Page 64 - Mechanism and Theory in Organic Chemistry

Molecular Orbital Theory 53

Remember that there are N of these equations, one for each of the N possible
values of i. If we regard the 6,'s as unknowns, these equations constitute a set of N
linear homogeneous equations in N unknowns. The set has solutions for those
values of E for which the determinant

is zero; in more compact notation,
det JH,, - ES,,J = 0 (A2.15)
This is the secular equation.
Solution of the secular equation amounts to finding the roots of an Nth order
equation in E. The N roots are the energies of the N molecular orbitals; the
forms of the orbitals in terms of the basis atomic orbitals gpj are found by substi-
tuting each value of E, in turn, back into Equations A2.13 and solving for the cys
using the additional condition that each MO +, is to be normalized,

Electrons are then assigned, two to each molecular orbital starting from the
lowest energy.
Standard computer methods are available to solve Equation A2.15 if
numerical values can be found for Htj and St,. The St, can be determined easily
with the aid of the computer, but the Hij, which represent the interactions be-
tween the basis orbitals, are more difficult to obtain. A number of methods can
be used to deal with this problem.

The Hiickel method is the simplest of the quantitative MO techniques. It has the
following characteristics :

1. Only .rr electrons are treated.
2. The basis consists of a p orbital on each carbon atom of the T system.
3. All overlaps, Stj, i # j, are assumed to be zero; overlaps St, are unity
because the basis orbitals are normalized.
4. Interactions Hi,, i # j, are assumed to be zero except for pairs of basis
orbitals i and j that are on carbons directly bonded to each other. All H,, for
bonded pairs are assumed to have the same value, which is not calculated but is
simply called /3. (/3 represents an energy lowering and, therefore, is negative.)
5. Hi,, which represents the energy of an electron in the basis orbital i be-
fore any interaction with its neighbors occurs, is the same for all i (because all
basis orbitals are the same). It is called a.