# Ultrafast Fluorescence Depolarization in Conjugated Polymers

Isabel Gonzalvez Perez<EMAIL_ADDRESS>Department of
Chemistry, Physical and Theoretical Chemistry Laboratory, University of
Oxford, Oxford, OX1 3QZ, United Kingdom  William Barford
<EMAIL_ADDRESS>Department of Chemistry, Physical and
Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ,
United Kingdom

###### Abstract

We report on large-scale simulations of intrachain exciton dynamics in
poly(para-phenylene vinylene). Our coarse-grained model describes Frenkel
exciton coupling to both fast, quantized C-C bond vibrations and slow,
classical torsional modes. We also incorporate system-bath interactions. The
dynamics are simulated using the Time Evolution Block Decimation method, which
avoids the failures of the Ehrenfest approximation to describe decoherence
processes and nonadiabatic interstate conversion. System-bath interactions are
modeled using quantum trajectories and Lindblad quantum jump operators. We
find that following photoexcitation, the quantum mechanical entanglement of
the exciton and C-C bond phonons causes exciton-site decoherence. Next,
system-bath interactions cause the stochastic collapse of high-energy
delocalized excitons onto chromophores. Finally, torsional relaxation causes
additional exciton-density localization. We relate these dynamical processes
to the predicted fluorescence depolarization and extract the timescales
corresponding to them.

A wide range of time-resolved spectroscopic techniques, including fluorescence
depolarization1, 2, 3, 4, three-pulse photon-echo5, 6, 7, 8 and coherent
electronic two-dimensional spectroscopy9, indicate that exciton dynamics in
conjugated polymers spans multiple and overlapping timescales. Some of the
timescales extracted from these experiments, ranging from a few femtoseconds
to sub-ns, are listed in Table 1. Dynamics on timescales exceeding ca. 20 ps
are usually assumed to be associated with interchromophore exciton diffusion
in the condensed phase, while dynamics on timescales of less than ca. 10 fs
are so fast, they must be occur from intrinsic intrachain processes. Dynamics
on intermediate timescales, however, arise from coupling to torsional modes
and system-bath interactions. Wells and Blank’s7 analysis of two-color three-
pulse echo peak shifts concluded that for P3HT in chloroform solution the
driving mechanism for exciton relaxation within 200 fs is a coupling to
overdamped torsional modes. On the other hand, for MEH-PPV Yang et al.5 had
previously shown that the ultrafast dynamics in less than 30 fs corresponds to
a stochastic process of localization. Timescales of tens of ps have been
ascribed to a collective motion of an exciton and the rotational degrees of
freedom10, and indeed can be explained by intrachain exciton diffusion driven
by stochastic torsional fluctuations11, 12.

Polymer |  State |  Timescales |  Citation
---|---|---|---
MEH-PPV |  Solution |  $\tau_{1}=50$ fs, $\tau_{2}=1-2$ ps |  Ref2
MEH-PPV |  Solution |  $\tau_{1}=5-10$ fs, $\tau_{2}=100-200$ fs |  Ref13
PDOPT |  Film |  $\tau=0.5-4$ ps |  Ref14
PDOPT |  Solution |  $\tau_{1}<1$ ps, $\tau_{2}=15-23$ ps |  Ref14
P3HT |  Film |  $\tau_{1}=300$ fs, $\tau_{2}=2.5$ ps, $\tau_{3}=40$ ps |  Ref14
P3HT |  Solution |  $\tau_{1}=700$ fs, $\tau_{2}=6$ ps, $\tau_{3}=41$ ps, $\tau_{4}=530$ ps |  Ref15
P3HT |  Solution |  $\tau_{1}=60-200$ fs, $\tau_{2}=1-2$ ps, $\tau_{3}=14-20$ ps |  Ref16
P3HT |  Solution |  $\tau_{1}<100$ fs, $\tau_{2}=1-10$ ps |  Ref3
Table 1: Some of the excitonic dynamical timescales observed in conjugated
polymers.

In this letter, we report on large-scale simulations of intrachain exciton
dynamics in poly(para-phenylene vinylene) that can explain the sub-ps
processes. Our model describes exciton coupling to both fast, quantized C-C
bond vibrations and slow, classical torsional modes. We also incorporate
system-bath interactions. We relate these dynamical processes to the predicted
fluorescence depolarization and extract the timescales corresponding to them.
Crucially, because the exciton dynamics spans 10 to 1000 femtoseconds and over
100 monomers, coarse-grained models of excitons and nuclei must be employed.
Furthermore, since the first two processes to be discussed are associated with
exciton decoherence and nonadiabatic energy relaxation via multiple potential
energy surfaces, an Ehrenfest treatment of the nuclear dynamics is not
appropriate.

The Ehrenfest method fails to describe ultrafast dynamics because of its two
key assumptions17, 18. The first is to treat the nuclei classically. This
means that nuclear quantum tunneling and zero-point energies are neglected,
and that exciton-polarons are not correctly described. The second assumption
is that the total wavefunction is a product of the electronic and nuclear
wavefunctions. This means that there is no entanglement between the electrons
and nuclei, and so the nuclei cannot cause decoherence of the electronic
degrees of freedom. A simple product wavefunction also implies that the nuclei
move in a mean field potential determined by the electrons. This means that a
splitting of the nuclear wave packet when passing through a conical
intersection or an avoided crossing does not occur, and that there is an
incorrect description of energy transfer between the electronic and nuclear
degrees of freedom.

To avoid the failures of the Ehrenfest approximation in describing ultrafast
process we use the Time Evolution Block Decimation (TEBD) method to
numerically evolve the system’s state vector under the action of the Frenkel-
Holstein model in which the fast C-C bond vibrations are fully quantized. (The
TEBD method is briefly described in the SI and implemented in the Tensor
Network Theory Library19.)

