# Room Temperature Quantum Coherence vs. Electron Transfer in a Rhodanine
Derivative Chromophore

Duvalier Madrid-Úsuga<EMAIL_ADDRESS>Centre for
Bioinformatics and Photonics (CIBioFi), Calle 13 No. 100-00, Edificio E20 No.
1069, Universidad del Valle, 760032 Cali, Colombia  Departamento de Física,
Universidad del Valle, 760032 Cali, Colombia  Cristian E. Susa
<EMAIL_ADDRESS>Centre for Bioinformatics and Photonics
(CIBioFi), Calle 13 No. 100-00, Edificio E20 No. 1069, Universidad del Valle,
760032 Cali, Colombia  Departamento de Física y Electrónica, Universidad de
Córdoba, 230002 Montería, Colombia  John H. Reina
<EMAIL_ADDRESS>Centre for Bioinformatics and Photonics
(CIBioFi), Calle 13 No. 100-00, Edificio E20 No. 1069, Universidad del Valle,
760032 Cali, Colombia  Departamento de Física, Universidad del Valle, 760032
Cali, Colombia

###### Abstract

Understanding electron transfer in organic molecules is of great interest in
quantum materials for light harvesting, energy conversion and integration of
molecules into solar cells. This, however, poses the challenge of designing
specific optimal molecular structure for which the processes of ultrafast
quantum coherence and electron transport are not so well understood. In this
work, we investigate subpicosecond time scale quantum dynamics and electron
transfer in an efficient electron acceptor Rhodanine chromophoric complex. We
consider an open quantum system approach to model the complex-solvent
interaction, and compute the crossover from weak to strong dissipation on the
reduced system dynamics for both a polar (Methanol) and a non polar solvent
(Toluene). We show that the electron transfer rates are enhanced in the strong
chromophore-solvent coupling regime, being the highest transfer rates those
found at room temperature. Even though the computed dynamics are highly non-
Markovian, and they may exhibit a quantum character up to hundreds of
femtoseconds, we show that quantum coherence does not necessarily optimise the
electron transfer in the chromophore.

###### pacs:

03.65.Yz, 87.15.ht, 82.20.Xr. 87.15.ag, 04.25.-g

##  I Introduction

Electron transfer (ET) mechanisms and dynamics are ubiquitous to fundamental
processing of energy in biological and chemical systems that are essential to
life. They are at the core of light harvesting for energy production in a
variety of systems; e.g., biomolecular photosynthetic complexes, biosensors,
and solar cells engel2007 ; panitchayangkoon2010 ; collini2009 ; hwang2010 ;
song2016 ; hedley2016  to name but a few. From a perspective of novel
materials and technologies for solar energy conversion, organic photovoltaic
(OPV) sytems have attracted significant attention sauve2015 ; pandey2010 ;
ho2015  due to the enhancement of their power conversion efficiency (PCE)
song2016 ; hedley2016 , flexibility,manufacturing reduction costs, and high-
temperature production of cells hughes2014 ; Insuasty2011385 ; madrid2018 ;
cabrera2018 ; cabrera2017 . On a different facet, but complementary to this
development, quantum coherence has, over the past decade, been experimentally
linked to early-stage energy transfer in light harvesting photosynthetic
complexes engel2007 ; panitchayangkoon2010 ; collini2010 ; lee2007C ;
collini2009 ; hwang2010 ; tempelaar2015 ; roscioli2017 ; strumpfer2012e ; in
particular, time-resolved ultrafast optical and electronic spectroscopies have
evidenced oscillations of exciton state populations lasting up to a few
hundred femtoseconds, which have been attributed to quantum coherence emerging
as a result of initially prepared laser pulses engel2007 ;
panitchayangkoon2010 ; collini2010 ; lee2007C ; collini2009 ; hwang2010 ;
tempelaar2015 ; roscioli2017 . This two-way breakthrough poses an intriguing
question about the possible functional role of quantum coherent effects as a
precursor of efficient ET in materials and biological systems under structural
and energetic disorder at physiological conditions.

The above physical insight requires that ET reactions in such chemical or
biological light harvesters must consider the effects due to electronic
coupling, interactions with the environment, and intramolecular vibrational
modes.In particular, photoinduced ultrafast energy and ET processes in OPV
devices require a quantum-mechanical treatment of such fundamental microscopic
processes. Here, electron transfer occurs between electron-donor and electron-
acceptor molecules immersed in a solvent, and such process can be modelled by
considering a quantum system in interaction with a local environment
(solvent), whose dynamics, control, and correlations are amenable to a
description within the framework of open quantum systems Weiss ; breuer2002 ;
RevModPhys ; reina2014 ; reina2002 ; koch2016 ; eckel2009 ; thorwart2009 ;
chen2009 ; reina2018 ; susa2014 ; melo2017 ; forgy2014 ; mazziotti2012 ;
rebentrost2009 . Donor-acceptor chromophoric complexes convey electronic
coupling between sites on a scale that relies on the solvent reorganisation
energy and defines different regimes for the complex coherent/incoherent
dynamics. Due to the interplay between energy fluctuation and transport, here
we account for the crossover from weak to strong dissipation and consider
memory effects for the chromophore-solvent quantum dynamics yuan2009 ;
farhat2017 ; these can be described by means of intensive computational
techniques such as path-integral based formalisms makarov1994 ; makri1995 ;
eckel2009 ; thorwart2009 , Monte Carlo algorithms muhlbacher2003 ,
Hierarchical Equations of Motion (HEOM) Ishizaki ; tanimura2006 ; ishizaki2009
; chen2015 , and reaction-coordinate methods bolhuis2000 .

