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Electronic and Optical Properties of Conjugated Polymers$

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

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

H = m H m + m n H m n ,
(C.1)
where H m is the intramonomer one-particle Hamiltonian and H mn is the intermonomer one-particle Hamiltonian. In particular,
H m = σ c _ m σ · H _ _ m · c _ m σ = σ ( c m σ 1 c m σ 2 ) · ( t m 11 t m 12 t m 21 t m 22 ) · ( c m σ 1 c m σ 2 ) ,
(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,

H m n = σ c _ m σ · H _ _ m n · c _ n σ = σ ( c m σ 1 c m σ 2 ) · ( t m n 11 t m n 12 t m n 21 t m n 22 ) · ( c n σ 1 c 2 σ 2 ) ,
(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,

c _ m σ = 1 N k c _ k σ exp ( i k m d ) ,
(C.4)
and (p.248)
c _ m σ = 1 N k c _ k σ exp ( i k m d ) ,
(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,

H = k σ c _ k σ · H _ _ m · c _ k σ + k σ n c _ k σ · H _ _ m n · c _ k σ exp ( i k n d ) = k σ c _ k σ · H _ _ 0 · c _ k σ ,
(C.6)
where
H _ _ 0 = H _ _ m + n H _ _ m n exp ( i k n d ) .
(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

H = k σ c _ k σ · S _ _ · S _ _ · H _ _ 0 · S _ _ · S _ _ · c _ k σ = k σ c ̃ _ k σ · H ̃ _ _ 0 · c ̃ _ k σ ,
(C.8)
where
H ̃ _ _ 0 = S _ _ · H _ _ 0 · S _ _
(C.9)
is the diagonal representation of the Hamiltonian and
c ̃ _ k σ = c _ k σ · S _ _
(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

H = k σ ( c k σ 1 c k σ 2 ) · ( 0 t d + t s exp ( i 2 k a ) t d + t s exp ( i 2 k a ) 0 ) · ( c k σ 1 c k σ 2 ) ,
(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

H = k σ ( c k σ v c k σ c ) · ( ϵ k v 0 0 ϵ k c ) · ( c k σ v c k σ c ) ,
(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

H = k σ ( c k σ 1 c k σ 2 c k σ 3 c k σ 4 c k σ 5 c k σ 6 ) ( 0 t 0 t s exp ( i k d ) 0 t t 0 t 0 0 0 0 t 0 t 0 0 t s exp ( i k d ) 0 t 0 t 0 0 0 0 t 0 t t 0 0 0 t 0 ) · ( c k σ 1 c k σ 2 c k σ 3 c k σ 4 c k σ 5 c k σ 6 ) ,
(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.