# Mapping Electronic Decoherence Pathways in Molecules

Ignacio Gustin  Department of Chemistry, University of Rochester, Rochester,
New York 14627, USA  Chang Woo Kim  Department of Chemistry, Chonnam National
University, Gwangju 61186, South Korea  David W. McCamant  Department of
Chemistry, University of Rochester, Rochester, New York 14627, USA  Ignacio
Franco<EMAIL_ADDRESS>Department of Chemistry, University of
Rochester, Rochester, New York 14627, USA  Department of Physics, University
of Rochester, Rochester, New York 14627, USA

###### Abstract

Establishing the fundamental chemical principles that govern molecular
electronic quantum decoherence has remained an outstanding challenge.
Fundamental questions such as how solvent and intramolecular vibrations or
chemical functionalization contribute to the decoherence remain unanswered and
are beyond the reach of state-of-the-art theoretical and experimental
approaches. Here we address this challenge by developing a strategy to isolate
electronic decoherence pathways for molecular chromophores immersed in
condensed phase environments that enables elucidating how electronic quantum
coherence is lost. For this, we first identify resonance Raman spectroscopy as
a general experimental method to reconstruct molecular spectral densities with
full chemical complexity at room temperature, in solvent, and for fluorescent
and non-fluorescent molecules. We then show how to quantitatively capture the
decoherence dynamics from the spectral density and identify decoherence
pathways by decomposing the overall coherence loss into contributions due to
individual molecular vibrations and solvent modes. We illustrate the utility
of the strategy by analyzing the electronic decoherence pathways of the DNA
base thymine in water. Its electronic coherences decay in $\sim 30$ fs. The
early-time decoherence is determined by intramolecular vibrations while the
overall decay by solvent. Chemical substitution of thymine modulates the
decoherence with hydrogen-bond interactions of the thymine ring with water
leading to the fastest decoherence. Increasing temperature leads to faster
decoherence as it enhances the importance of solvent contributions but leaves
the early-time decoherence dynamics intact. The developed strategy opens key
opportunities to establish the connection between molecular structure and
quantum decoherence as needed to develop chemical strategies to rationally
modulate it.

Quantum Dynamics, Open Quantum Systems, Chemical Design, Resonance Raman,
Spectral Densities

Chemistry builds up from the idea that molecular structure determines the
chemical and physical properties of matter. This principle guides the modern
design of molecules for medicine, agriculture, and energy applications.
However, chemical design principles have mostly remained elusive for emerging
quantum technologies that exploit the fleeting but transformative properties
of quantum coherence responsible for the wave-like interference of microscopic
particles. Specifically, currently, it is not understood how chemical
structure should be modified to rationally modulate quantum coherence and its
loss. Wasielewski  _et al._ (2020); Atzori and Sessoli (2019)

To leverage Chemistry’s ability to build complex molecular architectures for
the development of next-generation quantum technologies, there is a critical
need to identify methods to tune quantum coherences in molecules and better
protect them from quantum noise (or decoherence) that arises due to
uncontrollable interactions of the molecular degrees of freedom of interest
with its quantum environment.Zhu  _et al._ (2022); Bayliss  _et al._ (2022);
Hu  _et al._ (2022); Viola  _et al._ (1999) Protecting and manipulating
molecular coherences is also necessary to unshackle chemical processes from
the constraints of thermal Boltzmann statistics, as needed to enhance
molecular function through coherence,Scholes  _et al._ (2017) develop novel
routes for the quantum control of chemical dynamics,Shapiro and Brumer (2012);
Rice  _et al._ (2000) and the design of optical spectroscopies with enhanced
resolution capabilities.Wang  _et al._ (2019); Mukamel (1995)

In particular, electronic coherences generated by photoexcitation with
ultrafast laser pulses in single-molecules and molecular arrays have received
widespread attention.Biswas  _et al._ (2022); Dani and Makri (2022) These
coherences dictate the photoexcited dynamics of molecules and, as such, play a
pivotal role in our elementary description of photochemistry, photophysics,
charge and energy transport, and in our understanding of vital processes such
as photosynthesis and vision.De Sio and Lienau (2017); Nelson  _et al._
(2018); Kreisbeck and Kramer (2012); Popp  _et al._ (2019); Blau  _et al._
(2018); Ramos  _et al._ (2019); Hahn and Stock (2000); Marsili  _et al._
(2019); Chuang and Brumer (2021) These coherences decay on ultrafast
(femtosecond) timescales due to the interaction of the electronic degrees of
freedom with intramolecular vibrational modes and solvent.Hu  _et al._ (2018);
Gu and Franco (2018); Vacher  _et al._ (2017); Arnold  _et al._ (2017);
Matselyukh  _et al._ (2022) Despite considerable experimental and theoretical
efforts to study electronic decoherence in molecules Salvador  _et al._
(2003); Colonna  _et al._ (2005); Yang  _et al._ (2005); Biswas  _et al._
(2022); Hwang and Rossky (2004a, b); Shu and Truhlar (2023); Hu  _et al._
(2018); Gu and Franco (2018); Jasper and Truhlar (2005); Matselyukh  _et al._
(2022), its basic chemical principles are still not understood. For instance,
how does electronic coherences in, say, the DNA base thymine decohere in
water? How do the different vibrations in the molecule and solvent contribute
to the overall coherence loss and which one is dominant? How do chemical
functionalization and isomerism influence the different contributions to the
overall decoherence? Addressing these basic questions requires a general
strategy to connect chemical structure of both molecule and solvent to quantum
decoherence phenomena.

From an experimental perspective, the task requires a method that enables the
decomposition of the overall decoherence into individual contributions by
chemical groups. This is beyond what can be done using optical absorption or
fluorescence,Heller (1981) photon-echoSalvador  _et al._ (2003); Colonna  _et
al._ (2005); Yang  _et al._ (2005) and even multidimensional laser
spectroscopies.Biswas  _et al._ (2022) To varying extents, these techniques
enable characterizing the overall decoherence time scale and the molecular
Hamiltonian, but do not offer means to disentangle the contributions by
specific functional chemical groups.

Figure 1: Mapping electronic decoherence pathways. (a) Photoexcitation of
molecules creates electronic superposition states with coherences that decay
on ultrafast timescales due to interactions of the electronic chromophore with
the surrounding nuclei (solvent and intramolecular vibrations). Here we
advance a method to quantify the overall decoherence and decompose it into
contributions due to specific vibrational modes (b) and solvent (c), thus
establishing decoherence pathways that link chemical structure with quantum
decoherence. This is done by extracting the highly structured spectral
densities for solvated molecules directly from resonance Raman experiments and
using the theory of decoherence functions to isolate individual contributions
by solvent and vibrational modes. The strategy opens exciting opportunities to
unlock the chemical principles of electronic quantum decoherence.

From a theory perspective, the challenge is that accurately capturing the
decoherence requires a fully quantum quantitative description of the
energetics and dynamics of the molecule in its chemical environment, and the
computational cost of doing so increases exponentially with the size of the
molecule and the environment. For this reason, most of our theoretical
understanding of decoherence arises from model problems that, while extremely
useful, do not fully capture the complexities of realistic chemical
systems.Tomasi  _et al._ (2005); Olbrich  _et al._ (2011) For instance, recent
efforts to quantify electronic decoherence by explicitly propagating
multidimensional quantum dynamics of molecules have so far been limited to
systems at zero temperature and in vacuum,Dey  _et al._ (2022); Arnold  _et
al._ (2018, 2017); Hu  _et al._ (2018); Golubev  _et al._ (2020); Vester  _et
al._ (2023) and thus are not informative of quantum decoherence in solvent and
other condensed phase environments. In turn, efforts to include the influence
of thermal environments implicitly through numerically exact quantum master
equations, such as the HEOMTanimura (2020); Ikeda and Scholes (2020) and
TEMPOStrathearn  _et al._ (2018), have thus far been mostly limited to simple
models that do not necessarily capture the complexity of chemical
environments.Zhang  _et al._ (2020); Duan  _et al._ (2020); Varvelo  _et al._
(2023)

Specifically, in quantum master equations the interaction between the
molecular chromophore and its thermal environment is characterized by a
spectral density, $J(\omega)$Leggett  _et al._ (1987), which quantifies the
frequencies of the nuclear environment, $\omega$, and their coupling strength
with the electronic excitations. Unfortunately, the quantitative determination
of $J(\omega)$ has remained elusive to theoryMaity  _et al._ (2021); Kim  _et
al._ (2018); Cignoni  _et al._ (2022); Jang and Mennucci (2018); Lee and Coker
(2016); Kell  _et al._ (2013); Chen  _et al._ (2023), and its experimental
characterization is limited.Rätsep and Freiberg (2007); Rätsep  _et al._
(2008); Pieper  _et al._ (2009); Freiberg  _et al._ (2009); Gryliuk  _et al._
(2014) For this reason, most studies of molecular decoherence based on quantum
master equations rely on simple models of the spectral density that do not
capture the interactions of realistic chemical problems.Zhang  _et al._
(2020); Duan  _et al._ (2020); Varvelo  _et al._ (2023)

Here we introduce a general strategy to quantitatively characterize the
electronic decoherence dynamics of molecules in realistic chemical
environments and to map decoherence pathways, see Fig. 1. The strategy is
based on reconstructing the spectral density $J(\omega)$ from resonance Raman
experiments Mukamel (1995); Cina (2022) and decomposing the overall coherence
decay into contributions by individual vibrational and solvent modes. Using
this strategy, we can now address previously inaccessible questions, such as
the interplay of solvent and vibrational contributions and the influence of
chemical substitution on quantum decoherence. The reconstructed spectral
densities open opportunities to investigate molecular decoherence dynamics in
realistic chemical environments using state-of-the-art methods in quantum
dynamics. Further, the overall strategy connects chemical structure to quantum
decoherence dynamics and opens unique opportunities to develop the chemical
principles of quantum decoherence.