In this work, we consider an OPV light harvester (an efficient Rhodanine
derivative chromophore Insuasty2011385 ; madrid2018 ), and focus on
theoretical aspects regarding its ultrafast carrier dynamics and the interplay
between the solvent, electronic coupling, and temperature conditions on both
quantum coherence and electronic transfer. We consider Methanol, the polar
solvent used in Insuasty2011385 , and, for completeness, we also quantify the
effects due to a non polar solvent, Toluene. In both cases, the chromophore-
solvent coupling is found to be in the strong regime and the corresponding non
trivial non-Markovian dynamics is treated within a non-perturbative open
quantum system approach, with the numerically exact HEOM method chen2015 , on
a spin-boson model of the complex-solvent RevModPhys ; eckel2009 ;
thorwart2009 .

This paper is organised as follows: the chromophore and solvent properties are
developed in Sec. II; the chemical structure and main parameters of the
organic molecule are presented in Sec. II.1, and the mathematical model and
chromophore+solvent physical parameters are given in Sec. II.2. The main
results and discussions are presented in Sec. III: solvent and temperature
effects on the quantum dynamics (Sec. III.1), and electronic transfer (Sec.
III.2). Finally, conclusions are given in Sec. IV.

##  II Chromophore$+$Solvent System: Chemical and Physical Properties

The considered push-pull chromophore derived from Rhodanine, the
Dicyanorhodanine+2-formyltetratiafulvalene (D2F) complex Insuasty2011385 ;
madrid2018 , has a chemical structure as shown in Fig. 1(a): it consists of a
2-formyltetratiafulvalene as the electron-donor (D) and a Dicyanorhodanine as
the electron-acceptor (A) linked by Propylene. The compound exhibits appealing
photo- and electron-chemical properties; a D$-$A intramolecular charge
transfer in the absorption band of the donor at $\lambda_{max}=498$ nm
(visible region). This was confirmed by solvatochromics studies using Acetone,
Ethanol, and Methanol as solvents Insuasty2011385 : the “push-pull” compound
displays a clear electrochemically amphoteric behaviour and reveals a
significant D$-$A electronic communication through the $\pi$-conjugated core,
a feature that can be useful for ET integration into molecular systems of
greater complexity.

Figure 1: (a) Molecular structure for the push-pull chromophore derived from
Rhodanine; Dicyanorhodanine (Acceptor) +2-Formyltetratiafulvalene (Donor). (b)
Electronic density associated to the EX and CT states involved in the ET
process between D and A.

###  II.1 D2F Chemical Structure

The geometry of the D2F molecule was optimised by means of Gaussian 09
gaussian09 . The hybrid functional B3LYP and the 6-31G++ basis are used to
estimate the energy values of D and A electronic states, as well as the
coupling between them. We consider a scenario where the system is firstly
photoexcited from the ground electronic state to a $\pi$-$\pi^{*}$ state
localised on D. This is followed by a nonradiative process that corresponds to
an electronic transition from the $\pi$-$\pi^{*}$ state to a charge-transfer
(CT) state that involves a significant D$-$A charge transfer. The molecular
orbital analysis of both ground and excited states of the optimised geometry
indicates that the HOMO (LUMO) is mainly located on D (A). The two
aforementioned states are illustrated in Fig. 1(b); the first transition that
gives the excited state (EX) corresponds to the photoexcitation D–A
$\rightarrow$ D∗–A at an excitation energy of $2.85$ eV. This exhibits an
oscillator strength $f_{osc}=0.2348$ and a percentage contribution of 83%
(HOMO $\rightarrow$ LUMO+5 transition in Fig. 1(b)). The second one that
generates the CT state (D-–A+) has an energy of $2.73$ eV, and this state has
$f_{osc}=0.0002$, with a percentage contribution of 75% for the HOMO
$\rightarrow$ LUMO+4 transition (Fig. 1(b)). These values were computed for
Methanol via the time dependent density functional theory, by using the B3LYP
functional. This scenario corresponds to an ET process from the
2-formyltetratiafulvalene compound to the Dicyanorhodanine acceptor.

The energies associated to the two main states involved in the ET mechanism
are sketched in Fig. 2. According to the B3LYP/6-31G++ optimisation, the
estimated site energies for Methanol are $E_{D}=-4323$ cm-1 and $E_{A}=-5291$
cm-1. On the other hand, for Toluene $E_{D}=-5201$ cm-1 and $E_{A}=-5838$
cm-1. Other physical parameters of interest are given in Fig. 2 and Table 1.