The fate of the exciton is controlled by a series of fascinating and complex
quantum mechanical processes. Before describing this, however, we need to
establish the optically prepared state. For a conformationally disordered
polymer, the delocalization of the exciton center-of-mass wavefunction is
determined by the disordered Frenkel model (as given by Eq. (1) in the SI). As
described elsewhere20, 21 and in the SI, exciton eigenstates may be classified
as low-energy local exciton ground states (LEGSs) and high-energy
quasiextended exciton states (QEESs). LEGSs are non-overlapping, space-filling
and quasinodeless. Their spatial extent defines chromophores. In contrast, as
illustrated in Fig. 2 of the SI, QEESs are spatially extended, nodeful and
spatially overlap lower-lying LEGSs. In general, depending on the laser pulse,
an initial state will be a non-stationary state composed of a linear
superposition of these Frenkel model eigenstates.

The initial dynamical process following photoexcitation is the quantum
mechanical correlation of the exciton with phonons associated with high-
frequency C-C bond vibrations. Since the C-C bond vibrations are fast (one
period is 21 fs) they must be treated quantum mechanically, which we do using
the fully quantized Frenkel-Holstein model (as given by Eq. (7) in the SI). As
shown in Fig. 7 of ref22, the instantaneous force on the nuclei causes local
nuclear displacements and the creation of an exciton-polaron within 10 fs of
photoexcitation.

Crucially, the system wavefunction, which at photoexcitation is a direct
product of the exciton and nuclear wavefunctions, is now a quantum
mechanically entangled exciton-nuclear wavefunction. As a consequence, the
exciton decoheres23, meaning that there is a loss of off-diagonal long range
order (ODLRO) in the exciton’s reduced density matrix. To quantify this
statement, we introduce the coherence correlation function24, 25

$C_{n}^{\textrm{coh}}(t)=\sum_{m}\left|\rho_{m,m+n}\right|,$ (1)

where $\rho_{m,m^{\prime}}$ is the exciton reduced density matrix obtained by
tracing over the vibrational degrees of freedom. Here, the density matrix is
defined within the local Frenkel exciton basis $\\{|m\rangle\\}$, where the
index $m$ labels a moiety (equivalent to a site in the Frenkel model). The
loss of site-basis coherence is further quantified by the coherence length,
defined by

$L_{\textrm{coh}}=\sum_{n}C_{n}^{\textrm{coh}}.$ (2)

As illustrated in Fig. 1, $L_{\textrm{coh}}$ decays to ca. 4 moieties in ca.
10 fs, reflecting the localization of exciton coherence resulting from the
short range exciton-phonon correlations.

Figure 1: The exciton coherence length $L_{\textrm{coh}}$ (in number of
moieties) as a function of time in PPV chains of 40 moieties. These results
are calculated from an ensemble of delocalized high-energy states (i.e.,
QEESs) at $t=0$.

The second dynamical process is associated with system-bath interactions. For
an exciton to dissipate energy it must first couple to fast internal degrees
of freedom (as just described) and then these degrees of freedom must couple
to the environment to expell heat. For a low-energy exciton (i.e., a LEGS)
this process will cause adiabatic relaxation on a single potential energy
surface, forming a vibrationally relaxed state26, 27, 28, 29. However, for a
kinetically hot exciton (i.e., a QEES) this relaxation is through a dense
manifold of states and is necessarily a nonadiabatic interstate conversion
between different potential energy surfaces. As already mentioned, the
Ehrenfest approximation fails to correctly describe this process30, so again
it is necessary to treat the exciton and fast C-C bond vibrations on the same
quantum mechanical footing.

Dissipation of energy from an open quantum system arising from system-
environment coupling is commonly described by a Lindblad master equation31,
23, as discussed in the SI. In practice, a direct solution of the Lindblad
master equation is usually prohibitively expensive, as the size of Liouville
space scales as the square of the size of the associated Hilbert space.
Instead, Hilbert space scaling can be maintained by performing ensemble
averages over quantum trajectories (evaluated via the TEBD method), where the
action of the Linblad dissipator is modeled by quantum jumps.32

Ultrafast exciton-site decoherence occurs via the coupling of the exciton to
fast internal degrees of freedom, namely the C-C bond vibrations. We note that
this coupling does not cause exciton density localization. However,
dissipation of energy to the environment causes an exciton in a higher energy
QEES to relax onto a lower energy LEGS (i.e., onto a chromophore) and thus the
exciton density becomes localized. The spatial extent of the exciton density,
averaged over an ensemble of quantum trajectories and different realizations
of the disorder, is quantified by the localization length, defined by20

$L_{\text{loc}}=5.6\left(\langle L^{2}\rangle-\langle
L\rangle^{2}\right)^{1/2}.$ (3)

Fig. 2 shows the time dependence of $L_{\text{loc}}$ with an external
dissipation time $\gamma^{-1}=100$ fs.

Figure 2: The exciton localization length $L_{\textrm{loc}}$ (in number of
moieties) as a function of time in PPV chains of 40 moieties. These results
are calculated from an ensemble of delocalized high-energy states (i.e.,
QEESs) at $t=0$. The horizontal dashed line at 14 moieties corresponds to the
average chromophore size in the absence of exciton-torsional coupling (i.e.,
to $L_{\text{loc}}$ averaged over all LEGSs). The asymptotic value of 9
moieties is a consequence of additional localization caused by torsional
relaxation. The curve is a one-exponential fit with a time constant of 100 fs.

Figure 2 is obtained by averaging over an ensemble of quantum trajectories
starting from a distribution of different QEESs. To understand the physical
process of localization onto a chromophore, Fig. 12 of ref22 illustrates the
exciton density of a single quantum trajectory for a photoexcited QEES. At a
time ca. 20 fs a ‘quantum jump’ caused by the stochastic application of a
Lindblad jump operator causes the exciton to localize onto a LEGS, i.e., the
high-energy extended state has randomly localized onto a chromophore because
of a ‘measurement’ by the environment.

