The Cyclopropenyl System Handling Degeneracies

SOLUTION HMO theory favors neither. The difference between linear and bent allyl shows up as a difference in Ci - C2 - C3 angle and in Ci to C3 distance. The HMO method has no angular-dependent features and explicitly omits interactions between non-neighbor carbons, like Cl and C3.  

The allyl system results when three Jt AOs internet in a linear arrangement wherein ffi2 = H23 = P, but H 2, = 0. We ean also treat the situation where the three tt AOs approach each other on vertices of an ever-shrinking equilateral triangle. In this case, each AO interacts equally with the other two. This triangular system is the cyclopropenyl system C3H3 shown in Fig. 8-6. 

For this MO, the coefficients are all of the same sign, and so the AOs show phase agreement across all bonds and all interactions are bonding. 

To find 02 and 03 is trickier. We begin by inserting x = d=l into our simultaneous equations. This gives 

With three unknowns and two equations, an infinite number of solutions is possible. Let us pick a convenient one c = —c 2, C3 = 0. The normalization requirement then gives c = I/V2, C2 = — I/V2, C3 = 0. Let us call this solution 02  

---