Figure 2: Energy representation of the $\pi$-$\pi^{*}$ and CT states of interest in the D2F complex. $\epsilon\equiv|E_{D}-E_{A}|$ is the energy difference between the $EX$ and $CT$ states, and $V_{DA}$ stands for the D$-$A electronic coupling. Methanol: $E_{D}=-4323$ cm-1, $E_{A}=-5291$ cm-1, $V_{DA}=280$ cm-1; Toluene: $E_{D}=-5201$ cm-1, $E_{A}=-5838$ cm-1, $V_{DA}=317$ cm-1. Table 1: Chromophore$+$solvent physical parameters. Energy values are given by the B3LYP/6-31G++ optimisation. Dielectric constants and dipole relaxation times are taken from Horng ; Shcherbakov ; barthel1991  for Methanol, and from Cecilie2000 ; Vincent1967  for Toluene. $\Delta\mu_{M}=1.1058\times 10^{-29}$ Cm, $\Delta\mu_{T}=1.3455\times\leavevmode\nobreak\ 10^{-28}$ Cm, $a_{0}=5.07\times 10^{-10}$ m, $\epsilon_{0}=8.8542\times 10^{-12}$ ${\rm C^{2}/Nm^{2}}$. Subscripts $M$ and $T$ in the relative dipole moment, $\Delta\mu$, stand for Methanol and Toluene, respectively. Solvent | Parameter
---|---
$\epsilon$ (cm-1) |  $V_{DA}$ (cm-1) | $\varepsilon_{s}$ | $\varepsilon_{\infty}$ |  $\tau_{D}$ (ps) |  $\tau_{s}$ (ps)
Methanol | $968$ | $280$ | $32.7$ | $1.8$ | $49.6$ | $0.1$
Toluene | $637$ | $317$ | $2.4$ | $2.2$ | $5.1$ | $0.2$
Figure 3: Weak coupling regime. Population inversion
$\langle\sigma_{z}(t)\rangle$ (panels (a) and (c)) and real-part coherence
$\Re{(\rho_{DA})}$ (panels (b) and (d)) dynamics as a function of the
chromophore-solvent coupling at $T=300$ K. Subscripts $M$ and $T$ stand for
Methanol and Toluene (top and bottom plots). In each figure, turquoise,
violet, and black curves correspond to $\eta=0.005$, $\eta=0.010$,
$\eta=0.025$, respectively. Hereafter, populations are considered to be
localised on the donor at $t=0$, i.e., $\langle\sigma_{z}(0)\rangle=1$, and a
logarithmic scale for the time axis is used in Figs. 3, 4 and 5. Figure 4:
Strong coupling regime. Population inversion $\langle\sigma_{z}(t)\rangle$
(panels (a) and (c)) and real-part coherence $\Re{(\rho_{DA})}$ (panels (b)
and (d)) dynamics as functions of the chromophore-solvent coupling at $T=300$
K. Methanol (top) and Toluene (bottom), for $\eta=0.10$ (turquoise),
$\eta=0.25$ (violet), and $\eta=0.33$ (black) curves.

###  II.2 Hamiltonian and Chromophore Quantum Dissipative Dynamics

We consider a spin-boson model to treat the chromophore-solvent ultrafast
dynamics in a quantum fashion Brisker ; Malka ; RevModPhys ; eckel2009 ;
thorwart2009 . The total Hamiltonian can be written as
$\widehat{H}=\widehat{H}_{S}+\widehat{H}_{B}+\widehat{H}_{SB}$, where the
subscripts stand for system (the chromophore), bath (the solvent), and for the
system-bath interaction, respectively. The solvent is modelled as a bath of
harmonic oscillators with
$\widehat{H}_{B}=\sum_{j}\hbar\omega_{j}\left(\widehat{a}_{j}^{\dagger}\widehat{a}_{j}+\frac{1}{2}\right)$,
where $\widehat{a}_{j}^{\dagger}$ ($\widehat{a}_{j}$) denotes the creation
(annihilation) operator of the $j$-th bosonic mode with frequency
$\omega_{j}$.

The $N$-partite donor-bridge-acceptor chromophore Hamiltonian is described in
terms of the local (diabatic) electronic states
$\left|\varphi_{n}\right\rangle$ at site $n$:

$\widehat{H}_{S}=\sum_{n=1}^{N}E_{n}|\varphi_{n}\rangle\langle\varphi_{n}|+\sum_{n\neq
m=1}^{N}V_{nm}|\varphi_{n}\rangle\langle\varphi_{m}|,$ (1)

where $E_{n}$ corresponds to the energy of the $n$-th site and $V_{nm}$ is the
coupling energy between sites $n$ and $m$. Here, we consider a reduced two-
site model ($N=2$) between the $EX$ and $CT$ states with energies as shown in
Fig. 2, following the prominent D$-$A charge transfer due to the
$\pi$-$\pi^{*}$ state described in Sec. II.1. Thus, an effective two-level
(D$-$A) system Hamiltonian can be identified for $\widehat{H}_{\text{S}}$:

$\widehat{H}_{\text{TLS}}=\frac{\epsilon}{2}\widehat{\sigma}_{z}+V_{DA}\widehat{\sigma}_{x},$
(2)

where $V_{DA}=\epsilon\mu_{12}/[(\mu_{1}-\mu_{2})^{2}+4\mu_{12}^{2}]^{1/2}$ is
the electronic coupling, here computed by means of the generalised Mulliken-
Hush model cave1996 ; zheng2005 , $\epsilon\equiv|E_{D}-E_{A}|$ is the energy
difference between donor and acceptor orbitals, and $\widehat{\sigma}_{x,z}$
are Pauli’s operators. For Methanol, $\mu_{12}=6.6990$ D is the transition
dipole moment connecting the two adiabatic states in the charge transition,
and $\Delta\mu_{12}=\leavevmode\nobreak\ \mu_{1}-\mu_{2}=18.8447$ D is the
difference in adiabatic state dipole moments. A similar procedure gives the
corresponding values for Toluene.

