?The energy levels of N atoms in the tight-binding Hückel approximation are the roots of

Chapter 15, Problem P15C.6

(choose chapter or problem)

The energy levels of N atoms in the tight-binding Hückel approximation are the roots of a tridiagonal determinant (eqn 15C.1):

\(E_{k}=\alpha+2 \beta \cos \frac{k \pi}{N+1}\) \quad \(k=1,2, \ldots, N\)

                   

If the atoms are arranged in a ring, the solutions are the roots of a ‘cyclic’ determinant:

\(E_{k}=\alpha+2 \beta \cos \frac{2 k \pi}{N}\) \quad \(k=0, \pm 1, \pm 2, \ldots, \pm \frac{1}{2} N\)

(for N even). Discuss the consequences, if any, of joining the ends of an initially straight length of material.

Text Transcription:

E_k=alpha+2 beta cos k pi/N+1

k=1,2, ldots, N

E_k=alpha+2 beta cos 2k pi/N

k=0, pm 1, pm 2, ldots, pm1/2 N