The secular determinant for this system becomes
a À E b
0
b
aÀ E b
0
b
aÀ E
¼ 0
The zero terms arise from carbon atoms 1 and 3 which are unconnected.
Dividing by b and letting (a − E)/b = X, the secular determinant is
X 1 0
1 X 1
0 1 X
¼ 0
resolved as
X
X 1
1 X
À 1
1 1
0 X
¼ 0
X
3
– 2X = 0 with solutions X = 0 X ¼ Æ
ffiffi ffi
2
p
that is
ða À EÞ=b ¼ 0; þ
ffiffi ffi
2
p ; À
ffiffi ffi
2
p
These three roots produce the three energies
E 1 ¼ a þ
ffiffi ffi
2
p b
E ¼ a
E 3 ¼ a À
ffiffi ffi
2
p b
The secular equations can be written by taking the general equations and multiplying the column matrix [C 1 C 2 C 3 ] by the 3 Â 3 determinant.
C 1 X þ C 2 ¼ 0
C 1 þ C 2 X þ C 3 ¼ 0
C 2 þ C 3 X ¼ 0
When X = − √2
C 2 ¼ C 2 =
ffiffi ffi
2
p
C 3 ¼ C 2 =
ffiffi ffi
2
p
1.3 The Hückel Approximation
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