The solvent is modelled by considering that each mode interacts independently
with the electronic sites. The explicit interaction in the simplified two-site
scenario, where $g_{j}$ accounts for the coupling strength to the $j$-th mode,
is written as:

$H_{SB}=\hbar\sigma_{z}\otimes\sum_{j}g_{j}(a_{j}^{\dagger}+a_{j}).$ (3)

The information about the solvent and its interaction with the chromophore is
encoded in the spectral density
$J(\omega)=\sum_{j}g_{j}^{2}\delta(\omega-\omega_{j})$ and the bath
correlation functions (see Appendix A for details). A precise description of
this information through the spectral density is usually not a straightforward
task. Here, we consider Onsager’s reaction model for describing the
D2F-solvent coupling. Geometrically, the D2F chromophore is assumed to occupy
a region of radius $a_{0}$ ($\sim$ the van de Waals size) surrounded by the
solvent Onsager . Within this model, a spectral density with Lorentzian shape
can be derived Gilmore :

$J(\omega)=\frac{(\Delta\mu)^{2}}{2\pi\varepsilon_{0}a_{0}^{3}}\frac{6(\varepsilon_{s}-\varepsilon_{\infty})}{(2\varepsilon_{s}+1)(2\varepsilon_{\infty}+1)}\frac{\omega\tau_{s}}{\omega^{2}\tau_{s}^{2}+1},$
(4)

where $\Delta\mu=\mu_{e}-\mu_{g}$ is the relative dipole moment between the
excited and ground state dipole moments of the D$-$A molecule.
$\varepsilon_{0}$, $\varepsilon_{s}$ and $\varepsilon_{\infty}$ stand for
vacuum, solvent static and high frequency dielectric constants, respectively.
$\tau_{s}$ is the solvent relaxation time (cutoff frequency $\omega_{c}\equiv
1/\tau_{s}$). The reorganisation energy
$\lambda:=\frac{2}{\pi}\int\frac{J(\omega)}{\omega}d\omega$ is simply given by
$\lambda=2\hbar\alpha\omega_{c}$, where
$\alpha=\frac{(\Delta\mu)^{2}}{4\pi\epsilon_{0}a_{0}^{3}}\frac{6(\epsilon_{s}-\epsilon_{\infty})}{(2\epsilon_{s}+1)(2\epsilon_{\infty}+1)\omega_{c}}$
is the frequency-independent factor of Eq. (4).

The quantum dissipative dynamics of the chromophore can be described within
the density operator formalism Weiss ; breuer2002 ; RevModPhys . If
$\widehat{\rho}_{T}(t)$ is the time-dependent density operator associated to
the total (chromophore$+$solvent) system (considered to be isolated), the
reduced density operator for the chromophore is given by
$\widehat{\rho}_{S}(t)=\text{Tr}_{B}(\widehat{\rho}_{T}(t))$. Here, we employ
the HEOM technique Yoshitaka ; tanaka2010  to numerically compute the temporal
evolution of $\widehat{\rho}_{S}(t)$, as explained in Appendix A.

The coherent nature of the ET depends on several factors such as temperature
and chromophore-solvent coupling. Here, we consider two different solvents;
Methanol (polar) and Toluene (non polar) and present numerical results for
their effects on the D2F dynamics and the electronic transfer. Important
physical parameters that characterise the D2F system are given in Table 1.

##  III Results and Discussion

The spectral density (4) can be written in a simpler way as
$J(\omega)=2\alpha\omega/(\omega^{2}\tau_{s}^{2}+1)$, where the $\alpha$
values are determined from the parameters in Table 1, for each solvent. In
fact, this chromophore-solvent coupling gives reorganisation energies of
$86.18$ cm-1 and $2.20$ cm-1 for Methanol and Toluene, respectively. In order
to directly compare the effects of both solvents on the D2F dynamics, we
introduce a dimensionless system-bath coupling:

$\eta:=\frac{\alpha}{\hbar}=\frac{\lambda}{2\hbar\omega_{c}}.$ (5)

A system-bath coupling falls within the weak regime if $\eta\ll 1$; in the
strong regime $\eta\sim 1$ Weiss ; breuer2002 ; RevModPhys . We obtained,
according to the solvent properties given in Table 1, $\eta=0.25$ and $0.33$
for Methanol and Toluene, respectively: these indicate that the strong
coupling regime dictates the chromophore dynamics and ET in both cases.

Figure 5: Temperature dependence of D2F-solvent dynamics. Population inversion
$\langle\sigma_{z}(t)\rangle$ ((a) and (c)) and real-part coherence
$\Re{(\rho_{DA}(t))}$ ((b) and (d)) dynamics as functions of temperature, for
Methanol ($\eta_{M}=0.25$) and Toluene ($\eta_{T}=0.33$). In each panel,
turquoise, violet, orange and black curves correspond to $T=10$ K, $50$ K,
$100$ K and $300$ K, respectively.

###  III.1 Solvent and Temperature Effects

Having determined the main properties of the considered chromophore-solvent
system, we investigate the solvent effects on the D2F dynamics in both weak
and strong regimes. For doing so, we vary the value of $\eta$ and compute the
population inversion $\langle\sigma_{z}(t)\rangle=\rho_{D}(t)-\rho_{A}(t)$, as
well as the real-part coherence $\Re{(\rho_{DA}(t))}$ associated to the
chromophore, where $\rho_{D}$, $\rho_{A}$ (populations) and
$\rho_{DA}=\rho_{AD}^{*}$ (coherence) are the matrix elements of the density
operator $\widehat{\rho}_{S}(t)$. In all the figures, populations are
considered to be localised on the donor at $t=0$, i.e.,
$\langle\sigma_{z}(0)\rangle=1$, and a logarithmic time scale is used in Figs.
3, 4 and 5.