## Quantum Decoherence Basics

Quantum coherences are usually defined as the off-diagonal elements of the
density matrix $\sigma(t)$ expressed in a given basis. For molecular quantum
dynamics, it is useful to think about coherences in the energy basis
$\sigma_{eg}(t)=\bra{e}\sigma(t)\ket{g}$ between an electronic ground
$\ket{g}$ and excited state $\ket{e}$ as they lead to quantum beatingsWang
_et al._ (2019) that are visible in laser spectroscopies. During decoherence
electronic superposition states with density matrix
$\sigma=|\psi\rangle\langle\psi|$, where
$\ket{\psi}=c_{g}\ket{g}+c_{e}\ket{e}$, decay to a statistical mixture of
states $\sigma(t)=p_{g}\ket{g}\bra{g}+p_{e}\ket{e}\bra{e}$ where $p_{i}\geq 0$
are the state populations. This decay is ubiquitous and arises due to the
interaction of the electrons with the surrounding nuclei (vibrations,
torsions, and solvent).Breuer and Petruccione (2002); Schlosshauer (2007);
Joos  _et al._ (2013)

At zero temperature, electronic decoherence is understood as arising due to
nuclear wavepacket evolution in alternative potential surfaces. Hwang and
Rossky (2004a, b); Prezhdo and Rossky (1998); Bittner and Rossky (1995);
Jasper and Truhlar (2005); Hu  _et al._ (2018); Fiete and Heller (2003); Shu
and Truhlar (2023) In this view, the wavefunction of electrons and their
nuclear environment is in an entangled state
$\ket{\Psi(t)}=\ket{g}\ket{\chi_{g}(t)}+\ket{e}\ket{\chi_{e}(t)}$, where
$\ket{\chi_{n}}$ is the nuclear wavepacket evolving in the ground $n=g$ or
excited $n=e$ potential energy surface $E_{n}(\textbf{x})$ and
$\\{\textbf{x}\\}$ are the nuclear coordinates. In this case, the electronic
coherences $\sigma_{eg}(t)=\bra{\chi_{e}(t)}\chi_{g}(t)\rangle$ decay with the
overlap between the nuclear wavepackets in the ground and excited state. At
finite temperatures $T$, for early times $t$ Gu and Franco (2018, 2017);
Prezhdo and Rossky (1998); Bittner and Rossky (1995) the electronic coherences
$|\sigma_{eg}(t)|^{2}=|\sigma_{eg}(0)|^{2}\exp{(-t^{2}/{\tau}_{eg}^{2})}$
decay like a Gaussian with a time scale
$\tau_{eg}=\frac{\hbar}{\sqrt{\langle\delta^{2}{\mathcal{E}}_{eg}\rangle}}$
dictated by fluctuations of the energy gap
$\mathcal{E}_{eg}(\textbf{x})=E_{e}(\textbf{x})-E_{g}(\textbf{x})$ due to
thermal or quantum fluctuations of the nuclear environment. At high
temperatures, $\tau_{eg}=\frac{\hbar}{\sqrt{sk_{\mathrm{B}}T}}$ is simply
connected to the Stokes shift $s$ (the difference between positions of the
maxima of absorption and fluorescence due to a given electronic transition).

## Reconstructing Spectral Densities from Resonance Raman

We first note that $J(\omega)$ can be reconstructed from resonance Raman (RR)
spectroscopy as this technique provides detailed quantitative structural
information about molecules in solution. While RR experiments are routinely
used to investigate the vibrational structure and photodynamics of
moleculesDietze and Mathies (2016); Piontkowski  _et al._ (2019); Fang and
Tang (2020), their utility for decoherence studies had not been appreciated
before. A leading technique used to reconstruct $J(\omega)$ for molecules is
difference fluorescence line narrowing spectra ($\Delta$FLN).Rätsep and
Freiberg (2007); Rätsep  _et al._ (2008); Pieper  _et al._ (2011, 2009);
Freiberg  _et al._ (2009); Gryliuk  _et al._ (2014) The advantage of RR over
$\Delta$FLN is that it can be used in both fluorescent and non-fluorescent
moleculesMcCamant  _et al._ (2004, 2003); Kukura  _et al._ (2007); Lee  _et
al._ (2004) (instead of only fluorescent), in solvent (instead of glass
matrices), at room temperature (instead of cryogenic temperatures), and that
it offers the possibility of varying the nature of the solvent and the
temperature.

In RR spectroscopy, incident light $E_{\text{L}}$ of frequency $\omega$ chosen
to be at resonance with an electronic transition is scattered inelastically,
yielding a Stokes and an anti-Stokes signal $E_{\text{S}}$ that changes the
molecular vibrational state. In contrast to non-resonance Raman where
selection rules restrict the transitions to vibrational modes that change the
molecular polarizability, in RR all vibrational modes that are coupled to the
electronic transition yield a signal.McHale (2002) The theory of RR cross
sections (see Supplementary Information and Refs. Myers  _et al._ (1982);
Shreve and Mathies (1995); Mukamel (1995); Tannor and Heller (1982); Page and
Tonks (1981)) is based on a two-surface molecular model, with Hamiltonian
$H_{M}=H_{g}\left\lvert g\right\rangle\left\langle
g\right\rvert+H_{e}\left\lvert e\right\rangle\left\langle e\right\rvert$.
Here,
$H_{g}=\sum_{\alpha}\left(\frac{p^{2}_{\alpha}}{2m_{\alpha}}+\frac{1}{2}m_{\alpha}\omega^{2}_{\alpha}x^{2}_{\alpha}\right)$
is the ground-state state nuclear Hamiltonian, where $x_{\alpha}$ and
$p_{\alpha}$ are the position and momentum operators of the $\alpha$-th mode
of effective mass $m_{\alpha}$ and frequency $\omega_{\alpha}$. In turn, the
excited potential energy surface consists of the same set of nuclear modes but
displaced in conformational space, i.e.,
$H_{e}=\hbar\omega_{eg}+\sum_{\alpha}\left(\frac{p^{2}_{\alpha}}{2m_{\alpha}}+\frac{1}{2}m_{\alpha}\omega^{2}_{\alpha}\left(x_{\alpha}-d_{\alpha}\right)^{2}\right)$
where $\hbar\omega_{eg}$ is the electronic excitation energy. The displacement
along the $\alpha$-th mode, $d_{\alpha}$, determines the strength of the
electron-nuclear coupling as measured by the Huang-Rhys factor
$S_{\alpha}=\frac{1}{2\hbar}m_{\alpha}\omega_{\alpha}d^{2}_{\alpha}$ or
reorganization energy $\lambda_{\alpha}=\hbar S_{\alpha}\omega_{\alpha}$. The
overall Stokes shift $s$ is determined by
$s=2\lambda=2\sum_{\alpha}\lambda_{\alpha}$, where $\lambda$ is the overall
reorganization energy.

The RR experiments yield the frequencies of the vibrational modes
$\\{\omega_{\alpha}\\}$ and also the Huang-Rhys factors $S_{\alpha}$ which are
extracted by fitting the signal to the RR cross section.Myers  _et al._
(1982); Shreve and Mathies (1995); Mukamel (1995); Tannor and Heller (1982);
Page and Tonks (1981) The analysis also yields contributions due to solvent,
and other (sub 200 cm-1) nuclear modes that cannot be experimentally resolved,
to the overall line shape and Stokes shift which are captured by an overdamped
oscillator with correlation time $\frac{1}{\Lambda}$ and reorganization energy
$\lambda_{0}$.Li  _et al._ (1994) The procedure that is routinely done to
extract these parameters is detailed in the supplementary information.

This information is exactly what is needed to reconstruct the spectral density
of molecules in solvent with full chemical complexity. In this case, the
spectral density consists of a broad low-frequency feature $J_{0}(\omega)$
describing the influence of solvent and a series of discrete high-frequency
peaks $J_{\alpha}(\omega)$ ($\alpha=1,...,N$) due to interaction of the
chromophore with intramolecular vibrational modes. The basic functional forms
for the spectral density have been isolated through physical models in which
the thermal environment is described as a collection of harmonic oscillators
(the so-called Brownian oscillator model).Mukamel (1995) This yields,

$\begin{split}J(\omega)&=\sum_{\alpha=0}^{N}J_{\alpha}(\omega),\text{ where
}\\\
J_{\alpha}(\omega)&=\frac{2}{\pi}\lambda_{\alpha}\omega^{2}_{\alpha}\frac{\omega\gamma_{\alpha}}{\left(\omega^{2}_{\alpha}-\omega^{2}\right)^{2}+\omega^{2}\gamma^{2}_{\alpha}}\end{split}$
(1)

and $\\{\gamma_{\alpha}^{-1}\\}$ are the vibrational lifetimes. The solvent
spectral density can be adequately modeled via a Drude-Lorentz functional form
$J_{0}(\omega)=\frac{2}{\pi}\lambda_{0}\frac{\omega\Lambda}{\omega^{2}+\Lambda^{2}}$
which arises from $J_{\alpha}(\omega)$ when the lifetimes are short (i.e. when
$\omega_{\alpha}/\gamma_{\alpha}\ll 1$). Thus, all the parameters defining the
spectral density can be extracted from the RR experiments. While the specific
value of $\gamma_{\alpha}$ cannot be directly resolved using RR, its value
plays no significant role in electronic decoherence as vibrational relaxation
occurs on much longer time scales for physical values of
$\hbar\gamma_{\alpha}\in(1,20)$cm-1 (see Fig. S1).Maity  _et al._ (2021);
Bennett  _et al._ (2013); Novoderezhkin  _et al._ (2004)

