a circle of diameter 4/3 with one comer at the lowest point, the points at which the comers of the polygon touch the circle define the energy levels. The energy levels obtained for benzene and cyclobutadiene with Frost s circle are shown in Fig. 1.12. 

For the special case of cyclic polyenes, (CH) , Frost and Musulin65 obtained the general circle mnemonic for the HMO eigenvalues in the form... 

Figure 3.48 The Frost-Musulin circle mnemonic (Eq. (3.141)) for HMO orbital energies ej of benzene, n = 6.

Fig. 4.22 A useful mnemonic for getting the simple Hiickel method pattern for cyclic n systems. Setting the radius of the circle at 2ipi, the energy separations from the nonbonding level can even be calculated by trigonometry...

Whereas in the Frost mnemonic for Hiickel systems the polygon is inscribed with a vertex down, in the Zimmerman mnemonic for Mobius systems the inscription is with the polygon side down. Three examples of each type are shown in Figure 9. Note that each intersection of the polygon with the inscribed circle corresponds to an MO and that the vertical positioning of the intersection gives the MO energy analytically. Thus, all of the Hiickel systems, with one vertex at the bottom, have in common one MO at —2 P. Also the odd-sized arrays have their Hiickel and Mobius relatives turned upside down from one another, while in the even series there is no such relationship. 

This operation is called the Frost-Htickel mnemonic or simply the circle trick. 

The appropriate correlation diagrams can also be constructed for the Hiickel and Mobius closures in pericyclic processes where the system maintains some symmetry, and the method is capable of extension to include for unsymmetrical systems (Zimmerman, 1966, 1971). The molecular orbital energies of Hiickel- and Mobius-type cyclic polyenes are readily derived from the simple circle mnemonic discussed earlier (see pp. 43 and 55). 

Frosts circle (Section 11 19) A mnemonic that gives the Huckel TT MOs for cyclic conjugated molecules and 10ns Functional class nomenclature (Section 4 2) Type of lUPAC nomenclature in which compounds are named according to functional group families The last word in the name... 

Frost s circle (Section 11.19) A mnemonic that gives the Hiickel -tr MOs for cyclic conjugated molecules and ions. 

The stereochemistry of any pericyclic reaction can be predicted by counting the total number of electron pairs (bonds) involved in bond reorganization and then applying the mnemonic "The Electrons Circle Around. " That is, thermal (ground-slate) reactions involving an even number of electron pairs occur with either conrotatory or antarafacial stereochemistry. Exactly the opposite rules apply to photochemical (excited-state) reactions. 

The stereochemistry of any pericyclic reaction can be predicted by counting the total number of electron pairs (bonds) involved in bond reorganization and then applying the mnemonic The Electrons Circle... 

As in the case of Huckel-type tt systems, it is possible to give a convenient mnemonic form for the closed formulas 118> which is based on the relation of cosine and sine functions to motion on a circle. 

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