By dissipating energy into the environment on sub-ps timescales, hot excitons
relax into localized LEGSs, i.e., onto chromophores. In our model the final
intrachain relaxation and localization process now takes place, namely
exciton-polaron formation via coupling to the torsional degrees of freedom.
For this relaxation to occur bond rotations must be allowed, which means that
this process is highly dependent on the precise chemical structure of the
polymer and its environment. Assuming that bond rotations are not sterically
hindered, their coupling to the excitons is conveniently modeled by
supplementing the Frenkel-Holstein model by $\hat{H}_{\textrm{rot}}$33 (as
given by Eq. (9) of the SI).

Unlike C-C bond vibrations, being over 10 times slower torsional oscillations
can be treated classically33. Furthermore, since we are now concerned with
adiabatic relaxation on a single potential energy surface, we may employ the
Ehrenfest dynamics, meaning that the exciton dynamics is described by the
time-dependent Schrödinger equation while the torsional modes are subject to
Newton’s equation of motion with a mean field given by the exciton density (as
described further in the SI).

An exciton delocalized along a polymer chain in a chromophore couples to
multiple rotational oscillators resulting in collective oscillator dynamics.
Since the torsional motion is slow, the self-trapped exciton-polaron is
‘heavy’ and in the under-damped regime becomes self-localized on a timescale
of a single torsional period (i.e., ca. $200$ fs in our calculations). In this
limit the relaxed staggered angular displacement mirrors the exciton density.
Thus, the exciton is localized precisely as for a ‘classical’ Landau polaron
and is spread over ca. 9 moieties33, as shown by the asymptotic behavior of
$L^{\text{loc}}$ in Fig. 2.

The horizontal dashed line at 14 moieties in Fig. 2 corresponds to the average
chromophore size in the absence of torsional relaxation, i.e., to
$L_{\text{loc}}$ averaged over all LEGSs. The asymptotic value of ca. 9
moieties is a consequence of the additional density localization caused by
torsional planarization. (Some authors argue that planarization causes exciton
delocalization14. We discuss this point in the SI.)

For general polymer conformations, the loss of ODLRO (or the localization of
the exciton coherence function) causes a reduction and rotation of the
transition dipole moment. The rotation is quantified by the fluorescence
anisotropy, defined by34

$r=\frac{I_{\parallel}-I_{\perp}}{I_{\parallel}+2I_{\perp}},$ (4)

where $I_{\parallel}$ and $I_{\perp}$ are the intensities of the fluorescence
radiation polarised parallel and perpendicular to the incident radiation,
respectively. For an arbitrary state of a quantum system, $|\Psi\rangle$, the
integrated fluorescence intensity polarised along the $x$-axis is related to
the $x$-component of the transition dipole operator, $\hat{\mu}_{x}$, by

$\displaystyle
I_{x}\propto\sum_{v}\left|\langle\Psi|\hat{\mu}_{x}|\textrm{GS},v\rangle\right|^{2},$
(5)

where $|\textrm{GS},v\rangle$ corresponds to the system in the ground
electronic state, with the nuclear degrees of freedom in the state
characterised by the quantum number $v$.

The averaged fluorescence anisotropy is defined by

$\langle
r\left(t\right)\rangle=0.4\times\frac{\sum_{i}I_{i}\left(t\right)r_{i}\left(t\right)}{\sum_{i}I_{i}\left(t\right)},$
(6)

where $I_{i}\left(t\right)$ is the total fluorescence intensity and
$r_{i}\left(t\right)$ is the fluorescence anisotropy, associated with a
particular conformation $i$ at time $t$. The factor of 0.4 is included on the
assumption that the polymers are oriented uniformly in the bulk material.34
Fig. 3 shows the simulated $\langle r\left(t\right)\rangle$ for an ensemble of
initial high energy QEESs over an ensemble of conformationally disordered
polymers.

Figure 3: The fluorescence anisotropy $\langle r\left(t\right)\rangle$ as a
function of time calculated from an ensemble of delocalized high-energy states
(QEESs) at $t=0$. The curve is a two-exponential fit given by Eq. (8).

To understand the sub-10 fs depolarization, it is instructive to express Eq.
(5) as

$\displaystyle I_{x}\propto\sum_{m,n}s_{m}^{x}s_{n}^{x}\rho_{mn},$ (7)

where $s^{x}_{m}$ is the $x$-component of the unit vector for the $m$th moiety
and $\rho_{mn}$ is the exciton reduced density matrix. Then, using Eq. (1),
Eq. (28), and Eq. (7), we observe that the emission intensity, $I_{x}$, is
related to the coherence length, $L_{\textrm{coh}}$. Thus, not surprisingly,
the initial dynamics of $\langle r\left(t\right)\rangle$ resembles that of
$L_{\textrm{coh}}(t)$ shown in Fig. 1. In particular, we observe a loss of
fluorescence anisotropy within 10 fs, mirroring the reduction of
$L_{\textrm{coh}}$ in the same time. This predicted loss of fluorescence
anisotropy within 10 fs has been observed experimentally, as shown in Fig. 7
of ref3.

Slower sub-ps depolarization occurs because of subsequent exciton density
localization as discussed above, first by stochastic localization and second
by coupling to torsional modes. This density localization also causes a change
in the transition dipole moments2.

The inset of Fig. 3 indicates that the majority of sub-ps fluorescence
depolarization occurs within a few fs. This is confirmed by our best
functional fit of the data, which is of the form

$\langle
r\left(t\right)\rangle=r_{1}\exp(-t/\tau_{1})+r_{2}\exp(-t/\tau_{2})+r_{\infty},$
(8)

where $\tau_{1}=2$ fs and $\tau_{2}=37$ fs; and $r_{1}=0.122$, $r_{2}=0.024$
and $r_{\infty}=0.254$. The slower, less significant depolarization, has a
time constant of 37 fs.