The spectral densities that can be extracted from RR are due to “pure
dephasing” processes where there is no net exchange of energy between the
system and the environment. That is, for processes in which the environment
does not lead to electronic transitions or relaxation. In molecules, these
pure dephasing processes have been the subject of intense interest as they
typically occur in time scales that are much faster than the overall
relaxation and dominate the decoherence dynamics. Von der Linde  _et al._
(1997); Brinks  _et al._ (2014)

Naturally, this $J(\omega)$ reconstruction is limited by experimental
constraints and the scope of validity of the displaced harmonic oscillator
model. For instance, internal conversion due to conical intersections
occurring on time scales comparable or faster than the dephasing, or
unresolvable electronic state congestion, will invalidate the analysis. In
addition, anharmonicities, frequency shifts and Duschinsky rotations (mode
mixing) in the potential energy surfaces can introduce additional time scales
to the decoherence that are beyond the displaced harmonic oscillator
model.Hwang and Rossky (2004b); Hu  _et al._ (2018) However, numerical
evidence thus far suggest that their influence occurs on time scales that are
much slower than the overall dephasing in the condensed phase and thus they
are expected to play a minor role. Hwang and Rossky (2004b); Hu  _et al._
(2018); Zuehlsdorff  _et al._ (2020)

## Decoherence Pathways

Once a spectral density is extracted from RR experiments, the task is then to
use it to capture the realistic quantum dynamics of the molecular chromophore.
The challenge is that the computational cost of doing so increases
exponentially with the number of features $N+1$ in the environment.

Second, we note that pure-dephasing processes occurring in thermal harmonic
environments admit an exact solution without explicit propagation. In this
case Unruh (1995); Palma  _et al._ (1996); Breuer and Petruccione (2002), the
decay of the off-diagonal elements of the density matrix
$|\sigma_{eg}(t)|=|\sigma_{eg}(0)|\exp[-\Gamma(t)]$ is determined by the
decoherence function

$\Gamma(t)=\frac{1}{\hbar}\int_{0}^{\infty}d\omega\,J(\omega)\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\frac{1-\cos(\omega
t)}{\omega^{2}}$ (2)

where $k_{B}T$ is the thermal energy. Thus, computing the decoherence dynamics
amounts to accurately calculating $\Gamma(t)$ via numerical integration thus
avoiding explicit time propagation of $\sigma(t)$.

Third, remarkably, in this context, it is possible to extract decoherence
pathways in molecules by decomposing the overall decoherence into
contributions due to solvent and specific vibrational modes. Inserting Eq. (1)
into Eq. (2) leads to a decomposition of the decoherence function
$\Gamma(t)=\sum_{\alpha=0}^{N}\Gamma_{\alpha}(t)$ into individual
contributions by specific nuclear modes

$\Gamma_{\alpha}(t)=\frac{1}{\hbar}\int_{0}^{\infty}d\omega
J_{\alpha}(\omega)\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\frac{1-\cos(\omega
t)}{\omega^{2}}.$ (3)

Alternative methods to recover the overall decoherence time such as photon-
echo experimentsSalvador  _et al._ (2003); Colonna  _et al._ (2005); Yang  _et
al._ (2005), optical absorption cross-sectionHeller (1981) and even 2D
electronic spectroscopyBiswas  _et al._ (2022) cannot decompose the overall
signal into contributions by solvent and specific vibrational modes. This
unique aspect of our strategy opens a remarkable opportunity to develop
chemical insights on decoherence dynamics by isolating contributions due to
particular nuclear modes.

Figure 2: Molecular structure of DNA base thymine (T) its nucleoside (T+S) and
nucleotide (T+S+P) Figure 3: Decoherence dynamics of thymine nucleotide in
water at 298 K. (a) Spectral density $J(\omega)$ reconstructed from RR
experiments in Ref. Yarasi  _et al._ (2007); Ng  _et al._ (2008); Billinghurst
_et al._ (2012) with a vibrational damping factor $\hbar\gamma=4$ cm-1. (b)
Electronic decoherence (red line) and its contributions due to solvent (blue
line) and intramolecular vibrations (dashed line). The remaining curves depict
decoherence due to solvent plus selected vibrational modes (orange: 1663 cm-1;
yellow: 1663 cm-1\+ 1376 cm-1\+ 1243 cm-1).

## Electronic Decoherence in a DNA nucleotide

### Spectral Density and Overall Decoherence

To illustrate the power of the strategy, based on a RR analysis by
LoppnowYarasi  _et al._ (2007); Ng  _et al._ (2008); Billinghurst  _et al._
(2012) we have reconstructed the spectral density for a DNA nucleotide
(thymidine 5’-monophospate), see Fig. 2, in water at 298 K and used it to
scrutinize its decoherence dynamics. Figure 3 shows the reconstructed spectral
density and the overall decoherence dynamics. The parameters used to
reconstruct the spectral density are listed in Table S1 of the supplementary
information. The spectral density (Fig. 3a) consists of a broad low-frequency
feature (blue line, inset) due to solvent and several sharp peaks due to
intramolecular vibrations. The overall decoherence (Fig. 3b, red line) shows
an initial Gaussian decay profile, followed by a partial recurrence at $\sim
20$ fs and a complete decay of coherence in $\sim 30$ fs. The Gaussian feature
of the decoherence is a universal feature of initially separable system-bath
states Gu and Franco (2018) and can be seen as arising due to the quantum Zeno
effect. The recurrence signals the partial recovery of the nuclear wavepacket
overlap $|\bra{\chi_{e}(t)}\chi_{g}(t)\rangle|$ due to the wavepacket dynamics
in the excited state potential energy surface.Hu  _et al._ (2018)

It is known that this DNA nucleotide also exhibits internal conversion due to
the presence of a conical intersection. Such dynamics occurs on time scales of
100s of fs Onidas  _et al._ (2002); Gustavsson  _et al._ (2006) that are
significantly longer than those required for the pure dephasing process to
complete. This disparity in time scales makes the pure dephasing analysis to
remain applicable even in the presence of such non-radiative electronic
transitions.

### Decoherence Pathways

To understand the role of solvent and intramolecular vibrations on the
decoherence, Fig. 3b shows the contributions due to solvent (blue line),
vibrations (dashed line), and solvent plus specific vibrational modes (yellow
and orange lines). As shown, the early-time decoherence dynamics is dominated
by vibrational contributions while the overall coherence decay is dictated by
the solvent. Out of all vibrational modes in the molecule, we find that the
mode at $1663$ cm-1, plays the most important role. In fact, taking into
account the solvent and just three peaks due to intramolecular vibrations (at
$1663$ cm-1, $1376$ cm-1 and $1243$ cm-1) accounts for most of the decoherence
dynamics (yellow line). Overall, these results demonstrate the close interplay
between solvent and intramolecular vibrations in the decoherence dynamics.

Figure 4: (a) Frontier molecular orbitals for thymine and (b) vibrational
normal modes most important for electronic decoherence. HOMO/LUMO stand for
highest occupied/lowest unoccupied molecular orbital. The computed normal
modes account to the three most prominent peaks in the spectral density in
Fig. 3. The color convention is as follows: carbon (black), hydrogen (white),
nitrogen (cyan) and oxygen (red).

### Normal Mode Analysis

To understand the nature of the intramolecular vibrational modes most
responsible for the decoherence, we have modeled the resonance Raman spectra
using time-dependent density functional theory (TD-DFT)Perdew  _et al._
(1996); Grimme  _et al._ (2010); Rappoport and Furche (2010); Macak  _et al._
(2000) (computational details are included in the Methods). To keep the
computations tractable, we focus on the RR spectra of thymine as the spectral
features of the nucleoside and nucleotide are nearly identicalYarasi  _et al._
(2007); Ng  _et al._ (2008); Billinghurst  _et al._ (2012) and thus
vibrational assignments in thymine are expected to also apply in the other two
cases. Quantitative agreement between theory and experiment is obtained by
including four explicit water molecules in the first solvation shell that are
an active part of the vibrational motion of the molecule in its microsolvated
environment and that are significantly coupled to the electronic transition.

The experimental and computed RR spectra are shown in Fig. S3. The excellent
agreement between theory and experiments enables us to make vibrational
assignments to all the peaks in the spectral density. Figure 4 details the
frontier molecular orbitals for the electronic transition sampled by the RR
experiment and the 4 normal modes responsible for the most prominent peaks in
the spectral density (orange/yellow peaks in Fig. 3a). Table S6 details the
computed frequencies/Huang-Rhys factors for all 75 normal modes in the complex
and Fig. S4 provides an animation of the 10 normal modes most responsible for
decoherence. The normal modes that contribute most importantly to the
electronic decoherence are those that generate distortion around the thymine
ring and/or the carbonyl bonds where the frontier orbitals involved in the
electronic transition are located.

Figure 5: Temperature dependence of the electronic decoherence for thymine
nucleotide in water. (a) Full decoherence and (b) contributions due to
vibrations only.

### Temperature Dependence

To understand the role of temperature in electronic decoherence, we repeated
the analysis in the 283-313 K range every 5 K while taking into account that
for harmonic environments $J(\omega)$ is temperature independent. The results
in Fig. 5 show that solvent effects become increasingly more important with
temperature. In turn, the vibrational contributions (Fig. 5b) remain unaltered
because thermal energy does not excite the vibrational modes that dictate
decoherence. For this reason, the early-time decoherence dynamics is
approximately independent of temperature as it is controlled by vibrational
motion, while the overall decay is accelerated with increasing temperature
leading to a reduction in the visibility of the coherence recurrence at $\sim
20$ fs. Additional experiments are needed to isolate the importance of
emergent temperature dependence of $J(\omega)$ that can arise from differences
in the mapping of chemical environments to harmonic models with varying
temperature.

