# Appendix E (p.257) Derivation of the effective-particle Schrödinger equation

# Appendix E (p.257) Derivation of the effective-particle Schrödinger equation

An exciton is a bound electron-hole pair which can be described by two effective particles: a centre-of-mass particle that describes the centre-of-mass motion of the electron and hole, and a relative particle that describes the relative motion of the electron and hole. In this appendix we derive the Schrödinger equation for the relative particle introduced in Chapter 6, and qualitatively explain the essential physics that it describes.

First we need to derive an exciton Hamiltonian that describes an electron in the conduction band bound to a hole in the valence band. To do this it is necessary to recast the Pariser-Parr-Pople model in a molecular orbital basis acting within this restricted electron-hole subspace. The Pariser-Parr-Pople model (defined in Section 2.8.3) is

*V*

_{ij}, is defined in eqn (2.54).

As a simplification, we assume that the Wannier molecular orbitals for a linear, dimerized chain are localized on a particular dimer, i.e.,

Substituting eqns (E.4) and (E.5) into the Pariser-Parr-Pople model, eqn (E.1), we obtain the molecular orbital exciton Hamiltonian,^{1}

We now describe the terms in *H* _{exciton} in turn.

*H* _{1} is the single-electron Hamiltonian,

*γ*indicates the conduction (

*γ*=

*c*) or valence (

*γ*=

*v*) orbitals. The first term on the right-hand side of eqn (E.7) represents the transfer of electrons between nearest neighbour dimers. The second term is the one-electron energy of the HOMO and LUMO states. 2Δ = (

*ϵ*

_{c}−

*ϵ*

_{v}) is the HOMO-LUMO gap on a dimer.

Next, *H* _{2} describes the Coulomb interactions,

The first two terms are the Coulomb interactions between electrons on the same dimer, while the third term is the Coulomb interaction between electrons on dimers ${\ell}^{\prime}$ units apart.

Finally, *H* _{3} describes the exchange interaction,

*γ*≠

*γ*′,

*σ*are the Pauli spin matrices. This interaction arises from the usual Pauli exclusion principle mechanism that the electrons in a triplet state

*on the same dimer*avoid each other, whereas electrons in a singlet state do not. Thus, the exchange energy is

(This is precisely the energy difference between the ${}^{1}{B}_{u}^{-}$ and ${}^{3}{B}_{u}^{+}$ states on a dimer, as shown in Section 5.5.1.) We point out that this term is local: it only applies if the electron and hole are in the same dimer.

(p.259) In terms of the atomic orbital parameters the remaining molecular orbital parameters are

We now seek a solution to the Schrödinger equation,

*E*is the exciton energy relative to the ground state and |Φ

^{MW}〉 is the exciton eigenstate,

|*R* + *r*/2, *R* − *r*/2〉 is the electron-hole basis state, defined in eqn (6.6), and Φ(*r*,*R*) is the exciton wavefunction.

The scalar product $\u3008R+\frac{r}{2},R-\frac{r}{2}|{H}_{\text{exciton}}|{\Phi}^{\text{MW}}\u3009$ gives the following difference equation for Φ(*r*,*R*):

*δ* _{S} _{0} = 1 for singlet excitons and *δ* _{S} _{0} = 0 for triplet excitons. *δ* _{r} _{0} = 1 when *r* = 0. *d* is the contour length between repeat units (e.g., 2*a* for a dimerized chain, where *a* is the lattice spacing), and

Equation (E.19) is the Schrödinger equation for the two particle system of the electron and hole. To derive an effective-particle model we need to separate the centre-of-mass and relative degrees of freedom. For translationally invariant systems we assume that the centre-of-mass wavefunction is a travelling wave, and thus

*K*is the centre-of-mass momentum, satisfying: −

*π*/

*d*≤

*K*≤

*π*/

*d*.

(p.260) For linear chains we assume that the centre-of-mass wavefunction is standing wave, and thus

*β*

_{j}is the centre-of-mass pseudomomentum,

Substituting eqn (E.21) into eqn (E.19), we obtain the following Schrödinger difference equation for the relative wavefunction, *ψ* _{n}(*r*):

*ψ* _{n}(*r*) is the wavefunction for the effective particle that describes the relative motion of the electron and hole, where *r* is their separation. (An identical equation is obtained using eqn (E.22), except that *K* is replaced by *β* _{j}.)

