## William Barford

Print publication date: 2013

Print ISBN-13: 9780199677467

Published to Oxford Scholarship Online: May 2013

DOI: 10.1093/acprof:oso/9780199677467.001.0001

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# Appendix C (p.247) Single-particle eigensolutions of a periodic polymer chain

Source:
Electronic and Optical Properties of Conjugated Polymers
Publisher:
Oxford University Press

In this appendix we derive the eigenfunctions and eigenvalues of a noninteracting periodic polymer chain. We use a straightforward generalization of the method employed in Section 3.4.1 for the uniform cyclic chain.

We write the Hückel Hamiltonian for a polymer composed of a periodic sequence of monomers as

$Display mathematics$
(C.1)
where H m is the intramonomer one-particle Hamiltonian and H mn is the intermonomer one-particle Hamiltonian. In particular,
$Display mathematics$
(C.2)
where $c m σ i †$ creates an electron on the ith site of the mth unit cell and $t m i j$ is the intramonomer transfer integral between sites i and j.

Similarly,

$Display mathematics$
(C.3)
where $t m n i j$ is the intermonomer transfer integral between sites i and j on monomers m and n, respectively.

Exploiting the translational invariance of the polymer we introduce the Bloch transforms,

$Display mathematics$
(C.4)
and (p.248)
$Display mathematics$
(C.5)
where the Bloch wavevector, k = 2πj/Nd, d is the repeat distance, N is the number of unit cells and the quantum number j satisfies, −N/2 ≤ jN/2.

Substituting the Bloch transforms into eqn (C.2) and eqn (C.3), and following the same procedure as in Section 3.4.1, we obtain the momentum space representation,

$Display mathematics$
(C.6)
where
$Display mathematics$
(C.7)

Now, if $S _ _$ is the unitary matrix that diagonalizes $H _ _ 0$ (and $S _ _ †$ is its adjoint), we may write eqn (C.6) as

$Display mathematics$
(C.8)
where
$Display mathematics$
(C.9)
is the diagonal representation of the Hamiltonian and
$Display mathematics$
(C.10)
are the diagonalized Bloch operators.

We now use this procedure to find the eigensolutions of the dimerized chain and poly(para-phenylene) as examples of the method.

# C.1 Dimerized chain

With the sites of the unit cell labelled as shown in Fig. 3.4, eqn (C.6) becomes

$Display mathematics$
(C.11)
where t s and t d are the transfer integrals for the single and double bonds, respectively.

By the similarity transformation (eqn (C.9)) H is diagonalized to

$Display mathematics$
(C.12)
where $c k σ v †$ and $c k σ c †$ are defined by eqns (3.34) and (3.35), respectively, and $ϵ k v$ and $ϵ k v$ by eqns (3.38) and (3.39), respectively.

# C.2 (p.249) Poly(para-phenylene)

With the sites of the phenyl ring labelled as shown in Fig. 9.4, we have

$Display mathematics$
(C.13)
where t and t s are the intraphenyl and bridging bond transfer integrals, respectively. The eigenvalues are given by eqn (9.13) and plotted in Fig. 9.8.