### Early Time Decoherence

Recent efforts to understand electronic decoherence in molecules have focused
on the influence of vibrational motion upon photoexcitation at 0 K. Arnold
_et al._ (2017, 2018); Hu  _et al._ (2018); Dey  _et al._ (2022); Golubev  _et
al._ (2020); Vester  _et al._ (2023) Figures 3 and 5 show that this strategy
accurately captures the early time coherence decay, which is dominated by
high-frequency vibrations, but will not be able to correctly capture the
overall coherence loss, which requires taking into account the solvent and the
temperature dependence of the decoherence.

A useful formula for electronic decoherence time scales arises by focusing on
early times, which yield a Gaussian decay
$|\sigma_{eg}(t)|=|\sigma_{eg}(0)|e^{-t^{2}/\tau_{d}^{2}}$. Prezhdo and Rossky
(1998); Hwang and Rossky (2004a); Gu and Franco (2018) For harmonic
environments,
$\tau_{d}^{-2}=\frac{1}{2\hbar}\int_{0}^{\infty}d\omega\,J(\omega)\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)=\frac{\hbar}{\sqrt{\langle\delta^{2}{\mathcal{E}}_{eg}\rangle}}$
as can be seen by expanding the decoherence function $\Gamma(t)$ in Eq. (2) up
to second order in time. Figure 5a contrasts the exact decoherence dynamics
with this Gaussian form (dashed line). As shown, this simple estimate
accurately captures the early time decay but does not capture possible
recurrences and overestimates the overall decoherence by a factor of $\sim 2$.
Thus, such formulas are useful to understand the early stages of decoherence
but exaggerate the decoherence rate for electrons in molecules.

Nevertheless, to capture the electronic decoherence for molecules immersed in
solvent and other condensed phase environments, Fig. 5 underscores the utility
of the simple Gaussian estimate as it captures the early-time decoherence
dynamics correctly while avoiding explicitly solving the quantum dynamics. By
contrast, computationally expensive efforts to capture the decoherence based
on propagating the correlated quantum dynamics of electrons and nuclei at 0 K
using Multi Configurational Time-Dependent Hartree (MCTDH) Manthe  _et al._
(1992); Beck  _et al._ (2000); Arnold  _et al._ (2017, 2018); Dey  _et al._
(2022); Vester  _et al._ (2023) are only relevant for early times where finite
temperature effects due to solvent do not play a role. Figure 5 shows that
this early-time region of the decoherence can be accurately captured while
avoiding propagating the quantum dynamics all together.

### Effect of Chemical Substitution

How does varying chemical structure influences quantum coherence loss? To
address this question, we reconstructed the spectral densities of thymine (T),
its nucleoside (T + sugar (S)) and nucleotide (T + S + phosphate (P)) in water
at 298 K from RR data in Refs. Billinghurst  _et al._ (2012); Yarasi  _et al._
(2007). Figure 6 details the overall decoherence and the contributions of
intramolecular vibrations and solvent to the decoherence in this series, and
Fig. 2 their chemical structure. The reconstructed $J(\omega)$ for thymine and
its nucleoside, as well as all the needed parameters, are included in Fig. S2
and Table S2-S3 of the supplementary information, respectively.

Surprisingly, the overall decoherence in thymine is the fastest, even when it
is the smallest and least polar molecule in the set and admits fewer possible
hydrogen-bond interactions with the solvent. This can be rationalized by
noting that the electronic excitation is localized in the thymine ring (see
Fig. 4) and thus interactions with the solvent with atoms in this area are
expected to have the strongest impact on the decoherence. In fact, in T the
N-H termination in the ring, which is absent in T+S and T+S+P, strongly
interacts with water through hydrogen bondingBeyere  _et al._ (2004),
accelerating the overall decoherence.

Coherence recurrences are prominent in T+S because solvent-induced decoherence
is the slowest in this case (see Figure 6c). By contrast, T shows no
recurrences due to ultrafast decoherence due to the solvent. Figure 6b details
the decoherence due to intramolecular vibrations. The trend for the early time
decoherence due to vibrations is opposite to the overall behavior with T
showing the slowest decoherence, followed by T+S+P and then T+S. Thus, the
overall decoherence trends in Figure 6a are due to both contributions of
solvent and intramolecular vibrations, even at early times.

Figure 6: Effect of chemical substitution on electronic decoherence. (a) Full
electronic decoherence for the series (See Fig. 2). (b) Electronic decoherence
due to vibrations only. (c) Electronic decoherence due to solvent only.
Spectral densities were reconstructed from data in Ref. Billinghurst  _et al._
(2012); Yarasi  _et al._ (2007).

## Conclusions

In conclusion, we have advanced a strategy that can be used to establish the
chemical principles that underlie electronic quantum coherence loss in
molecules. Previously inaccessible fundamental questions such as how chemical
functionalization or solvent character contribute to the overall electronic
decoherence, or how to develop chemical strategies to rationally modulate the
decoherence can now be systematically addressed. The strategy is based on
reconstructing spectral densities from resonance Raman experiments on
molecular chromophores in solvent, and using the theory of decoherence
functions to decompose the overall decoherence dynamics into contributions by
individual solvent and vibrational modes, thus establishing decoherence
pathways in molecules.

Using this strategy we were able to scrutinize, for the first time, the
electronic decoherence dynamics of the DNA base thymine in room temperature
water. In this case, the decoherence occurs in $\sim 20-30$ fs with the early
stages dictated by intramolecular vibrations, while the overall decoherence
time due to interactions with solvent. Chemical substitution of the thymine
with sugar and sugar phosphate reveals a recurrence in the coherence dynamics
that is suppressed by hydrogen bonding between thymine and water. The
vibrational modes most responsible for the decoherence are those that distort
the thymine ring and carbonyl bonds. Increasing the temperature accelerates
the contributions of solvent to the decoherence but leaves the early-time
decoherence dynamics intact. For pure-dephasing, this early stage of the
decoherence is accurately captured by the theory of decoherence time scales Gu
and Franco (2018, 2017); Prezhdo and Rossky (1998); Bittner and Rossky (1995)
which does not require propagating the quantum dynamics.

We envision the application of this strategy to elucidate the role of diverse
chemical and biological environments to the decoherence, and to establish
decoherence pathways for localized and delocalized, spin-specific and charge-
transfer electronic excitations, and in molecules of varying size and
rigidity. Further, the extracted spectral densities will be of general utility
to characterize the dynamics of molecular chromophores with full chemical
complexity using quantum master equations Tanimura (2020); Ikeda and Scholes
(2020); Strathearn  _et al._ (2018) of increasing sophistication, and to
advance our computational capabilities to accurately capture electron-nuclear
interactions in condensed phase environments.Chen  _et al._ (2023); Maity  _et
al._ (2021); Cignoni  _et al._ (2022); Kim  _et al._ (2018).

## Materials and Methods

### RR spectra from TD-DFT

To compute the resonance Raman spectra for thymine we adopted the following
computational strategy. First, to determine the solvation structure around
thymine we used Autosolvate to create microsolvated cluster structures.Hruska
_et al._ (2022) Using a two-layer ONIOM, we performed a preliminary analysis
of the normal modes of the microsolvated thymine. In the ONIOM
analysisDapprich  _et al._ (1999); Chung  _et al._ (2015) the molecule was
treated at a quantum mechanical (QM) level while the solvent was captured
through a classical force field. For the QM computation we used Density
Functional Theory (DFT) with the PBE0 functional and 6-311+G(d,p) as a basis.
The normal mode analysis revealed four water molecules that most strongly
couple to the molecule. The geometry of the thymine and the 4 water molecules
were then optimized using DFT using both PBE0/6-311+G(d,p) and
CAM-B3LYP/6-311+G(d,p). The overall dielectric environment was captured
through a self-consistent reaction field (SCRF) Tomasi  _et al._ (2005) as the
polarizable continuum model. The XYZ coordinates of the optimized geometries
are shown in Table S4 and S5.

Using these structures, the resonance Raman spectra for thymine was computed
using the vertical gradientMacak  _et al._ (2000) approach. Ground state
potential energy surfaces were determined through DFT while electronic
transitions and excited-state gradients through time-dependent DFT.Perdew  _et
al._ (1996); Grimme  _et al._ (2010); Rappoport and Furche (2010) The solvent
was captured through the SCRF and the four explicit water molecules. The
spectra was computed using both PBE0/6-311+G(d,p) and CAM-B3LYP/6-311+G(d,p).
All computations were done using the Gaussian 16 simulation package.Frisch
_et al._ (2016) The computed frequencies and Huang-Rhys factors for the 75
normal modes in the complex are included in Table S6. Figure S3 compares the
computed RR spectra with the experimental results in Refs. Yarasi  _et al._
(2007); Ng  _et al._ (2008)

### Integration of the Decoherence Function.

To accurately compute the integral in Eq. (2) we employed the trapezoidal rule
with a frequency step of 0.6 cm-1. Recognizing the integrand as even, we
extended the integration to both negative and positive frequencies. This
approach effectively limited the error in solvent reorganization energy to
below 0.1$\%$ and in total vibrational reorganization energy to under
0.001$\%$.

###### Acknowledgements.

This work was supported by a Pump Primer II research award of the University
of Rochester. I.F. is supported by the National Science Foundation under Grant
No. CHE-2102386.

