In this section we outline a simple method to obtain the energy levels in a band . This is similar to the method needed to obtain the
orbital energies of butadiene. Let us consider the bonding ( ) and antibonding orbitals ( ) of a diatomic.
= c 1b + c 2b
= c 1a
+ c 2a
(20.1)
We need to determine the coefficients c 1b, c 2b, c 1a and c 2a as well as the energies of and . Here and are atomic orbitals. Consider a general MO
defined as
= c 1 + c 2
(20.2)
The normalization of is (since and are normalized individually)
+ 2 c1 c2S with
(20.3)
S =
If H is the hamiltonian operator of the molecule, the energy of the molecule is given by
E =
(20.5)
E =
[
] /
(20.6)
Here
,
and
(20.7)
To find the optimum coefficients c1 and c 2 for bonding and antibonding orbitals, we minimize E with respect to c 1 and c 2 by setting
E/
c 1 = 0 and
E / c2 = 0.
orbital energies of butadiene. Let us consider the bonding ( 
) and antibonding orbitals ( 
) of a diatomic. 
+ c 2b
defined as 
+ 2 c1 c2S with 
] /
,
and 
E/
c 1 = 0 and
E / 