We begin by considering room temperature effects on the D2F dynamics, since
possible applications of this chromophore as an efficient electron acceptor
are expected to work at such condition. As shown in Fig. 3 (weak regime) and
Fig. 4 (strong regime), these are computed for Methanol and Toluene in terms
of the chromophore-solvent coupling $\eta$. As expected, the oscillatory
behaviour of population inversion and coherence persists for longer times in
the weak regime. However, it is worth noting that those oscillations are more
favoured in Toluene (than in Methanol), regardless the strength of the
coupling regime.

In the strong regime, say $\eta=0.25$ (see violet line in Fig. 4) the
population inversion and coherence oscillations disappear around $100$ fs in
Methanol, while they continue to occur up to $\sim 220$ fs in Toluene. The
process of electronic energy transfer is more coherent in Toluene as the
interplay between the reorganisation energy and the inter-chromophore coupling
lies in the coherent transfer regime $\lambda\ll V_{DA}$Ishizaki:2010 .
Indeed, $\lambda/V_{DA}\sim 0.7\times 10^{-2}$ for Toluene while
$\lambda/V_{DA}\sim 0.3$ for Methanol (values of $\lambda$ are taken from
Table 2).

In spite of the fact that the dynamics in Methanol becomes incoherent faster
than in Toluene, the final population inversion is more efficient in the
former solvent; i.e., the stronger the coupling $\eta$ the higher the D-to-A
transfer. In particular, the stationary population inversion in Toluene is
$\sim 25\%$, and independent of $\eta$ (panel (c) in Figs. 3 and 4). For
Methanol, although the standing population inversion starts at a $\sim 15\%$
in the weak regime, it increases with $\eta$ and reaches about a $\sim 40\%$
in the strong regime (panel (a) of Figs. 3 and 4). This implies that a higher
ET can be obtained in Methanol at room temperature. This is in agreement with
the ET rate computed below in Section III.2 in terms of the density operator.
We stress that this phenomenon takes place at room temperature and that the ET
does not increase at lower temperatures, as shown below in Fig. 6.

The studied D2F complex exhibits a long-term coherence that is comparable to
the ones experimentally found in some light-harvesting and organic materials
lee2007C ; collini2009 ; hwang2010 ; tempelaar2015 ; roscioli2017 . As shown
in Fig. 4(b) for Methanol, coherence persists above $\sim 100$ fs in the
strong coupling regime $\eta_{M}=0.25$, where the system is found according to
the computed reorganisation energy. Furthermore, the coherence lifetime is
longer for Toluene ($\gtrsim 220$ fs, see Fig. 4(d)). As expected, the
decoherence process is slower for both solvents in the weak coupling regime
(Figs. 3(b) and 3(d)).

In addition to the time scale introduced by the solvent relaxation
($\tau_{s}\approx 100$ fs; $\hbar\omega_{c}=334$ cm-1 for Methanol, and
$\tau_{s}\approx 220$ fs; $\hbar\omega_{c}=167$ cm-1 for Toluene), other
relevant physical scales of interest are set by thermal fluctuations, the
reorganisation energy, and the intrinsic electronic coupling. In particular,
thermal fluctuations set by $k_{B}T/\hbar$ introduce a time scale
$\tau_{T}=\omega_{T}^{-1}\equiv\hbar/k_{B}T$ that affects the chromophore
dynamics for $t>\tau_{T}$. On the other hand, quantum vacuum fluctuations
contribute to the dephasing process in the regime $\tau_{s}<t<\tau_{T}$.
Solvent polarity also plays a crucial role as it affects the electronic energy
levels of the states involved in the transfer process. In our particular case,
Methanol exhibits a separation between the considered electronic states higher
than Toluene ($\epsilon_{M}>\epsilon_{T}$). A full numerical comparison of the
different time scales is given in Table 2 by means of the associated energy
ratios.

Table 2: Relevant energy scales for the D2F-solvent: ratio among thermal, decoherence, and inter-chromophore coupling energies. The involved parameters are as in Table 1. Solvent |  $T$(K) | $\frac{\hbar\omega_{T}}{\hbar\omega_{c}}$ | $\frac{\epsilon}{\hbar\omega_{c}}$ | $\frac{V_{DA}}{\hbar\omega_{c}}$ | $\frac{\hbar\omega_{T}}{\epsilon}$ | $\frac{\hbar\omega_{T}}{V_{DA}}$ | $\frac{\epsilon}{V_{DA}}$
---|---|---|---|---|---|---|---
Methanol $\eta=0.25$ $\lambda=85.80$ cm-1 |  |  |  |  |  |  |
300 | 0.62 | 2.90 | 0.84 | 0.22 | 0.75 | 3.46
50 | 0.10 | 2.90 | 0.84 | 0.04 | 0.12 | 3.46
10 | 0.02 | 2.90 | 0.84 | 0.01 | 0.03 | 3.46
Toluene $\eta=0.33$ $\lambda=2.20$ cm-1 |  |  |  |  |  |  |
300 | 1.25 | 3.81 | 1.90 | 0.33 | 0.66 | 2.01
50 | 0.21 | 3.81 | 1.90 | 0.06 | 0.11 | 2.01
10 | 0.04 | 3.81 | 1.90 | 0.02 | 0.02 | 2.01
Figure 6: Electron transfer rates as functions of the D2F-solvent coupling
strength. Methanol, $T=300$ K (a), $T=50$ K (b), and Toluene, $T=300$ K (c),
$T=50$ K (d). Turquoise, violet, black, orange and green curves correspond to
$\eta=0.08$, $\eta=0.25$, $\eta=0.45$, $\eta=0.50$ and $\eta=0.60$,
respectively. Other parameters are as in Table 1.