In summary, we have simulated the various sub-ps exciton dynamical processes
in conjugated polymers. First, following photoexcitation, the initial
dynamical process is the quantum mechanical correlation of the exciton and
phonons associated with high-frequency C-C bond vibrations. This quantum
entanglement causes exciton-site decoherence, which manifests itself as sub-10
fs fluorescence depolarization. Next, the energy that is transferred from the
exciton to the nuclei is dissipated into the environment on a timescale
determined by the strength of the system-bath interactions. For a hot exciton
(i.e., a QEES) the system-bath interactions cause the entangled exciton-
nuclear wavefunction to stochastically ‘collapse’ into a particular LEGS,
causing the exciton density to be localized on a ‘chromophore’. Finally, the
fate of an exciton on a chromophore is now strongly dependent on the polymer
chemical structure and the type of environment. For underdamped, freely
rotating moieties, the coupling of the exciton to the low-frequency classical
torsional modes creates an exciton-polaron, with associated planarization and
exciton-density localization.

In our simulations, the first of these processes is indentifiable by a rapid
fluorescence depolarization with a time constant of ca. 2 fs. However, because
the last two processes both correspond to intrachain exciton density
localization and occur on similar timescales, we have only been able to
extract one time constant for the associated depolarization, namely 37 fs.
Experimentally, it is observed that there is significant further
depolarization on post-ps timescales; the most significant component being
caused by interchromophore exciton diffusion.

I.G.P. acknowledges the EPSRC Centre for Doctoral Training, Theory and
Modelling in Chemical Sciences under Grant No. EP/L015722/1 for financial
support.

Table of contents:

  1. 1.

Excitons in Conjugated Polymers

  2. 2.

Fast C-C Bond Vibrations

  3. 3.

Slow Bond Rotations

  4. 4.

Open Quantum Systems

  5. 5.

Numerical Techniques

##  1 1\. Excitons in Conjugated Polymers

###  1.1 The Frenkel model

An exciton is a Coulombically bound electron-hole pair formed by the linear
combination of electron-hole excitations (for further details see20, 35, 36).
In a one-dimensional conjugated polymer an exciton is described by the two-
particle wavefunction, $\Phi_{mj}(r,R)=\psi_{m}(r)\Psi_{j}(R)$. The relative
wavefunction, $\psi_{m}(r)$, describes a particle bound to a screened Coulomb
potential, where $r$ is the electron-hole separation and $m$ is the principal
quantum number.

Frenkel excitons, for which $m=1$, are the primary photoexcited states of
light-emitting conjugated polymers. The center-of-mass wavefunction,
$\Psi_{j}(R)$, is described by the Frenkel Hamiltonian,

$\displaystyle\hat{H}_{F}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\epsilon_{n}\hat{N}_{n}+\sum_{n=1}^{N-1}J_{n}\left(\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right),$
(9)

where $N$ is the number of moieties, $n=(R/d)$ labels a moiety, and $d$ is the
intermoiety separation. The energy to excite a Frenkel exciton on moiety $n$
is $\epsilon_{n}$, where $\hat{N}_{n}=\ket{n}\bra{n}$ is the Frenkel exciton
number operator.

In principle, excitons delocalize along the chain via two mechanisms37, 36.
First, for even-parity (odd $m$) singlet excitons there is a Coulomb-induced
(or through space) mechanism. This is the familiar mechanism of Förster energy
transfer. In the point-dipole approximation

$J_{DA}^{mn}=\frac{\kappa_{mn}\mu_{0}^{2}}{4\pi\varepsilon_{r}\varepsilon_{0}R_{mn}^{3}},$
(10)

where $\mu_{0}$ is the transition dipole moment of a single moiety and
${R}_{mn}$ is the distance between the moieties $m$ and $n$. $\kappa_{mn}$ is
the orientational factor,

$\kappa_{mn}=\bm{\hat{r}_{m}}\cdot\bm{\hat{r}_{n}}-3(\bm{\hat{R}_{mn}}\cdot\bm{\hat{r}_{m}})(\bm{\hat{R}_{mn}}\cdot\bm{\hat{r}_{n}}),$
(11)

where $\bm{\hat{r}_{m}}$ is a unit vector parallel to the dipole on moiety $m$
and $\bm{\hat{R}_{mn}}$ is a unit vector parallel to the vector joining
moieties $m$ and $n$. For colinear moieties, the nearest neighbor through
space transfer integral is

$J_{DA}=-\frac{2\mu_{0}^{2}}{4\pi\varepsilon_{r}\varepsilon_{0}d^{3}}.$ (12)

Figure 4: A graphical depiction of our coarse-grained model of PPV. Each site
corresponds to a moiety along the polymer chain, with the connection between
sites characterised by the torsional (or dihedral) angle,
$\theta_{n}=(\phi_{n+1}-\phi_{n})$. The nuclear degrees of freedom, both
vibrational (blue) and torsional (red), are represented via harmonic
oscillators at each site, which are in turn locally coupled to the exciton.

Second, for all excitons there is a super-exchange (or through bond)
mechanism, whose origin lies in a virtual fluctuation from a Frenkel exciton
on a single moiety to a charge-transfer exciton spanning two moieties back to
a Frenkel exciton on a neighboring moiety. The energy scale for this process,
obtained from second order perturbation theory20, is

$J_{SE}(\theta)\propto-\frac{t(\theta)^{2}}{\Delta E},$ (13)

where $t(\theta)$ is proportional to the overlap of $p$-orbitals neighboring a
bridging bond, i.e., $t(\theta)\propto\cos\theta$ and $\theta$ is the
torsional (or dihedral) angle between neighboring moieties. $\Delta E$ is the
difference in energy between a charge-transfer and Frenkel exciton. The total
exciton transfer integral is thus

$J_{n}=J_{\text{DA}}+J_{\text{SE}}^{0}\cos^{2}{\theta_{n}}.$ (14)

Eq. (9) represents a ‘coarse-graining’ of the exciton degrees of freedom. The
key assumption is that we can replace the atomistic detail of each moiety and
replace it by a ‘coarse-grained’ site, as illustrated in Fig. 4.

