For butadiene, assuming
and all s to be the same, the secular determinant becomes
(20.13)
and for N atomic orbitals in a linear chain, the secular determinant gets generalized to
(20.14)
The solutions for the above secular determinant are the large number of roots of E of eq 20.14. With some
algebraic
manipulations, they can be shown to be
E k =
+2
cos (k
/(N+1)), k = 1,2 ...N
(20.15)
For large N, the energy difference between adjacent levels is very small as illustrated in fig 20.7
Figure 20.7 Energy levels of eq (20.15) in a band
As N becomes very large, the energy difference between EN and E1 becomes
EN - E1 = 4
(20.16)
Although this section is sketchy and we have made many approximations (which need to be improved in real solids), we have an expression for the energy levels in a band).
and all 
s to be the same, the secular determinant becomes 
+2
/(N+1)), k = 1,2 ...N 