Temperature effects on the population inversion and coherence for a fixed
D2F-solvent strong coupling are shown in Fig. 5. We consider the system-bath
coupling that naturally identifies each solvent, i.e., $\eta_{M}=0.25$ for
Methanol and $\eta_{T}=0.33$ for Toluene. In these Figures, the temperature is
varied from $10$ K to $300$ K. $\langle\sigma_{z}(t)\rangle$ and
$\Re{(\rho_{DA})}$ exhibit oscillations showing the quantum character of the
D2F dynamics even at room temperature along tens of femtoseconds. Despite the
shortness of this quantum effect, experimental techniques involving
femtosecond pump-probe spectroscopy and quantum coherence control are in place
to detect and control the reported ultrafast quantum dynamics lee2007C ;
collini2009 ; hwang2010 ; tempelaar2015 ; roscioli2017 .

###  III.2 Electron Transfer in the D2F Complex

Electron transfer in the chromophore can be analysed throughout the chemical
reaction rate, which is defined in terms of the flux-flux correlation function
between the population of reactant and product states QUA56 . Thus, it is
possible to relate the ET speed with the time-dependent density matrix
elements previously computed in Section III.1. In general, for an $N-2$-bridge
system, the charge transfer from site $m$ to site $n$ can be described, within
a charge hopping model blumberger2015 , by a jump rate $k_{mn}\equiv
k_{m\rightarrow n}$, and the occupation probability (population) for each site
can be written in terms of the ET rates by means of the following set of
coupled differential equations:

$\displaystyle\frac{d\rho_{D}}{dt}=-k_{D2}\rho_{D}+k_{2D}\rho_{2}$ (6)
$\displaystyle\vdots$
$\displaystyle\frac{d\rho_{N-1}}{dt}=k_{N-2N-1}\rho_{N-2}+k_{AN-1}\rho_{A}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\,\,-(k_{N-1N-2}+k_{N-1A})\rho_{N-1}$
$\displaystyle\frac{d\rho_{A}}{dt}=k_{N-1A}\rho_{N-1}-k_{AN-1}\rho_{A}.$

In our $N=2$-site model, this set reduces to:

$\frac{d\rho_{D}}{dt}=-k_{DA}\rho_{D}+k_{AD}\rho_{A},$ (7)

for the donor, and the corresponding equation for the acceptor is given by the
constrain $\rho_{D}+\rho_{A}=1$. The behaviour of the ET rate is quantified by
considering the deviation from equilibrium of the density operator, which for
the donor reads $\Delta\rho_{D}(t)=\rho_{D}(t)-\rho_{D}(t\rightarrow\infty)$.
It is known that a time-dependent ET rate can be obtained for the expected
exponential decay of the deviation approaching to zero. By taking the time
derivative to $\Delta\rho_{D}(t)$, applying Eq. (7), and assuming $k_{AD}=0$
for the sake of simplicity, according to the D2F chromophore site energies,
the ET rate can be written as:

$k_{DA}=-\frac{1}{\rho_{D}(t)}\frac{d\rho_{D}(t)}{dt}.$ (8)

Although the value for the ET rate comes from taking the limit to infinity in
Eq. (8), this equation allows us to interpret the ET speed in terms of the
quantum-mechanical dynamics of the chromophoric system. In so doing, we plot
the time dependence of $k_{DA}$ as a function of the chromophore-solvent
coupling strength in Fig. 6. An oscillatory behaviour is observed for $k_{DA}$
before the ET rate reaches its stationary value. We point out that beyond the
hopping model here described, the asymptotic $k_{DA}$ relies on the physical
properties of the system: the D$-$A coupling, the reorganization energy, etc.
Indeed, we have compared our findings for the charge hopping model with the
results found by means of the flickering resonance model, and corroborated
that these converge to the same result for the ET rate (for a discussion on
both methods see e.g. blumberger2015 ; zhang2014 ). Thus, the matching with
the computed values in the insets of Fig. 6 is confirmed (numerics not shown).
As anticipated in Sec. III.1, the ET rate is favoured by a strong D2F-solvent
coupling, for both solvents. In particular, for Methanol, the ET rate
increases up to one order of magnitude as the strength of the D2F-solvent
coupling increases, as shown in the insets of Figs. 6(a) and (b). On the other
hand, such a variation is less appreciable for Toluene, as can be seen in
Figs. 6(c) and (d), which is in agreement with the quantum dynamics reported
in Figs. 3 and 4.

The results obtained in Fig. 6 are a strong indication that non-Markovianity
assists the electronic transfer in the considered organic molecules, and we
stress that such a transfer is optimised at room temperature. On the other
hand, our calculations also show that cooling down the system (see Figs. 6(b)
and (d)), although enhances the chromophore quantum coherence, translates into
a slight reduction of the coupling-assisted ET rate, as can directly be
compared between left ($T=300$ K) and right ($T=50$ K) panels, for both
Methanol and Toluene. Thus, we conclude that ET rate enhancement in the
chromophore is a direct consequence of the natural strong coupling between the
complex and the solvent, and that this is not necessarily due to its quantum
coherence properties (cfr. Figs. 5 and 6).