###  1.2 Local exciton ground states (LEGSs) and quasiextended exciton states
(QEESs)

Figure 5: The density of the four local exciton ground states (LEGSs) and one
of the quasiextended exciton states (QEES) for one particular static
conformation of a PPV polymer chain of 40 moieties (i.e., 20 monomers).

Polymers are subject to conformational, chemical and environmental disorder.
Disorder localizes a quantum particle and determines their energetic and
spatial distributions. Malyshev and Malyshev38, 39 further observed that in
one-dimensional systems there are a class of states in the low energy tail of
the density of states that are superlocalized, named local exciton ground
states (LEGSs38, 39, 40). LEGSs are essentially nodeless, nonoverlapping
wavefunctions that together spatially span the entire chain. They are _local_
ground states, because for the individual parts of the chain that they span
there are no lower energy states. A consequence of the essentially nodeless
quality of LEGSs is that the square of their transition dipole moment scales
as their size40. Thus, LEGSs define chromophores (or spectroscopic segments),
namely the irreducible parts of a polymer chain that absorb and emit light.
Fig. 5 illustrates the four LEGSs for a particular conformation of PPV with 40
moieties (i.e., 20 monomers).

Higher energy lying states are also localized, but are nodeful and generally
spatially overlap a number of low-lying LEGSs. These states are named
quasiextended exciton states (QEESs) and an example is illustrated in Fig. 5.

##  2 2\. Fast C-C Bond Vibrations

The coupling of local normal modes (e.g., vinyl-unit bond stretches or phenyl-
ring symmetric breathing modes) to a Frenkel exciton is conveniently described
by the Frenkel-Holstein model41, 21,

$\displaystyle\hat{H}_{FH}$ $\displaystyle=$
$\displaystyle\hat{H}_{F}-A\hbar\omega_{\textrm{vib}}\sum_{n=1}^{N}\tilde{Q}_{n}\hat{N}_{n}+\frac{\hbar\omega_{\textrm{vib}}}{2}\sum_{n=1}^{N}\left(\tilde{Q}_{n}^{2}+\tilde{P}_{n}^{2}\right).$
(15)

$\hat{H}_{F}$ is the Frenkel Hamiltonian, defined in Eq. (9), while
$\tilde{Q}=(K_{\textrm{vib}}/\hbar\omega_{\textrm{vib}})^{1/2}Q$ and
$\tilde{P}=(\omega_{\textrm{vib}}/\hbar K_{\textrm{vib}})^{1/2}P$ are the
dimensionless displacement and momentum of the normal mode. The second term on
the right-hand-side of Eq. (15) indicates that the normal mode couples
linearly to the local exciton density. $A$ is the dimensionless exciton-phonon
coupling constant which introduces the local Huang-Rhys factor $S=A^{2}/2$.
The final term is the sum of the elastic and kinetic energies of the harmonic
oscillator, where $\omega_{\textrm{vib}}$ and $K_{\textrm{vib}}$ are the
angular frequency and force constant of the oscillator, respectively.

Exciton-nuclear dynamics is often modeled via the Ehrenfest approximation,
which treats the nuclear coordinates as classical variables moving in a mean
field determined by the exciton. However, (as discussed in the main text) the
Ehrenfest approximation fails to correctly describe ultrafast dynamical
processes. A correct description of the coupled exciton-nuclear dynamics
therefore requires a full quantum mechanical treatment of the system. This is
achieved by introducing the harmonic oscillator raising and lowering
operators, $\hat{b}_{n}^{\dagger}$ and $\hat{b}_{n}$, for the normal modes
i.e.,
$\tilde{Q}_{n}\rightarrow\hat{\tilde{Q}}_{n}=(\hat{b}_{n}^{\dagger}+\hat{b}_{n})/\sqrt{2}$
and
$\tilde{P}_{n}\rightarrow\hat{\tilde{P}}_{n}=i(\hat{b}_{n}^{\dagger}-\hat{b}_{n})\sqrt{2}$.
The time evolution of the quantum system can then conveniently be simulated
via the TEBD method, as described in Section 5.

###  2.1 Exciton-polarons

Since the photoexcited system has a different electronic bond order than the
ground state, an instantaneous force is established on the nuclei. This force
creates an exciton-polaron, whose spatial size is quantified by the exciton-
phonon correlation function42

$C_{n}^{\textrm{ex-
ph}}(t)\propto\sum_{m}\langle\hat{N}_{m}\hat{\tilde{Q}}_{m+n}\rangle.$ (16)

$C_{n}^{\textrm{ex-ph}}$ correlates the local phonon displacement, $Q$, with
the instantaneous exciton density $n$ moieties away. $C_{n}^{\textrm{ex-
ph}}(t)$, illustrated in Fig. 7 of ref22, shows that the exciton-polaron is
established within 10 fs (i.e., within half the period of a C-C bond
vibration) of photoexcitation. The temporal oscillations, determined by the
C-C bond vibrations, are damped as energy is dissipated into the vibrational
degrees of freedom, which acts as a heat bath for the exciton. The exciton-
phonon spatial correlations decay exponentially, extending over ca. 10
moieties. This short range correlation occurs because the C-C bond can respond
relatively quickly to the exciton’s motion.