The first term in eqn (E.24) represents the kinetic energy of the relative particle. To further understand the essential physics behind this equation, consider the limit $\stackrel{\u0303}{t}=0$. Then, when *r* = 0 the potential energy of the relative particle is

(because $\stackrel{\u0303}{U}=\stackrel{\u0303}{V}(r=0)$). Thus, when the electron and hole are bound in the same repeat unit, the energies are $(2\Delta +\stackrel{\u0303}{X})$ and $(2\Delta -\stackrel{\u0303}{X})$, for the singlet and triplet excitations, respectively. Similarly, for a general *r* 〉 0, the potential energy is

The potential energy required to separate the electron and hole by a distance *r* is therefore,^{2}

Since Δ*E*(*r* = 0) = 0 and $\stackrel{\u0303}{V}(r)\to 0$ as *r* → ∞, Δ*E*(*r*) is an increasing function of *r*. We therefore interpret $-\stackrel{\u0303}{V}(r)$ as the Coulomb potential binding the relative particle to *r* = 0. Neglecting the particle's kinetic energy, the total energy required to separate the electron and hole is $(\stackrel{\u0303}{U}-2\stackrel{\u0303}{X}{\delta}_{S0})$, but notice that the singlet is bound less strongly than the triplet.

(p.261)
From elementary quantum mechanics, the energy gained in localizing a particle in an attractive potential well is compensated by the increase in its kinetic energy. However, in one-dimension an attractive, symmetric potential has at least one bound state, and thus we deduce that a conjugated polymer will exhibit at least one exciton. (In fact, as shown in Appendix F, for a 1/*r* potential there is an infinite progression of bound states.)

As *r* → ∞

*V*(

*r*) is the Ohno potential, defined in eqn (2.54) and

*ϵ*

_{eff}is the effective relative permittivity arising from the polymer geometry. This scale factor arises because the electron-hole separation, or the relative coordinate,

*r*, is measured as a contour length along the polymer chain (so,

*r*/

*d*is the number of repeat units between the electron and hole). However, the Coulomb interaction is determined by the geometrical separation between the electron and hole. The scaling between these length scales is determined by

*ϵ*

_{eff}. (For example, in the

*trans*-polyacetylene structure, if the contour-distance between the electron and hole is

*r*the geometrical separation is $r/\sqrt{3}$ and thus, ${\u03f5}_{\text{eff}}=1/\sqrt{3}$. For poly(

*para*-phenylene), however,

*ϵ*

_{eff}= 1.)

The exciton Hamiltonian, defined by eqns (E.6), (E.7), (E.9), and (E.10), is general for any two-level exciton model. The parameters, given here by eqns (H.12)–(E.15), are derived for a *trans*-polyacetylene structure using the mapping between atomic orbitals and molecular orbitals given by eqns (E.2) and (E.3). However, the two-level exciton model can equally be applied, for example, to poly(*para*-phenylene) using the HOMO and LUMO defined by eqn (9.2.1) and Fig. 9.4, giving a different set of molecular orbital parameters.

Equation (E.24) is solved in the following appendix in the effective-mass limit. In Appendix H the two-level exciton model is further simplified to give the Frenkel exciton model.

## Notes:

(^{1})
The molecular orbital Hamiltonian also contains terms that change the occupancy of the valence and conduction bands. Such terms are neglected here, however, as they do not connect basis states within the exciton subspace.

(^{2})
Equation (E.27) applies to even parity excitons, i.e., *ψ* _{n}(*r*) = +*ψ* _{n}(−*r*). For odd parity excitons, i.e., *ψ* _{n}(*r*) = −*ψ* _{n}(−*r*), the equivalent expression is $\Delta E(r\u30091)=\stackrel{\u0303}{U}-\stackrel{\u0303}{V}(r)$, which implies that for this class of excitons singlets and triplets are degenerate.