Although a large number of molecules is required for direct applications in
functional OPV devices, our results contribute to the understanding of
fundamental ultrafast processes and reveal the importance of considering the
quantum nature of the involved early-stage electron transfer, whose dynamics
and efficiency determine the function of the OPV material. Further
investigations of solvent effects can take place by contemplating other
relevant aspects, e.g., different spectral densities. In the case of many-body
scenarios, techniques like machine learning hase2017  could also be applied to
further extend the scope of our findings.

##  IV Conclusions

We have investigated the ultrafast quantum dissipative dynamics and electronic
transfer processes in an electron acceptor push-pull Rhodanine derivative
chromophore (D2F complex). In so doing, we have computed the effects due to
the chromophore environment by considering polar (Methanol) and non polar
(Toluene) solvents, under different temperature conditions. The D2F chemical
structure (and the identification of suitable excited and charge transfer
states) allowed us to evaluate, within the Onsager model of solvation, the
complex-solvent coupling strengths $\eta$, which clearly set the dynamics
within the strong coupling regime. The corresponding non-Markovian features of
the solvent were treated within a non-perturbative open quantum system
approach, with the numerically exact HEOM method chen2015 .

Our findings demonstrate the quantum character (coherent oscillations of the
time-dependent density operator) of the D2F dynamics, even at room
temperature, for both solvents. Particularly, that coherence is favoured in
the case of Toluene thanks to the interplay between the reorganisation energy
and the chromophore-chromophore coupling, in both complex-solvent coupling
regimes. In contrast, Methanol exhibits a less coherent but higher ET rates,
as it turns out that the stationary population inversion increases with
coupling strength $\eta$. On the other hand, Toluene exhibits a steady
population inversion that is invariant with respect to $\eta$.

To give a simple description of the ET process, we computed the ET rate
$k_{DA}$ as the deviation from equilibrium of the density operator
probabilities. As mentioned before, this ET rate is significantly enhanced by
Methanol. Our simulations show that, at room temperature, the steady value of
$k_{DA}$ increases with the coupling strength $\eta$, and reaches values
almost two orders of magnitude higher in Methanol than in Toluene.
Furthermore, although prolonged coherence oscillations are observed at lower
temperatures, the ET rate optimises at room temperature for both solvents as
it reaches higher steady values (compare right and left panels in Fig. 6).

Finally, we conclude that the non polar solvent enhances the quantum
properties of the organic chromophore dynamics when compared to the polar
solvent; yet, and perhaps counterintuitive, it is the latter that exhibits the
higher ET rates. Even though the question as to whether quantum coherence is
crucial/helps electron transfer mechanisms in light harvesting chromophoric
systems remains open to debate, our findings reveal not to be the case in the
D2F-solvent complex here considered.

##  Appendix A HEOM for the reduced system dynamics

We briefly describe the Hierarchical Equations of Motion employed to obtain
the dissipative dynamics of the reduced chromophoric system. Considering the
total system to be formed by the D2F molecule (reduced system, $S$) plus the
solvent (bath, $B$), the time evolution of the associated density operator can
be computed as Weiss :
$i\hbar\dot{\widehat{\rho}}_{T}(t)=\mathcal{L}_{T}(t)\widehat{\rho}_{T}(t)\equiv\left[\widehat{H}_{T},\widehat{\rho}_{T}(t)\right]$,
where the total Hamiltonian $\widehat{H}_{T}\equiv\widehat{H}$ (Sec. II.2).
Assuming an uncorrelated initial condition
$\widehat{\rho}_{S}(0)\otimes\widehat{\rho}_{B}$, with the bath in the thermal
state $\rho_{B}=\exp(-\beta\widehat{H}_{B})/Tr(\exp(-\beta\widehat{H}_{B}))$,
$\beta=1/k_{B}T$ ($k_{B}$ is the Boltzmann constant), the reduced system
dynamics is obtained by tracing out over the bath degrees of freedom:

$\widehat{\rho}_{S}(t)=Tr_{B}\left(\widehat{U}(t)\widehat{\rho}_{S}(0)\otimes\widehat{\rho}_{B}\right),$
(9)

where Strumpfer1 ; May ; strumpfer2011
$\widehat{U}(t)=\exp(-\frac{i}{\hbar}\int_{0}^{t}d\tau\mathcal{L}_{T}(\tau))$.
Hence, bath effects are encoded in the correlation functions Strumpfer1 ;
Okamoto

$C_{a}(t)=\langle
u_{a}(t)u_{a}(0)\rangle_{B}=\frac{1}{\pi}\int_{0}^{\infty}d\omega
J_{a}(\omega)\frac{e^{-i\omega t}}{1-e^{\beta\hbar\omega}},$ (10)

where $u_{a}(t)=\sum_{\xi}b_{a\xi}\chi_{\xi}(t)$ are the time-dependent bath
coupling operators. They arise from the system-bath Hamiltonian (3) written in
terms of the bath position coordinates $\chi_{\xi}$. In general, this can be
written as
$H_{SB}=\sum_{a,\xi}b_{a\xi}|\varphi_{a}\rangle\langle\varphi_{a}|\chi_{\xi}$.
For the two-site model used in the main text, the coupling strength of Eq. (3)
simply reads $g_{\xi}=b_{\xi}/\sqrt{2\hbar m_{\xi}\omega_{\xi}}$.