##  3 3\. Slow Bond Rotations

Assuming that bond rotations are not sterically hindered, their coupling to
the excitons is conveniently modeled by supplementing the Frenkel-Holstein
model (i.e., Eq. (15)) by33

$\displaystyle\hat{H}_{\textrm{rot}}=-\sum_{n=1}^{N-1}B(\theta_{n}^{0})\times({\phi}_{n+1}-{\phi}_{n})\hat{T}_{n,n+1}+\frac{1}{2}\sum_{n=1}^{N}\left(K_{\textrm{rot}}{\phi}_{n}^{2}+{L}_{n}^{2}/I\right).$
(17)

Here, ${\phi}$ is the angular displacement of a from its groundstate
equilibrium value (see Fig. 4) and ${L}$ is the associated angular momentum of
a moiety around its bridging bonds. Eq. (17) is derived in ref33, assuming
that the angular displacements are small.

The first term on the right-hand-side of Eq. (17) indicates that the change in
the dihedral angle, $\Delta{\theta}_{n}=({\phi}_{n+1}-{\phi}_{n})$, couples
linearly to the bond-order operator,

$\hat{T}_{n,n+1}=\sum_{n=1}^{N-1}(|n\rangle\langle n+1|+|n+1\rangle\langle
n|),$ (18)

where

$B(\theta_{n}^{0})=J_{SE}^{0}\sin 2\theta_{n}^{0}$ (19)

is the exciton-roton coupling constant and $\theta_{n}^{0}$ is the groundstate
dihedral angle for the $n$th bridging bond. The final term is the sum of the
elastic and kinetic energies of the rotational harmonic oscillator.

The natural angular frequency of oscillation is
$\omega_{\textrm{rot}}=(K_{\textrm{rot}}/I)^{1/2}$, where $K_{\textrm{rot}}$
is the elastic constant of the rotational oscillator and $I$ is the moment of
inertia, respectively. $K_{\textrm{rot}}$ is larger for the bridging bond in
the excited state than the groundstate, because of the increase in bond order.
Also notice that both the moment of inertia (and thus $\omega_{\textrm{rot}}$)
of a rotating moiety and its viscous damping from a solvent are strongly
dependent on the side groups attached to it. As discussed in the next section,
this observation has important implications for whether the motion is under or
over damped and on its characterstic timescales.

###  3.1 A single torsional oscillator

Before considering a chain of torsional oscillators, it is instructive to
review the dynamics of a single, damped oscillator subject to both restoring
and displacement forces. The equation of motion for the angular displacement
is

$\frac{d^{2}\phi(t)}{dt^{2}}=-\omega_{\textrm{rot}}^{2}(\phi(t)-\phi_{\textrm{eq}})-\gamma\frac{d\phi(t)}{dt,}$
(20)

where $\phi_{\textrm{eq}}=\lambda/K_{rot}$ is proportional to the displacement
force and $\lambda$ is defined in Eq. (27).

In the _underdamped regime_43, defined by $\gamma<2\omega_{\textrm{rot}}$,

$\phi(t)=\phi_{\textrm{eq}}\left(1-\cos(\omega t)\exp(-\gamma t/2)\right),$
(21)

where $\omega=(\omega_{\textrm{rot}}^{2}-\gamma^{2}/4)^{1/2}$. In this regime,
the torsional angle undergoes damped oscillations with a period
$T=2\pi/\omega$ and a decay time $\tau=2/\gamma$.

Conversely, in the _overdamped regime_43, defined by
$\gamma>2\omega_{\textrm{rot}}$,

$\displaystyle\phi(t)=\phi_{\textrm{eq}}\left(1-\frac{1}{4\beta}\left(\gamma_{1}\exp(-\gamma_{2}t/2)-\gamma_{2}\exp(-\gamma_{1}t/2)\right)\right),$

where $\gamma_{1}=\gamma+2\beta$, $\gamma_{2}=\gamma-2\beta$ and
$\beta=(\gamma^{2}/4-\omega_{\textrm{rot}}^{2})^{1/2}$. Now, the torsional
angle undergoes damped biexponential decay with the decay times
$\tau_{1}=2/\gamma_{1}$ and $\tau_{2}=2/\gamma_{2}$. In the limit of strong
damping, i.e., $\gamma\gg 2\omega_{\textrm{rot}}$, there is a fast relaxation
time $\tau_{1}=1/\gamma=\tau/2$ and a slow relaxation time
$\tau_{2}=\gamma/\omega_{\textrm{rot}}^{2}\gg\tau$. In this limit, as the slow
relaxation dominates at long times, the torsional angle approaches equilibrium
with an effective mono-exponential decay.

For a polymer without alkyl side groups, e.g., PPP and PPV,
$\omega_{\textrm{rot}}\sim\gamma\sim 10^{13}$ s-1 and are thus in the
underdamped regime with sub-ps relaxation. However, polymers with side groups,
e.g., P3HT, MEH-PPV and PFO, have a rotational frequency up to ten times
smaller and a larger damping rate, and are thus in the overdamped regime7.

###  3.2 A chain of torsional oscillators

An exciton delocalized along a polymer chain in a chromophore couples to
multiple rotational oscillators resulting in collective oscillator dynamics.
Eq. (19) and Eq. (27) indicate that torsional relaxation only occurs if the
moieties are in a staggered arrangement in their groundstate, i.e.,
$\theta_{n}^{0}=(-1)^{n}\theta^{0}$. In this case the torque acts to planarize
the chain. Furthermore, since the torsional motion is slow, the self-trapped
exciton-polaron is ‘heavy’ and in the under-damped regime becomes self-
localized on a timescale of a single torsional period, i.e., $200-600$ fs. In
this limit the relaxed staggered bond angle displacement mirrors the exciton
density. Thus, the exciton is localized precisely as for a ‘classical’ Landau
polaron and is spread over $\sim 10$ moieties33.

