Page 66 - Mechanism and Theory in Organic Chemistry

Molecular Orbital Theory 55
other, as are 2 and 3; 1 and 3 are not bonded. The secular equation is A2.19.

Expansion of the determinant gives :
-E3 + 2E = 0

The roots are:

The energies, relative to a, are therefore - d/3, 0, and + d/3; because /3 is a
negative energy, the first of these is the highest energy and the third is the lowest.
To find the orbitals, we substitute each E, in turn, into the set of Equations
A2.23 :

E = - fi gives the relationships A2.24:

The other information we have about the coefficients is the normalization
condition :
c12 + cZ2 + c~~ = 1 (A2.25)
Combining Equations A2.24 and A2.25, we obtain the coefficients for the highest
energy MO :

The coefficients for the other orbitals are obtained in the same way starting with
E = 0 and E = + fi. The orbitals are, in order of decreasing energy, measured
relative to a as the zero of energy:
E3 = - fit3 #3 = 1/2vP1 - 1/d3vpZ + 1/2vP3
E2 = 0 #Z = 1/dPvpl - 1/d3~~3
E~ = + d2/3 = 1/2pp1 + 1/d2vp, + 1/2vp,
These are the orbitals shown in Figure 1.18.