## Supplementary Information

##  I Extracting the Spectral Density from Resonance Raman Spectra

For completeness, below we include standard expressions for the main cross
sections that result from the theory of absorption, fluorescence and Raman
scattering. We also sketch how the quantities involved are extracted in
practice from experiments. This procedure is needed to extract the spectral
density from resonance Raman experiments.

The cross-sections as a function of frequency $\omega$ for absorption,
fluorescence and Raman scattering are given by:Page and Tonks (1981); Shreve
and Mathies (1995); Tannor and Heller (1982); Myers  _et al._ (1982)

$\displaystyle\sigma_{A}(\omega)$ $\displaystyle=$ $\displaystyle\frac{4\pi
e^{2}\omega}{3c\eta\hbar}\int
d\omega_{eg}G(\omega_{eg})\text{Im}\left\\{i\int_{0}^{\infty}dt\>e^{it(\omega-\omega_{eg})-g(t)}A(t)\right\\};$
(4) $\displaystyle\sigma_{F}(\omega)$ $\displaystyle=$
$\displaystyle\frac{4\pi\eta e^{2}\omega}{3c\hbar}\int
d\omega_{eg}G(\omega_{eg})\text{Im}\left\\{i\int_{0}^{\infty}dt\>e^{it(\omega-\omega_{eg})-g^{*}(t)}A^{*}(t)\right\\};$
(5) $\displaystyle\sigma_{R}(\omega,\omega_{S})$ $\displaystyle=$
$\displaystyle\frac{8e^{4}\omega^{3}_{S}\omega}{9c^{4}\hbar^{2}}\int
d\omega_{eg}G(\omega_{eg})\times$
$\displaystyle\text{Im}\left\\{i\int_{0}^{\infty}d\tau
e^{i\tau(\omega-\omega_{eg})}d(\tau)\int_{0}^{\infty}dt_{1}e^{it_{1}(\omega-\omega_{eg})-g(t_{1})}\int_{0}^{\infty}dt_{2}e^{-it_{2}(\omega-\omega_{eg})-g^{*}(t_{2})}F(t_{1},t_{2},\tau)\right\\}.$

Here, $\omega_{eg}$ is the 0-0 transition energy between the ground and
excited electronic states, $\eta$ the refractive index, and $\omega_{S}$ the
frequency of the scattered radiation during the Raman process. The shape of
the absorption, emission, and Raman spectral features are determined by the
solvent-induced electronic line broadening function,

$g(t)=\frac{1}{\hbar}\int_{0}^{\infty}J_{0}(\omega)\left[\coth{\left(\frac{\beta\hbar\omega}{2}\right)}\frac{1-\cos{(\omega
t)}}{\omega^{2}}+i\frac{\sin{(\omega t)}-\omega t}{\omega^{2}}\right]d\omega.$
(7)

Here $\beta=\frac{1}{k_{B}T}$ is the inverse temperature and $J_{0}(\omega)$
is the solvent spectral density. The quantity $J_{0}(\omega)$ is well-
described by a Drude-Lorentz functional form (see main text) which, in turn,
is fully determined by the solvent correlation time $\frac{1}{\Lambda}$ and
reorganization energy $\lambda_{0}$. The Raman cross section has an additional
vibrational lineshape $d(\tau)$ which is usually described through a
normalized Lorentzian.

The effect of intramolecular electron-nuclear interactions on the cross-
sections is captured by the absorption $A(t)$ and resonance Raman
$F(t_{1},t_{2},\tau)$, time correlators:

$\displaystyle A(t)$ $\displaystyle=\sum_{i}P_{i}\sum_{\rho}\left\langle
i\right\rvert
e^{iH_{g}t/\hbar}\mu_{\rho}^{*}e^{-iH_{e}t/\hbar}\mu_{\rho}\left\lvert
i\right\rangle$ (8a) $\displaystyle F(t_{1},t_{2},\tau)$
$\displaystyle=\sum_{i}P_{i}\sum_{\rho\xi}\left\langle i\right\rvert
e^{-iH_{g}t_{2}/\hbar}\mu_{\xi}^{*}e^{iH_{e}t_{2}/\hbar}\mu_{\rho}e^{-iH_{g}\tau/\hbar}\mu_{\rho}^{*}e^{-iH_{e}t_{1}/\hbar}\mu_{\xi}e^{iH_{g}t_{1}/\hbar}e^{iH_{g}\tau/\hbar}\left\lvert
i\right\rangle.$ (8b)

Here $\mu$ is the electronic transition dipole moment along the $\rho$ or
$\xi$ cartesian axis, $P_{i}$ is the Boltzmann thermal occupation of
vibrational state $|i\rangle$, and $H_{g}$ and $H_{e}$ are the ground and
excited state vibrational Hamiltonians. In turn, the inhomogeneous broadening
function, $G(\omega_{eg})$, is expressed as a normalized distribution of 0-0
energies around an average energy $\bar{\omega}_{eg}$

$G(\omega_{eg})=\frac{1}{\theta\sqrt{2\pi}}e^{-\frac{(\bar{\omega}_{eg}-\omega_{eg})^{2}}{2\theta^{2}}}$
(9)

where $\theta$ is the standard deviation of the distribution.

The complexity of evaluating these cross sections is that they require quantum
dynamical propagation of vibrational wavepackets along multidimensional ground
and electronic potential energy surfaces. To make further progress, it is
useful to specialize considerations to the displaced harmonic oscillator model
where the ground and excited state potential energy surfaces are described by
$N$-dimensional harmonic surfaces with identical nuclear frequencies
$\\{\omega_{\alpha}\\}$ and masses $\\{m_{\alpha}\\}$ that are displaced in
conformational space, with displacement $d_{\alpha}$ along normal mode
$\alpha$. As shown by Shreve (Eqs. 15 and 17 in Ref. Shreve and Mathies
(1995)), for this model it is possible to analytically solve the quantum
dynamics in Eq. 8 and provide formulas for the cross sections that depend on
the normal mode frequencies $\omega_{\alpha}$, the Huang-Rhys factor
$S_{\alpha}=\frac{1}{2\hbar}m_{\alpha}\omega_{\alpha}d_{\alpha}^{2}$ or
reorganization energy $\lambda_{\alpha}=\hbar S_{\alpha}\omega_{\alpha}$, and
the remaining spectroscopic parameters that define the cross sections. The
total reorganization energy of the model is given by the sum of intramolecular
and solvent reorganization energy
$\lambda=\lambda_{0}+\sum_{\alpha}\lambda_{\alpha}$.

The absorption, emission, and resonance Raman excitation profiles can be
effectively modeled using equations Eqs. 4-I. The cross section units are
$\mathring{\text{A}}^{2}$/molecule, enabling direct comparison with
experiment. Adjusting the model’s parameters, {$\mu$, $\omega_{eg}$, $\theta$,
$\omega_{\alpha}$, $S_{\alpha}$, $\lambda_{0}$, $\Lambda$}, allows the
experimental profiles to be accurately fitted. Specifically, the temperature
and refractive index are fixed by the experimental conditions. Initial values
for the transition dipoles can be estimated from results for similar molecules
and the 0-0 transition frequency, $\omega_{eg}$, can be approximated by the
crossing point between absorption and fluorescence spectra. The solvent
parameters, $\lambda_{0}$ and $\frac{1}{\Lambda}$, are related by the ratio

$\kappa=\sqrt{\frac{\hbar\Lambda^{2}}{2\lambda_{0}k_{B}T}},$ (10)

with $\kappa$ found to be 0.1 for a polar solute in polar solventsMukamel
(1995); Li  _et al._ (1994); Yan and Mukamel (1987). Solvent parameters and
the full width at half maximum of the optical absorption lineshape, $\Gamma$,
are closely linked byMukamel (1995)

$\Gamma=\sqrt{\frac{2\lambda_{0}k_{B}T}{\hbar}}\frac{2.355+1.76\kappa}{1+0.85\kappa+0.88\kappa^{2}}.$
(11)

The initial $\Gamma$ value can be estimated from the behavior of similar
molecules. The vibrational frequencies, $\\{\omega_{\alpha}\\}$, are obtained
from the experimental resonance Raman spectra and the Huang-Rhys factors are
initially estimated under the approximation
$S_{\alpha}=\frac{I_{\alpha}}{2\omega_{\alpha}}$ where $I_{\alpha}$ is the
Raman integrated peak intensity.

In the fitting procedure, the parameters $\Gamma$, $\mu$, and $\omega_{eg}$
are optimized first as they have a large and uniform effect on all three
lineshapes. Then, the set $\\{S_{\alpha}\\}$ is adjusted to obtain the best
fit to the absorption, emission, and resonance Raman excitation profiles,
while accurately reproducing the experimental Stokes shift. The inhomogeneous
broadening parameter is neglected in the fitting procedure and is only
introduced when a better fit cannot be achieved. The parameters obtained from
the fitting procedure can be utilized to reconstruct the spectral density,
employing Eq. 1 in the main text.

##  II Effect of vibrational broadening on decoherence

Figure S1 shows the influence of changing the vibrational broadening $\gamma$
on the electronic coherence loss. As shown, for the physically accessible
range, $\gamma$ has a negligible influence on electronic decoherence.

Figure S1: Effect of the vibrational broadening factor $\gamma$ on electronic
decoherence. Coherence loss due to (a) solvent + vibrations and (b) vibrations
only for varying $\gamma$. As shown, the vibrational broadening in physically
meaningful ranges has a negligible effect on electronic coherence loss, as its
influence only becomes important at longer time scales.

##  III Extracted spectral densities for thymine and its derivatives

Tables 3-1 summarize the intramolecular and solvent parameters needed for
reconstructing the spectral density for all three molecules in Fig. 2. Figure
S2 details the spectral density for thymine and its nucleoside.