For a Drude type spectral density (Eq.(4)), the correlation functions can be
expanded as:

$C_{a}^{\infty}(t)=\sum_{k=0}^{\infty}c_{ak}e^{-\nu_{ak}t},$ (11)

where $\nu_{a0}=\omega_{c}$ and $\nu_{ak}=\frac{2\pi k}{\beta\hbar}$ ($k\geq
1$) are the Matsubara frequencies ishizaki2005 , while the correlation
coefficients Shi  $c_{a0}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{\lambda_{a}\omega_{c}}{\hbar}\left[\cot\left(\frac{\beta\hbar\omega_{c}}{2}\right)-i\right]$
and
$c_{ak}=\frac{4\lambda_{a}\omega_{c}}{\beta\hbar^{2}}\left(\frac{\nu_{ak}}{\nu_{ak}^{2}-\omega_{c}^{2}}\right)$,
for $k\geq 1$. The sum in Eq. (11) must be truncated at some finite level $K$;
$\nu_{aK}e^{-\nu_{aK}t}\approx\delta(t)$. Each exponential term in Eq. (11)
introduces a set of auxiliary density operators $\widehat{\rho}_{\textbf{n}}$
that are governed by the hierarchical structure of coupled equations of motion
Strumpfer1 ; Ishizaki :

$\displaystyle\frac{d\widehat{\rho}_{\textbf{n}}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\widehat{H}_{S},\widehat{\rho}_{\textbf{n}}\right]-\sum_{a=1}^{N}\sum_{k=0}^{K}n_{ak}\nu_{ak}\widehat{\rho}_{\textbf{n}}-i\sum_{a=1}^{N}\sum_{k=0}^{K}\left[|\varphi_{n}\rangle\langle\varphi_{n}|,\widehat{\rho}_{\textbf{n}_{ak}^{+}}\right]$
(12)
$\displaystyle-\sum_{a=1}^{M}\left(\frac{2\lambda_{a}}{\beta\hbar^{2}\omega_{c}}-\sum_{k=0}^{K}\frac{c_{ak}}{\nu_{ak}}\right)\big{[}|\varphi_{a}\rangle\langle\varphi_{a}|,\big{[}|\varphi_{a}\rangle\langle\varphi_{a}|,\widehat{\rho}_{\textbf{n}}\big{]}\big{]}$
$\displaystyle-\sum_{a=1}^{N}\sum_{k=0}^{K}n_{ak}\left(c_{a,k}|\varphi_{a}\rangle\langle\varphi_{a}|\widehat{\rho}_{\textbf{n}_{ak}^{-}}+\widehat{\rho}_{\textbf{n}_{ak}^{-}}|\varphi_{a}\rangle\langle\varphi_{a}|c_{a,k}^{*}\right),$

where each auxiliary density operator with nonnegative subscript
$\textbf{n}=(n_{10},\dots,n_{1K},\dots,n_{N0},\dots,n_{NK})$ is coupled to the
auxiliary density operators with subscripts
$\textbf{n}_{ak}^{\pm}=(n_{10},\dots,n_{ak}\pm 1,\dots,n_{NK})$. Each density
operator in the hierarchy is assigned to a hierarchy level
$L=\sum_{a=1}^{N}\sum_{k=0}^{K}n_{ak}$. The density operator
$\widehat{\rho}_{S}$ of the reduced system of interest corresponds to that one
with subscript $\textbf{n}=(0,\dots,0)$.

The amount of auxiliary density operators (which rapidly increases with level
$L$ and in principle is infinite) accounts for the non-Markovian character of
the system dynamics. Hence, a truncation such that
$L_{\text{max}}\gg\omega_{\text{max}}/\min{(\nu_{ak})}$, with
$\omega_{\text{max}}$ being a characteristic frequency of the system of
interest, must be taken. Convergence of the HEOM is finally achieved after
specifying the way to cutoff the terms beyond the truncation $L_{\text{max}}$.
For doing so, the so-called time-local truncation can be used such that the
Markovian approximation is employed for auxiliary density operators at level
$L_{\text{max}}$. This implies that, for all auxiliary density operators with
$L=L_{\text{max}}-1$,

$\sum_{k=0}^{K}\widehat{\rho}_{\textbf{n}_{ak}}^{+}\approx-i\left(\widehat{Q}_{a}^{K}(t)\widehat{\rho}_{\textbf{n}}-\widehat{\rho}_{\textbf{n}}\widehat{Q}_{a}^{K}(t)^{\dagger}\right),$
(13)

where
$\widehat{Q}_{a}^{K}(t)=\int_{0}^{t}C_{a}^{K}(\tau)e^{\left(-\frac{i}{\hbar}\widehat{H}_{S}\tau\right)}\left|\varphi_{a}\right\rangle\left\langle\varphi_{a}\right|e^{\left(\frac{i}{\hbar}\widehat{H}_{S}\tau\right)}d\tau$.

## Acknowledgements

We acknowledge J. Arce and A. Ortiz for helpful discussions. This work was
supported by the Colombian Science, Technology and Innovation Fund-General
Royalties System (Fondo CTeI-Sistema General de Regalías) under contract BPIN
2013000100007. C.E.S. thanks Universidad de Córdoba for partial support (Grant
No. CA-097).

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