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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 A (p.242) Dirac bra-ket operator representation of one-particle Hamiltonians

Appendix A (p.242) Dirac bra-ket operator representation of one-particle Hamiltonians

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

Throughout this book electronic models of conjugated polymers are developed within the number or second quantization representation. This representation is particularly powerful for treating many-body problems. However, as it is less familiar than a first quantization approach, this appendix explains the equivalence of the two approaches for single particle Hamiltonians. We take two examples: the fermion noninteracting (or Hückel) Hamiltonian and the Frenkel exciton model.

A.1 The Hückel Hamiltonian

The Hückel Hamiltonian, described in Chapter 3, is

H = n = 1 , σ N t n ( c n σ c n + 1 σ + c n + 1 σ c n σ ) ,
(A.1)
where t n = t(1 + δ n).

As explained in Section 2.4, c n σ creates an electron with spin σ in the spin-orbital, χ n(r,σ). We define the Dirac ket state as

| n , σ = c n σ | 0 ,
(A.2)
where |0〉 is the vacuum state. The ket state |n, σ〉 is formally equivalent to the spin orbital χ n(r, σ).

The bra state, 〈n, σ|, is the conjugate to the ket state. The scalar product of a bra and ket is defined as

m , σ | n , σ .
(A.3)

If the states form an orthonormal set then,

m , σ | n , σ = δ m n .
(A.4)

For an orthonormal set we notice that the operator

| m , σ n , σ |
(A.5)
projects the state |n, σ〉 onto the state |m, σ〉: (p.243)
| m , σ n , σ | k , σ = | m , σ δ n k .
(A.6)

Thus, |m, σ〉 〈n, σ| and c m σ c n σ are equivalent: both have the effect of transferring an electron from the spin-orbital χ n(r,σ) to the spin-orbital χ m(r, σ).

We can therefore express the Hückel Hamiltonian (eqn (A.1)) as

H = n = 1 , σ N t n ( | n , σ n + 1 , σ | + | n + 1 , σ n , σ | ) .
(A.7)

As described in Section 3.4.1, for cyclic undimerized chains this Hamiltonian is diagonalized by the Bloch states,

| k , σ = 1 N n | n , σ exp ( i k n a ) ,
(A.8)
where N is the number of sites. To demonstrate this, consider
H | k , σ = H 1 N n | n , σ exp ( i k n a ) = t N n , σ ( | n , σ exp ( i k ( n + 1 ) a ) + | n + 1 , σ exp ( i k n a ) ) ,
(A.9)
where we have used the properties of the projection operator, eqn (A.5). Resumming the second term on the right-hand side, we have
H | k , σ = t N n , σ ( | n , σ exp ( i k ( n + 1 ) a ) + | n , σ exp ( i k ( n 1 ) a ) ) , = 2 t cos ( k a ) | k , σ = ϵ k | k , σ .
(A.10)

A.2 The Frenkel exciton Hamiltonian

The Frenkel exciton Hamiltonian,

H = m n ( J m n E m E n + J n m E n E m ) + Δ m E m E m ,
(A.11)
was introduced in Section 10.4.2 and is derived in Appendix H. Here, E m creates an exciton on the mth molecule. The ket state
| m = E m n = 1 N | GS n
(A.12)
represents an exciton localized on the mth molecule. (In analogy to eqn (A.2), we may regard n = 1 N | GS n as the vacuum state.)

(p.244) Equation (A.11) is equivalent to

H = m n ( J m n | m n | + J n m | n m | ) + Δ m | m m | .
(A.13)

For a dimer it is easy to show that the symmetric and antisymmetric states

| + = 1 2 ( | 1 + | 2 )
(A.14)
and
| = 1 2 ( | 1 | 2 )
(A.15)
diagonalize H with energies (Δ + J) and (Δ − J), respectively (with J mn = J nm = J).