Table 1: Spectral density parameters for thymine nucleotide. Solvent parameters: $\lambda_{0}=715.7\text{ cm}^{-1}$, $\Lambda=54.5\text{ cm}^{-1}$. Data obtained from Ref. Billinghurst  _et al._ (2012) Mode Frequency [cm-1] | Huang-Rhys factor $S$ | Reorganization energy $\lambda$ [cm-1]
---|---|---
442 | 0.034 | 14.9
665 | 0.048 | 31.9
784 | 0.034 | 26.5
1193 | 0.065 | 77.3
1243 | 0.13 | 161.6
1376 | 0.135 | 186
1416 | 0.018 | 25.6
1663 | 0.198 | 330
Table 2: Spectral density parameters for thymine nucleoside. Solvent parameters: $\lambda_{0}=596.2\text{ cm}^{-1}$, $\Lambda=49.7\text{ cm}^{-1}$. Data obtained from Ref. Billinghurst  _et al._ (2012) Mode Frequency [cm-1] | Huang-Rhys factor $S$ | Reorganization energy $\lambda$ [cm-1]
---|---|---
439 | 0.036 | 16
780 | 0.051 | 39.9
1189 | 0.065 | 77
1240 | 0.162 | 201.4
1374 | 0.168 | 231.1
1413 | 0.029 | 40.7
1645 | 0.245 | 405.2
Table 3: Spectral density parameters for thymine. Solvent parameters: $\lambda_{0}=1302.9\text{ cm}^{-1}$, $\Lambda=73.5\text{ cm}^{-1}$. Data obtained from Ref. Yarasi  _et al._ (2007) Mode Frequency [cm-1] | Huang-Rhys factor $S$ | Reorganization energy $\lambda$ [cm-1]
---|---|---
567 | 0.028 | 16.3
641 | 0.045 | 28.8
762 | 0.045 | 34.3
811 | 0.039 | 31.8
1173 | 0.029 | 33.8
1237 | 0.097 | 119.7
1362 | 0.125 | 170.2
1423 | 0.024 | 34.4
1667 | 0.151 | 252.1
Figure S2: Reconstructed Spectral Densities for (a) thymine and (b) its
nucleoside in water at 298 K. Labels 1-9 correspond to the normal modes in
Fig. S5.

##  IV Computed resonance Raman spectra for thymine

The resonance Raman spectra for thymine was computed as described in the
Materials and Methods section. The XYZ coordinates of the optimized geometries
of the complexes used in the computations are shown in Table 4 and Table 5.
The computed frequencies and Huang-Rhys factors for the 75 normal modes in the
complex are included in Table 6. Figure S4 compares the computed RR spectra
with the experimental results in Refs. Yarasi  _et al._ (2007); Ng  _et al._
(2008) The excellent agreement between theory and experiment enables us to
assign specific vibrational modes to the peaks in the spectral density and
interpret the electronic decoherence pathways. Animated Figure S5 details
representative normal modes 1-9 that most strongly couples to the electronic
transition.

Figure S3: Comparison between calculated and experimental RR spectra for thymine.Billinghurst  _et al._ (2012) The Raman excitation wavelength is 266 nm, and the functional frequencies have been scaled according to literature scaling factors for CAM-B3LYP (0.9613) and PBE0 (0.9512).Merrick  _et al._ (2007) The quantitative agreements allows us to make vibrational assignments. The asterisk marks the experimental internal standard. Labels 1-9 correspond to the normal modes in Fig. S5. The experimental RR spectra was adapted with permission from Billinghurst  _et al._ (2012). Copyright 2012 American Chemical Society  Figure S4: Comparison between calculated and experimental RR spectra for thymine.Billinghurst  _et al._ (2012) The Raman excitation wavelength is 266 nm, and the functional frequencies have been scaled according to literature scaling factors for CAM-B3LYP (0.9613) and PBE0 (0.9512).Ng  _et al._ (2008); Merrick  _et al._ (2007) The quantitative agreements allows us to make vibrational assignments. The asterisk marks the experimental internal standard. Labels 1-9 correspond to the normal modes in Fig. S5. Table 4: Cartesian coordinates of thymine-water complex with ground-state energy optimized using PBE0/6-311+G(d,p) level of theory with SCRF. Atom | X | Y | Z
---|---|---|---
C | -2.298864 | -0.360076 | -0.319497
N | -1.003836 | -0.862288 | -0.354389
C | 0.108847 | -0.267634 | 0.138846
N | -0.093939 | 0.937181 | 0.716834
C | -2.439529 | 0.936629 | 0.319065
C | -1.330471 | 1.521347 | 0.805037
C | -3.794800 | 1.548191 | 0.399962
H | -4.477269 | 0.902513 | 0.953392
H | -4.216033 | 1.685889 | -0.596278
H | -3.752455 | 2.515516 | 0.896321
H | -0.849747 | -1.768425 | -0.800394
H | 0.712647 | 1.402454 | 1.099369
H | -1.345439 | 2.484616 | 1.293254
O | 1.225135 | -0.793560 | 0.063601
O | -3.211347 | -1.001981 | -0.804257
O | 0.378590 | -3.081638 | -1.434347
H | 0.438850 | -3.989602 | -1.141002
H | 1.072148 | -2.608714 | -0.964139
O | 3.494884 | 0.580578 | -0.829242
H | 4.255647 | 0.229532 | -0.369024
H | 2.738422 | 0.076340 | -0.498377
O | 3.102473 | 3.325965 | -0.766230
H | 3.069676 | 3.600623 | -1.680091
H | 3.253787 | 2.368987 | -0.800202
O | 2.627371 | -1.943346 | 2.277198
H | 2.126894 | -1.572948 | 1.542294
H | 2.249337 | -2.809914 | 2.413494
Table 5: Cartesian coordinates of thymine-water complex with ground-state energy optimized using CAM-B3LYP/6-311+G(d,p) level of theory with SCRF. Atom | X | Y | Z
---|---|---|---
C | -2.326166 | -0.341637 | -0.374253
N | -1.026666 | -0.849412 | -0.386018
C | 0.066204 | -0.281992 | 0.188660
N | -0.161152 | 0.889441 | 0.828755
C | -2.492730 | 0.921563 | 0.333224
C | -1.407144 | 1.473656 | 0.897765
C | -3.856109 | 1.532827 | 0.391440
H | -4.563462 | 0.857952 | 0.877734
H | -4.235904 | 1.730011 | -0.613122
H | -3.835763 | 2.471166 | 0.944968
H | -0.857326 | -1.733726 | -0.875809
H | 0.632447 | 1.333719 | 1.269400
H | -1.440905 | 2.408178 | 1.440566
O | 1.190447 | -0.805181 | 0.127573
O | -3.219883 | -0.953566 | -0.932662
O | 0.446328 | -3.009241 | -1.502304
H | 0.527831 | -3.940438 | -1.273746
H | 1.092445 | -2.534325 | -0.957143
O | 3.497294 | 0.521983 | -0.764408
H | 4.253624 | 0.153379 | -0.297437
H | 2.715598 | 0.046707 | -0.429374
O | 3.294370 | 3.280833 | -0.924304
H | 3.349177 | 3.501582 | -1.858136
H | 3.374030 | 2.308120 | -0.881284
O | 2.647056 | -1.837874 | 2.357708
H | 2.120987 | -1.512185 | 1.610298
H | 2.432709 | -2.772485 | 2.430093
Table 6: Thymine normal modes frequencies and Huang-Rhys factors from DFT. The Huang-Rhys factors have been normalized. The labels 1-9 in bold correspond to the normal modes in Fig. S5. Mode Frequency PBEO [cm-1] | Huang-Rhys factor S | Mode Frequency CAM-B3LYP [cm-1] | Huang-Rhys factor S
---|---|---|---
11.33 | 0.009 | 14.69 | 0.070
21.45 | 0.033 | 17.54 | 0.022
27.94 | 0.019 | 27.98 | 0.000
29.68 | 0.005 | 32.16 | 0.029
37.49 | 0.125 | 46.62 | 0.000
40.91 | 0.010 | 57.97 | 0.002
50.65 | 0.002 | 66.80 | 0.001
57.32 | 0.001 | 70.14 | 0.000
71.66 | 0.002 | 93.04 | 0.011
81.86 | 0.048 | 115.81 | 0.000
110.83 | 0.001 | 150.61 | 0.269
142.85 | 0.250 | 161.80 | 0.001
151.09 | 0.012 | 166.14 | 0.001
162.96 | 0.085 | 175.67 | 0.176
164.29 | 0.274 | 196.45 | 0.003
192.00 | 0.003 | 206.37 | 0.000
206.53 | 0.008 | 222.49 | 0.035
224.06 | 0.082 | 251.54 | 0.008
233.29 | 0.001 | 294.13 | 0.045
284.48 | 0.034 | 304.16 | 0.000
291.67 | 0.002 | 315.72 | 0.005
301.15 | 0.002 | 336.17 | 0.003
316.45 | 0.003 | 388.64 | 0.001
385.21 | 0.010 | 419.98 | 0.266
401.08 (1) | 0.237 | 420.80 | 0.007
414.55 | 0.000 | 442.93 | 0.001
419.05 | 0.006 | 499.92 | 0.192
482.19 | 0.099 | 556.47 | 0.000
512.30 | 0.001 | 558.91 | 0.009
527.44 | 0.000 | 592.24 | 0.194
572.02 (2) | 0.167 | 611.15 | 0.022
596.37 | 0.012 | 647.23 | 0.000
616.10 | 0.000 | 649.01 | 0.286
626.73 (3) | 0.288 | 773.30 | 0.006
747.26 | 0.013 | 789.86 | 0.167
761.55 | 0.000 | 807.95 | 0.000
770.65 (4) | 0.460 | 815.04 | 0.000
782.26 | 0.000 | 857.53 | 0.223
833.47 (5) | 0.284 | 877.88 | 0.000
848.87 | 0.000 | 981.67 | 0.000
943.41 | 0.000 | 1051.36 | 0.188
1018.46 | 0.233 | 1096.85 | 0.004
1065.56 | 0.002 | 1122.24 | 0.000
1078.05 | 0.000 | 1228.23 | 0.228
1202.78 | 0.106 | 1279.11 | 0.000
1256.67 | 0.005 | 1321.97 | 0.484
1299.59 (6) | 0.320 | 1459.14 | 0.960
1419.79 (7) | 0.974 | 1486.39 | 0.001
1435.58 | 0.012 | 1519.69 | 0.225
1480.09 | 0.000 | 1528.88 | 0.000
1481.75 (8) | 0.235 | 1544.22 | 0.001
1500.46 | 0.001 | 1553.01 | 0.000
1507.75 | 0.003 | 1590.18 | 0.000
1559.88 | 0.002 | 1654.80 | 0.005
1629.17 | 0.000 | 1675.82 | 0.000
1650.36 | 0.000 | 1681.49 | 0.000
1657.42 | 0.003 | 1708.83 | 0.002
1678.83 | 0.000 | 1783.09 | 0.183
1750.56 (9a) | 1.000 | 1804.90 | 1.000
1752.42 | 0.046 | 1815.27 | 0.377
1775.69 (9b) | 0.346 | 3183.84 | 0.002
Mode Frequency PBEO [cm-1] | Huang-Rhys factor S | Mode Frequency CAM-B3LYP [cm-1] | Huang-Rhys factor S
---|---|---|---
3099.66 | 0.005 | 3243.54 | 0.000
3166.16 | 0.000 | 3273.18 | 0.000
3195.53 | 0.000 | 3372.26 | 0.009
3282.88 | 0.007 | 3524.17 | 0.004
3432.97 | 0.000 | 3703.70 | 0.000
3661.04 | 0.009 | 3779.26 | 0.000
3700.04 | 0.006 | 3783.55 | 0.002
3726.16 | 0.000 | 3853.91 | 0.000
3802.59 | 0.004 | 3884.43 | 0.000
3840.61 | 0.008 | 4047.99 | 0.000
3982.62 | 0.002 | 4055.70 | 0.000
3991.37 | 0.001 | 4057.12 | 0.000
3992.55 | 0.003 | 4057.27 | 0.000
3993.70 | 0.005 |  |

