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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 L (p.287) Direct configuration interaction-singles calculations for the Pariser-Parr-Pople model

Appendix L (p.287) Direct configuration interaction-singles calculations for the Pariser-Parr-Pople model

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

This appendix describes an efficient direct configuration interaction-singles (CI-S) algorithm for the computation of the low-energy excited states of the Pariser-Parr-Pople model. First, the computation of the Hartree-Fock ground state is briefly described. (The reader is referred to the standard quantum chemistry text books, e.g., (Szabo and Ostlund 1996), for more details of the Hartree-Fock and CI-S methods.)

L.1 Hartree-Fock Solutions

Within the N-dimensional orthonormalized atomic orbital basis of the Pariser-Parr-Pople model, {|i〉}, the Hartree-Fock molecular orbitals,

| β = i = 1 N c i β | i ,
(L.1)
and energies, ϵ β, may be found by iterative solutions of the Roothan equations, defined by
F _ _ · c _ β = c _ β ϵ β .
(L.2)

The Fock matrix is

F i j = H i j c o r e + G i j ,
(L.3)
where
H i j c o r e = t i j δ i j ( U 2 + k i V i k ) δ i j ,
(L.4)

δ ij means that i and j are neighbouring sites, and V ij is the Coulomb integral, defined in eqn (2.54). The G matrix is

G i j = ( P i i U 2 + k i P k k V i k ) δ i j 1 2 P i j V i j ( 1 δ i j ) ,
(L.5)
where P ij is the density matrix, (p.288)
P i j = 2 β occupied N / 2 c i β c j β * .
(L.6)

Thus, for the Pariser-Parr-Pople model at half-filling, the Fock matrix has the particularly simple form

F i j = t i j δ i j 1 2 P i j V i j ( 1 δ i j ) ,
(L.7)

(using P ii ≡ 1 for electron-hole symmetric models at half-filling), reflecting the cancellation of electron-nuclear, nuclear-nuclear, and Hartree mean-field electron-electron interactions. The remaining nontrivial term on the right-hand side of eqn (L.7) arises from the exchange interaction.

L.2 Direct CI-singles method

As pointed out by Tomlinson and Yaron (2003), the direct CI-S method avoids the complete diagonalization of the full CI-S Hamiltonian matrix (Szabo and Ostlund 1996) by performing an iterative sparse matrix diagonalization (e.g., via the conjugate gradient method (Nightingale et al. 1993)). For reduced basis Hamiltonians, such as the Pariser-Parr-Pople and ZINDO models, this approach is particularly efficient. It can also be readily extended to symmetry-adapted trial states (such as electron-hole states), avoiding the need to block diagonalize the Hamiltonian.

This direct method requires the vector |Φ〉 obtained by the operation of H ̂ on an arbitrary vector, |Ψ〉. In particular, following Tomlinson and Yaron (2003), if

| Ψ = a occupied r virtual ψ a r | ψ a r
(L.8)
and
| Φ = a occupied r virtual φ a r | ψ a r ,
(L.9)
where | Φ = H ̂ | Ψ and the initial amplitudes, { ψ a r } , are known, then
φ a r = ( ϵ r ϵ a + E 0 ) ψ a r i j c i a ( V i j ψ ̃ i j ) c j r * + A i j c i a ( V i j ψ ̃ j j ) c i r * .
(L.10)

A = 2 for singlets and A = 0 for triplets, E 0 is the Hartree-Fock ground state energy,

E 0 = 1 2 i j P i j ( H j i c o r e + F j i ) ,
(L.11)
and
ψ ̃ i j = a r c i a * ψ a r c j r .
(L.12)

The Hartree-Fock eigenvectors and eigenvalues ( { c _ β } and {ϵ β}) are obtained via the prior solutions of eqn (L.2).