Appendix D (p.250) The Holstein model
Appendix D (p.250) The Holstein model
The Holstein model (Holstein 1959a, 1959b) was originally introduced to describe the coupling of charges to the vibrational degrees of freedom in molecular assemblies. The vibrational degrees of freedom are assumed to be normal modes localized on individual molecular components of the assembly. This coupling creates a charged polaron: a deformation of the nuclear coordinates around the charged particle. The Holstein model is readily extended to describe the coupling of Frenkel excitons to vibrational degrees of freedom, the only differences between the two cases being the precise mechanism for ‘particle’ transfer and ‘particle’phonon coupling. Partly for reasons of its simplicity, this model has thus widely been used to investigate charge and exciton dynamics in molecular assemblies. (See (Spano 2006) for a review.)
In molecular assemblies the charge or exitonic bandwidth is generally much smaller than the characteristic energy of the normal mode. In this limit the local nuclear displacement responds instantaneously to the particle. The particle is then dressed by the local displacement as it propagates through the molecular assembly, giving it an effective mass enhanced by the inverse of the 0 − 0 FranckCondon overlap factor (Holstein 1959b). This is the small, heavy polaron limit.
The Holstein model may also be applied to individual conjugated polymers, where now the polymer is equivalent to the molecular assembly and the moieties constituting the polymer (e.g., vinylene or phenylene units) are equivalent to the individual molecular components. The Holstein model thus provides a quantitative description of charged polarons (see Section 4.9) and Frenkel exciton polarons (see Section 9.7.4) in conjugated polymers. (For the case of Frenkel exciton polarons, the starting point is the derivation of the Frenkel exciton Hamiltonian, described in Appendix H. When applied to Frenkel excitons it is named the FrenkelHolstein model.)
In contrast to molecular assemblies, however, in conjugated polymers the charge or exitonic bandwidth is generally larger than the characteristic energy of the normal mode, and thus the local displacement responds slowly to the particle.^{1} In addition, as shown below in Section D.2.3, the polaron is ‘large’, i.e., the nuclear deformation extends over many monomers. In the adiabatic limit the probability of coherent motion of all the displaced oscillators encompassing the polaron vanishes, and thus the particle becomes selftrapped by the nuclear displacements. It is this limit that we consider here.^{2}^{3}
D.1 (p.251) The model
Suppose that the operator ${\widehat{a}}_{n}^{\u2020}$ creates a particle (e.g., an electron, hole, or Frenkel exciton) on the nth monomer. We also suppose that the particle couples to a local generalized normal coordinate, Q _{n}, on the same monomer. (For tPA the monomer is the vinylene bond and the normal mode is the symmetric vibration of this bond. For a Frenkel exciton confined to a single phenyl ring the normal mode is associated with the aromatic to quinoid distortion.)
Then the Holstein model reads
It is convenient to introduce the dimensionless variables,
We first consider the BornOppenheimer limit of the large polaron limit, defined by Mω → ∞, so that eqn (D.4) becomes
The general oneparticle eigenstate of eqn (D.5) is
To find the equilibrium normal coordinates we minimize the expectation value of the energy, $E(\{\stackrel{\u0303}{Q}\})\equiv \u3008\psi {H}_{\text{Holstein}}^{\text{BO}}\psi \u3009$, with respect to these coordinates. From eqn (D.5) and eqn (D.6),
Now, substituting eqn (D.8) into eqn (D.5), ignoring for the moment the elastic strain energy (as it is a cnumber), and projecting the Schrödinger equation,
Finally, adding 2Jψ _{n} to both sides of eqn (D.10) and replacing the second order difference (ψ _{n} _{+1}−2ψ _{n}+ψ _{n} _{−1}) by d ^{2} ψ _{n}/dn ^{2} gives the nonlinear Schrödinger equation,
D.2 (p.253) General solutions
D.2.1 Single monomer molecule, N = 1
Returning to eqn (D.8), for a single monomer molecule ψ = 1 and hence the equilibrium (dimensionless) displacement is ${\stackrel{\u0303}{Q}}^{\text{eq}}=A$. The relaxation energy, E _{r}, relative to the undistorted monomer is (via eqn (D.7))
D.2.2 Dimer, N = 2
For a dimer, ${\psi}_{1}=\pm {\psi}_{2}=1/\sqrt{2}$, which implies that ${\stackrel{\u0303}{Q}}_{1}^{\text{eq}}={\stackrel{\u0303}{Q}}_{2}^{\text{eq}}=A/2$ and hence ${E}_{r}=\hslash \omega S/2$. Notice that the relaxation energy on a dimer is half that of a single monomer molecule.
D.2.3 Polymer, N → ∞
In the continuum and asymptotic limits, eqn (D.11) has the exact solution (Holstein 1959a; Rashba 1982)
The associated energy eigenvalue of eqn (D.13) is E′ = −Jγ ^{2} and thus the energy of the polaron relative to the free particle is E′ plus the elastic strain energy, namely
(with ${\stackrel{\u0303}{Q}}_{n}^{\text{eq}}$ given by eqn (D.8)). The relaxation energy of the polaron with respect to the undistorted polymer is then E _{r} = −ΔE, or (using eqns (D.12), (D.14), and (D.15)),
From this analysis we see that:

• The equilibrium displacement of the nth oscillator, Q _{n}, exactly mirrors the particle probability density ψ _{n}^{2}, as shown by eqn (D.8).

• (p.254) The expression for the polaron wavefunction, eqn (D.13), quantitatively reproduces the chargedpolaron wavefunction obtained via the SuSchriefferHeeger model for weak polaron binding (Campbell et al. 1982). It also quantitatively reproduces the Frenkel excitonpolaron wavefunction for poly(paraphenylene) obtained via the CIS calculation, shown in Fig. 9.19.

• The relaxation energy in the continuum, asymptotic limit is E _{r}(N = 1)/3N _{p}, i.e., smaller than the single monomer molecule value and a decreasing function of the spread of the polaron wavefunction. In general, the relaxation energy is a decreasing function of polymer size, as shown by Agranovich and Kamchatnov (1999) and illustrated in Fig. 9.21. The physical explanation for this reduction can be traced to eqn (D.7), which shows that the particle's coupling to the nth normal coordinate is proportional to its probability of being on the nth monomer.

• Using eqn (D.14) and eqn (D.16), we see that the relaxation energy in the asymptotic limit is a decreasing function of the particle bandwidth, 4J. Thus, particles with the same HuangRhys parameter, but a higher kinetic energy exhibit a smaller relaxation energy.
D.3 Variational calculation
Further insight into polaron formation may be gathered via a variational calculation with respect to the particle wavefunction. Using eqn (D.7) and eqn (D.8) the Holstein model in the continuum limit has an associated energy functional,
The important observation of this analysis is that for small α, E(α) is a decreasing function of α. In other words, unlike the case in three dimensions, in onedimensional systems polaron formation is a barrierless process. This explains why polaron formation in conjugated polymers is an ultrafast process, occurring within the first CC bond oscillation.
D.4 Optical intensities
In this section we evaluate the transition dipole moments between the ground and excited states of neutral polymers, and so we now assume that the operator ${\widehat{a}}_{n}^{\u2020}$ creates a Frenkel exciton on monomer n.
(p.255) The transition dipole moment between the ground state, GS〉, (defined as the vacuum of the Frenkel exciton and with undisplaced oscillators) and the excited state, EX〉, (defined as the Frenkel exciton state and with displaced oscillators) is
D.4.1 0 − 0 transition
The ground state in the absence of vibrational excitations is
Evaluating eqn (D.21) gives
Assuming that oscillators on different sites are orthogonal, i.e.,
Summing ψ _{n} and ψ _{n}^{4} gives
In contrast (as shown in Section I.1.1), for ‘free’ excitons
Notes:
(^{1}) As shown in Appendix H, the 1^{1} B _{1u} exciton bandwidth in poly(paraphenylene) is estimated to be ~ 6 eV, whereas the relevant phonon energy is ~ 0.2 eV.
(^{2}) Spohn (1986) and Gerlach and Löwen (1991) have shown that the Holstein model does not exhibit a true delocalization to localization (or selftrapping) transition, in the sense that its ground state energy remains analytical for all parameter values. However, pinning potentials (caused, for example, by disorder) can pin the polaron. Thus, in practice, the prediction of selftrapping in the large polaron, adiabatic limit is valid.
(^{3}) The use of the term ‘selftrapping’ in the literature is rather imprecise. In this book it implies an immobile particle localized because of its coupling to the nuclear degrees of freedom. However, in the literature it can also imply a particle whose coupling to the nuclear degrees causes a local nuclear (polaronic) displacement that — for translationally invariant systems — is delocalized. See, also, the discussion in Section 4.11.