[loop,autoplay,width=]10Gif/something-023

Figure S5: Frontier molecular orbitals for thymine and vibrational normal
modes that couple most strongly with the electronic transition. HOMO/LUMO
stand for highest occupied/lowest unoccupied molecular orbital. The labels 1-9
for the normal modes correspond to peaks 1-9 in the spectral density (Fig. S2)
and the resonance Raman spectra (Fig. S4). The normal modes frequencies and
Huang-Rhys factors are mark in bold in Table 6 . The color convention is as
follows: carbon (black), hydrogen (white), nitrogen (cyan) and oxygen (red).
This is an animated figure that shows the normal mode motions and is best seen
in Acrobat or Okular.

## References

  * Wasielewski  _et al._ (2020) M. R. Wasielewski, M. D. Forbes, N. L. Frank, K. Kowalski, G. D. Scholes, J. Yuen-Zhou, M. A. Baldo, D. E. Freedman, R. H. Goldsmith, T. Goodson III, _et al._ , Nat. Rev. Chem. 4, 490 (2020).
  * Atzori and Sessoli (2019) M. Atzori and R. Sessoli, J. Am. Chem. Soc. 141, 11339 (2019).
  * Zhu  _et al._ (2022) G.-Z. Zhu, D. Mitra, B. L. Augenbraun, C. E. Dickerson, M. J. Frim, G. Lao, Z. D. Lasner, A. N. Alexandrova, W. C. Campbell, J. R. Caram, _et al._ , Nat. Chem. 14, 995 (2022).
  * Bayliss  _et al._ (2022) S. L. Bayliss, P. Deb, D. W. Laorenza, M. Onizhuk, G. Galli, D. E. Freedman, and D. D. Awschalom, Phys. Rev. X 12, 031028 (2022).
  * Hu  _et al._ (2022) W. Hu, I. Gustin, T. D. Krauss, and I. Franco, J. Phys. Chem. Lett. 13, 11503 (2022).
  * Viola  _et al._ (1999) L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).
  * Scholes  _et al._ (2017) G. D. Scholes, G. R. Fleming, L. X. Chen, A. Aspuru-Guzik, A. Buchleitner, D. F. Coker, G. S. Engel, R. Van Grondelle, A. Ishizaki, D. M. Jonas, _et al._ , Nature 543, 647 (2017).
  * Shapiro and Brumer (2012) M. Shapiro and P. Brumer,  _Quantum control of molecular processes_ (John Wiley & Sons, 2012).
  * Rice  _et al._ (2000) S. A. Rice, M. Zhao, _et al._ ,  _Optical control of molecular dynamics_ (John Wiley, 2000).
  * Wang  _et al._ (2019) L. Wang, M. A. Allodi, and G. S. Engel, Nat. Rev. Chem. 3, 477 (2019).
  * Mukamel (1995) S. Mukamel,  _Principles of Nonlinear Optical Spectroscopy_ , 6 (Oxford University Press, USA, 1995).
  * Biswas  _et al._ (2022) S. Biswas, J. Kim, X. Zhang, and G. D. Scholes, Chem. Rev. 122, 4257 (2022).
  * Dani and Makri (2022) R. Dani and N. Makri, Journal of Physical Chemistry B 126, 9361 (2022).
  * De Sio and Lienau (2017) A. De Sio and C. Lienau, Phys. Chem. Chem. Phys. 19, 18813 (2017).
  * Nelson  _et al._ (2018) T. R. Nelson, D. Ondarse-Alvarez, N. Oldani, B. Rodriguez-Hernandez, L. Alfonso-Hernandez, J. F. Galindo, V. D. Kleiman, S. Fernandez-Alberti, A. E. Roitberg, and S. Tretiak, Nat. Commun. 9, 2316 (2018).
  * Kreisbeck and Kramer (2012) C. Kreisbeck and T. Kramer, J. Phys. Chem. Lett. 3, 2828 (2012).
  * Popp  _et al._ (2019) W. Popp, M. Polkehn, R. Binder, and I. Burghardt, J. Phys. Chem. Lett. 10, 3326 (2019).
  * Blau  _et al._ (2018) S. M. Blau, D. I. Bennett, C. Kreisbeck, G. D. Scholes, and A. Aspuru-Guzik, Proc. Natl. Acad. Sci. 115, E3342 (2018).
  * Ramos  _et al._ (2019) F. C. Ramos, M. Nottoli, L. Cupellini, and B. Mennucci, Chem. Sci. 10, 9650 (2019).
  * Hahn and Stock (2000) S. Hahn and G. Stock, J. Phys. Chem. B 104, 1146 (2000).
  * Marsili  _et al._ (2019) E. Marsili, M. H. Farag, X. Yang, L. De Vico, and M. Olivucci, J. Phys. Chem. A 123, 1710 (2019).
  * Chuang and Brumer (2021) C. Chuang and P. Brumer, J. Phys. Chem. Lett. 12, 3618 (2021).
  * Hu  _et al._ (2018) W. Hu, B. Gu, and I. Franco, J. Chem. Phys. 148, 134304 (2018).
  * Gu and Franco (2018) B. Gu and I. Franco, J. Phys. Chem. Lett. 9, 773 (2018).
  * Vacher  _et al._ (2017) M. Vacher, M. J. Bearpark, M. A. Robb, and J. P. Malhado, Phys. Rev. Lett. 118, 083001 (2017).
  * Arnold  _et al._ (2017) C. Arnold, O. Vendrell, and R. Santra, Phys. Rev. A 95, 033425 (2017).
  * Matselyukh  _et al._ (2022) D. T. Matselyukh, V. Despré, N. V. Golubev, A. I. Kuleff, and H. J. Wörner, Nature Physics 18, 1206 (2022).
  * Salvador  _et al._ (2003) M. R. Salvador, M. A. Hines, and G. D. Scholes, J. Chem. Phys. 118, 9380 (2003).
  * Colonna  _et al._ (2005) A. E. Colonna, X. Yang, and G. D. Scholes, phys. status solidi B 242, 990 (2005).
  * Yang  _et al._ (2005) X. Yang, T. E. Dykstra, and G. D. Scholes, Phys. Rev. B 71, 045203 (2005).
  * Hwang and Rossky (2004a) H. Hwang and P. J. Rossky, J. Chem. Phys. 120, 11380 (2004a).
  * Hwang and Rossky (2004b) H. Hwang and P. J. Rossky, J. Phys. Chem. B 108, 6723 (2004b).
  * Shu and Truhlar (2023) Y. Shu and D. G. Truhlar, J. Chem. Theory Comput. 19, 380 (2023).
  * Jasper and Truhlar (2005) A. W. Jasper and D. G. Truhlar, J. Chem. Phys. 123, 064103 (2005).
  * Heller (1981) E. J. Heller, Acc. Chem. Res. 14, 368 (1981).
  * Tomasi  _et al._ (2005) J. Tomasi, B. Mennucci, and R. Cammi, Chem. rev. 105, 2999 (2005).
  * Olbrich  _et al._ (2011) C. Olbrich, J. Strümpfer, K. Schulten, and U. Kleinekathöfer, J. Phys. Chem. Lett. 2, 1771 (2011).
  * Dey  _et al._ (2022) D. Dey, A. I. Kuleff, and G. A. Worth, Phys. Rev. Lett. 129, 173203 (2022).
  * Arnold  _et al._ (2018) C. Arnold, O. Vendrell, R. Welsch, and R. Santra, Phys. Rev. Lett. 120, 123001 (2018).
  * Golubev  _et al._ (2020) N. V. Golubev, T. Begušić, and J. c. v. Vaníček, Phys. Rev. Lett. 125, 083001 (2020).
  * Vester  _et al._ (2023) J. Vester, V. Despré, and A. I. Kuleff, J. Chem. Phys. 158 (2023).
  * Tanimura (2020) Y. Tanimura, J. Chem. Phys. 153, 020901 (2020).
  * Ikeda and Scholes (2020) T. Ikeda and G. D. Scholes, J. Chem. Phys. 152, 204101 (2020).
  * Strathearn  _et al._ (2018) A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Nat. Commun. 9, 3322 (2018).