The time-evolution of the staggered angular displacement,
$\langle{\phi}_{n}\rangle\times(-1)^{n}$, is shown in Fig. 5 of ref33
illustrating that these displacements reach their equilibrated values after
two torsional periods (i.e., $t>400$ fs). The inset also displays the time-
evolution of the exciton density, $\langle N_{n}\rangle$, showing exciton
density localization after a single torsional period ($\sim 200$ fs).

We have described how exciton coupling to torsional modes causes a spatially
varying planarization of the moieties that acts as a one-dimensional potential
which self-localizes the exciton. The exciton ‘digs a hole for itself’,
forming an exciton-polaron44. Some researchers14, however, argue that
torsional relaxation causes an exciton to become more _delocalized_. A
mechanism that can cause exciton delocalization occurs if the disorder-induced
localization length is shorter than the intrinsic exciton-polaron size. Then,
in this case for freely rotating moieties, the stiffer elastic potential in
the excited state causes a decrease both in the variance of the dihedral
angular distribution, $\sigma_{\theta}^{2}=k_{B}T/K_{\textrm{rot}}$, and the
mean dihedral angle, $\theta^{0}$. This, in turn, means that the exciton band
width, $|4J|$, increases and the diagonal disorder21,
$\sigma_{J}=J_{SE}\sigma_{\theta}\sin 2\theta^{0}$, decreases. Hence, the
disorder-induced localization, $L_{\textrm{loc}}\sim(|J|/\sigma_{J})^{2/3}$,
increases.

##  4 4\. Open Quantum Systems

Dissipation of energy from an open quantum system arising from system-
environment coupling is commonly described by a Lindblad master equation31

$\frac{\partial\hat{\rho}}{\partial{t}}=-\frac{i}{\hbar}\left[\hat{H},\hat{\rho}\right]-\frac{{\gamma}}{2}\sum_{n}\left(\hat{L}_{n}^{\dagger}\hat{L}_{n}\hat{\rho}+\hat{\rho}\hat{L}_{n}^{\dagger}\hat{L}_{n}-2\hat{L}_{n}\hat{\rho}\hat{L}_{n}^{\dagger}\right),$
(22)

where $\hat{L}_{n}^{\dagger}$ and $\hat{L}_{n}$ are the Linblad operators, and
$\hat{\rho}$ is the system density operator. In practice, a direct solution of
the Lindblad master equation is usually prohibitively expensive, as the size
of Liouville space scales as the square of the size of the associated Hilbert
space. Instead, Hilbert space scaling can be maintained by performing ensemble
averages over quantum trajectories (evaluated via the TEBD method), where the
action of the Linblad dissipator is modeled by quantum jumps.32

Here we assume that the C-C bond vibrations couple directly with the
environment 22, in which case the Linblad operators are the associated raising
and lowering operators (i.e., $\hat{L}_{n}\equiv\hat{b}_{n}$, introduced in
Section 2). In addition,

$\hat{H}_{FH}^{\prime}=\hat{H}_{\text{FH}}+\frac{\hbar\gamma}{4}\sum_{n}\left(\hat{\tilde{Q}}_{n}\hat{\tilde{P}}_{n}+\hat{\tilde{P}}_{n}\hat{\tilde{Q}}_{n}\right).$
(23)

In practice, a direct solution of the Lindblad master equation is usually
prohibitively expensive, as the size of Liouville space scales as the square
of the size of the associated Hilbert space. Instead, Hilbert space scaling
can be maintained by performing ensemble averages over quantum trajectories
(evaluated via the TEBD method), where the action of the Linblad dissipator is
modeled by quantum jumps, as described in Section 5.32

##  5 5\. Numerical Techniques

###  5.1 Time Evolving Block Decimation (TEBD)

Given the initial condition, $|\Psi(t=0)\rangle$, the state of the system is
propagated in time via the evolution operator, i.e.,

$|\Psi(t+\delta t)\rangle=\exp(-i\hat{H}\delta t/\hbar)|\Psi(t)\rangle,$ (24)

where

$\hat{H}=\hat{H}_{FH}^{\prime}+\hat{H}_{\textrm{rot}}.$ (25)

Thus, since the C-C bond vibrations are quantized, this method avoids the
failures of the Ehrenfest approximation. In particular, exciton-site
decoherence and non-adiabatic state interconversion are correctly described.

We perform this procedure numerically using the Tensor Network Theory
Library19, which implements the Time Evolving Block Decimation (TEBD)
method45, 46. Details of the TEBD method are described in Appendix B of ref22.
Briefly, the TEBD method works by (1) representing $|\Psi\rangle$ as a matrix
product state47, (2) expressing the evolution operator via a Trotter
decomposition, and (3) compressing the action of the evolution operator on
$|\Psi\rangle$ via a singular value decomposition (SVD). Time-dependent
expectation values of relevant observables are determined via
$|\Psi(t)\rangle$.

Importantly, this approach is ‘numerically exact’ as long as the truncation
parameter exceeds $2^{S}$, where $S$ is the entanglement entropy, defined by
$S=-\sum_{\alpha}\omega_{\alpha}\textrm{ln}_{2}\omega_{\alpha}$ and
$\\{\omega\\}$ are the singular values obtained at the SVD. The TEBD method
permits the electronic and nuclear degrees of freedom to be treated as quantum
variables on an equal footing. It thus rectifies all of the failures of the
Ehrenfest method and it is not limited by the representation of the potential
energy surface.

###  5.2 TEBD and Ehrenfest dynamics

Unlike C-C bond vibrations, being over 10 times slower torsional oscillations
can be treated classically33. Furthermore, since we are now concerned with
adiabatic relaxation on a single potential energy surface, we may employ the
Ehrenfest approximation for these modes. Thus, using Eq. (17), the torque on
each ring is

$\displaystyle\Gamma_{n}$ $\displaystyle=$
$\displaystyle-\frac{\partial\langle\hat{H}_{rot}\rangle}{\partial{\phi}_{n}}$
(26) $\displaystyle=$ $\displaystyle-K_{\textrm{rot}}{\phi}_{n}+\lambda_{n}$

where we define

$\lambda_{n}=B(\theta_{n-1}^{0})\langle\hat{T}_{n-1,n}\rangle-B(\theta_{n}^{0})\langle\hat{T}_{n,n+1}\rangle.$
(27)

Setting $\Gamma_{n}=0$ gives the equilibrium angular displacements in the
excited state as ${\phi}_{n}^{\textrm{eq}}=\lambda_{n}/K_{\textrm{rot}}$.
${\phi}_{n}$ is subject to the Ehrenfest equations of motion,

$I\frac{d{\phi}_{n}}{d{t}}={L}_{n},$ (28)

and

$\displaystyle\frac{d{L}_{n}}{d{t}}=\Gamma_{n}-\gamma L_{n},$ (29)

where the final term represents the damping of the rotational motion by the
solvent. Eq. (28) and (29) are integrated using the velocity Verlet algorithm
and the classical values of $\\{\phi\\}$ are fed into
$\hat{H}_{\textrm{rot}}$. Our dynamics are in the underdamped regime.

###  5.3 The quantum jump trajectory method

Rather than solve the Linblad master equation, Eq. (22), directly, we use the
quantum jump trajectory method and average over quantum trajectories. In this
method the time evolution of a single trajectory is determined by the
effective Hamiltonian32

$\hat{H}_{\text{eff}}=\hat{H}-\frac{\textrm{i}{\hbar\gamma}}{2}\sum_{n}\hat{b}_{n}^{\dagger}\hat{b}_{n},$
(30)

where $\hat{H}$ is the system Hamiltonian given in Eq. (25), $\gamma$ is the
dissipation parameter and $\hat{b}_{n}$ is the Lindblad operator for site $n$.
Evolving this single quantum trajectory, initially in state $|\Psi(t)\rangle$,
under the effective Hamiltonian gives a trial state at time $t+\delta t$

$|\Psi_{\text{trial}}(t+\delta t)\rangle=\hat{U}(t+\delta
t,t)|\Psi(t)\rangle,$ (31)

where $\hat{U}(t+\delta t,t)$ is the evolution operator

$\hat{U}(t+\delta t,t)=\exp(-\textrm{i}\hat{H}_{\text{eff}}\delta t/\hbar).$
(32)

As the effective Hamiltonian in Eq. (30) is non-Hermitian, the time evolution
in Eq. (31) does not conserve the quantum state’s norm

$\langle\Psi_{\text{trial}}(t+\delta t)|\Psi_{\text{trial}}(t+\delta
t)\rangle=1-\delta p,$ (33)

where $\delta p$ is the amount the norm has decayed over the time step $\delta
t$. Using Eq. (33), the state of the single quantum trajectory at time
$t+\delta t$ is determined probabilistically as:

  * (i)

With probability $1-\delta p$, the state at time $t+\delta t$ is

$|\Psi(t+\delta t)\rangle=\frac{\hat{U}(t+\delta
t,t)|\Psi(t)\rangle}{\sqrt{1-\delta p}}.$ (34)

  * (ii)

With probability $\delta p$, the state at time $t+\delta t$ is

$|\Psi(t+\delta
t)\rangle=\frac{\hat{b}_{n}|\Psi(t)\rangle}{\sqrt{\langle\Psi(t)|\hat{b}_{n}^{\dagger}\hat{b}_{n}|\Psi(t)\rangle}},$
(35)

where $\hat{b}_{n}$ is the Lindblad operator for site $n$. The site at which
the Lindblad operator is applied in Eq. (35) is again determined
probabilistically, where the probability associated with applying the Lindblad
operator at site $n$ is given by

$P_{n}=\frac{\langle\Psi(t)|\hat{b}_{n}^{\dagger}\hat{b}_{n}|\Psi(t)\rangle}{\sum_{m}\langle\Psi(t)|\hat{b}_{m}^{\dagger}\hat{b}_{m}|\Psi(t)\rangle}.$
(36)

Averaging over a large number of these stochastically determined trajectories
accurately reproduces the time-dependent observables associated with the
Lindblad master equation.32

###  5.4 Ensemble averaging

We perform our calculations of the fluorescence anisotropy over an ensemble of
1,000 different polymer conformations achieved by assuming that the torsional
angle, $\theta$, is a Gaussian random variable with a mean of $25^{\circ}$ and
a standard deviation of $10^{\circ}$. We also assume that the probability of a
cis-trans defect is 8%. Furthermore, to include a general approach to the
decay of our initial high energy QEESs onto LEGSs, our results include an
averaging over 10 different QEESs. Finally, our method of treating open
quantum systems, via the quantum jump method, requires an averaging over
different quantum trajectories per initial QEES. The accuracy of this method
increases with the number of quantum trajectories. We have used 20 different
trajectories per initial QEES.

###  5.5 Parameters

Parameter | Value
---|---
$J_{\textrm{DA}}$ | -0.60 eV
$J_{\textrm{SE}}^{0}$ | -1.68 eV
$\epsilon_{\textrm{phenyl}}$ | 5.0 eV
$\epsilon_{\textrm{vinylene}}$ | 4.0 eV
$A$ | 4
$\theta^{0}$ | $25^{\circ}$
$T_{\textrm{vib}}=2\pi/\omega_{\textrm{vib}}$ | 21 fs
$T_{\textrm{rot}}=2\pi/\omega_{\textrm{rot}}$ | 210 fs
$\gamma^{-1}$ | 100 fs
$\sigma_{\theta}$ | $10^{\circ}$
$x_{\textrm{cis-trans}}$ | 0.8
Table 2: Parameters used in the modeling of exciton dynamics in PPV.

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