  * Zhang  _et al._ (2020) J. Zhang, R. Borrelli, and Y. Tanimura, J. Chem. Phys. 152, 214114 (2020).
  * Duan  _et al._ (2020) C. Duan, C.-Y. Hsieh, J. Liu, J. Wu, and J. Cao, J. Phys. Chem. Lett. 11, 4080 (2020).
  * Varvelo  _et al._ (2023) L. Varvelo, J. K. Lynd, B. Citty, O. Kühn, and D. I. Raccah, J. Phys. Chem. Lett. 14, 3077 (2023).
  * Leggett  _et al._ (1987) A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).
  * Maity  _et al._ (2021) S. Maity, V. Daskalakis, M. Elstner, and U. Kleinekathöfer, Phys. Chem. Chem. Phys. 23, 7407 (2021).
  * Kim  _et al._ (2018) C. W. Kim, B. Choi, and Y. M. Rhee, Phys. Chem. Chem. Phys. 20, 3310 (2018).
  * Cignoni  _et al._ (2022) E. Cignoni, V. Slama, L. Cupellini, and B. Mennucci, J. Chem. Phys. 156, 120901 (2022).
  * Jang and Mennucci (2018) S. J. Jang and B. Mennucci, Rev. Mod. Phys. 90, 035003 (2018).
  * Lee and Coker (2016) M. K. Lee and D. F. Coker, J. Phys. Chem. Lett. 7, 3171 (2016).
  * Kell  _et al._ (2013) A. Kell, X. Feng, M. Reppert, and R. Jankowiak, J. Phys. Chem. B 117, 7317 (2013).
  * Chen  _et al._ (2023) M. S. Chen, Y. Mao, A. Snider, P. Gupta, A. Montoya-Castillo, T. J. Zuehlsdorff, C. M. Isborn, and T. E. Markland, J. Phys. Chem. Lett. 14, 6610 (2023).
  * Rätsep and Freiberg (2007) M. Rätsep and A. Freiberg, J. Lumin. 127, 251 (2007).
  * Rätsep  _et al._ (2008) M. Rätsep, J. Pieper, K.-D. Irrgang, and A. Freiberg, J. Phys. Chem. B 112, 110 (2008).
  * Pieper  _et al._ (2009) J. Pieper, M. Rätsep, K.-D. Irrgang, and A. Freiberg, J. Phys. Chem. B 113, 10870 (2009).
  * Freiberg  _et al._ (2009) A. Freiberg, M. Rätsep, K. Timpmann, and G. Trinkunas, Chem. Phys. 357, 102 (2009).
  * Gryliuk  _et al._ (2014) G. Gryliuk, M. Rätsep, S. Hildebrandt, K.-D. Irrgang, H.-J. Eckert, and J. Pieper, Biochim. Biophys. Acta (BBA) Bioenerg. 1837, 1490 (2014).
  * Cina (2022) J. Cina,  _Getting Started on Time-Resolved Molecular Spectroscopy_ (OUP Oxford, 2022).
  * Breuer and Petruccione (2002) H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, New York, 2002).
  * Schlosshauer (2007) M. A. Schlosshauer,  _Decoherence: and the quantum-to-classical transition_ (Springer Science & Business Media, 2007).
  * Joos  _et al._ (2013) E. Joos, H. D. Zeh, C. Kiefer, D. J. Giulini, J. Kupsch, and I.-O. Stamatescu,  _Decoherence and the appearance of a classical world in quantum theory_ (Springer Science & Business Media, 2013).
  * Prezhdo and Rossky (1998) O. V. Prezhdo and P. J. Rossky, Phys. Rev. Lett. 81, 5294 (1998).
  * Bittner and Rossky (1995) E. R. Bittner and P. J. Rossky, J. Chem. Phys. 103, 8130 (1995).
  * Fiete and Heller (2003) G. A. Fiete and E. J. Heller, Phys. Rev. A 68, 022112 (2003).
  * Gu and Franco (2017) B. Gu and I. Franco, J. Phys. Chem. Lett. 8, 4289 (2017).
  * Dietze and Mathies (2016) D. R. Dietze and R. A. Mathies, ChemPhysChem 17, 1224 (2016).
  * Piontkowski  _et al._ (2019) Z. Piontkowski, D. J. Mark, M. A. Bedics, R. P. Sabatini, M. F. Mark, M. R. Detty, and D. W. McCamant, J. Phys. Chem. A 123, 8807 (2019).
  * Fang and Tang (2020) C. Fang and L. Tang, Annu. Rev. Phys. Chem. 71, 239 (2020).
  * Pieper  _et al._ (2011) J. Pieper, M. Rätsep, I. Trostmann, F.-J. Schmitt, C. Theiss, H. Paulsen, H. Eichler, A. Freiberg, and G. Renger, J. Phys. Chem. B 115, 4053 (2011).
  * McCamant  _et al._ (2004) D. W. McCamant, P. Kukura, S. Yoon, and R. A. Mathies, Rev. Sci. Instrum. 75, 4971 (2004).
  * McCamant  _et al._ (2003) D. W. McCamant, P. Kukura, and R. A. Mathies, J. Phys. Chem. A 107, 8208 (2003).
  * Kukura  _et al._ (2007) P. Kukura, D. W. McCamant, and R. A. Mathies, Annu. Rev. Phys. Chem. 58, 461 (2007).
  * Lee  _et al._ (2004) S.-Y. Lee, D. Zhang, D. W. McCamant, P. Kukura, and R. A. Mathies, J. Chem. Phys. 121, 3632 (2004).
  * McHale (2002) J. L. McHale,  _Resonance Raman Spectroscopy. In Handbook of Vibrational Spectroscopy; Chalmers, J. M.; Griffiths, P. R._ (John Wiley & Sons Ltd., 2002).
  * Myers  _et al._ (1982) A. B. Myers, R. A. Mathies, D. J. Tannor, and E. Heller, J. Chem. Phys. 77, 3857 (1982).
  * Shreve and Mathies (1995) A. P. Shreve and R. A. Mathies, J. Phys. Chem. 99, 7285 (1995).
  * Tannor and Heller (1982) D. J. Tannor and E. J. Heller, J. Chem. Phys. 77, 202 (1982).
  * Page and Tonks (1981) J. B. Page and D. L. Tonks, J. Chem. Phys. 75, 5694 (1981).
  * Li  _et al._ (1994) B. Li, A. E. Johnson, S. Mukamel, and A. B. Myers, J. Am. Chem. Soc. 116, 11039 (1994).
  * Bennett  _et al._ (2013) D. I. G. Bennett, K. Amarnath, and G. R. Fleming, J. Am. Chem. Soc. 135, 9164 (2013).
  * Novoderezhkin  _et al._ (2004) V. I. Novoderezhkin, M. A. Palacios, H. van Amerongen, and R. van Grondelle, J. Phys. Chem. B 108, 10363 (2004).
  * Von der Linde  _et al._ (1997) D. Von der Linde, K. Sokolowski-Tinten, and J. Bialkowski, Appl. Surf. Sci. 109, 1 (1997).
  * Brinks  _et al._ (2014) D. Brinks, R. Hildner, E. M. Van Dijk, F. D. Stefani, J. B. Nieder, J. Hernando, and N. F. Van Hulst, Chem. Soc.Rev. 43, 2476 (2014).
  * Zuehlsdorff  _et al._ (2020) T. J. Zuehlsdorff, H. Hong, L. Shi, and C. M. Isborn, J. Chem. Phys. 153 (2020).
  * Unruh (1995) W. G. Unruh, Phys. Rev. A 51, 992 (1995).
  * Palma  _et al._ (1996) G. M. Palma, K.-A. Suominen, and A. Ekert, Proc. R. Soc. Lond. A452, 567 (1996).
  * Yarasi  _et al._ (2007) S. Yarasi, P. Brost, and G. R. Loppnow, J. Phys. Chem. A 111, 5130 (2007).
  * Ng  _et al._ (2008) S. Ng, S. Yarasi, P. Brost, and G. R. Loppnow, J. Phys. Chem. A 112, 10436 (2008).
  * Billinghurst  _et al._ (2012) B. E. Billinghurst, S. A. Oladepo, and G. R. Loppnow, J. Phys. Chem. B 116, 10496 (2012).
  * Onidas  _et al._ (2002) D. Onidas, D. Markovitsi, S. Marguet, A. Sharonov, and T. Gustavsson, J. Phys. Chem. B 106, 11367 (2002).
  * Gustavsson  _et al._ (2006) T. Gustavsson, A. Bányász, E. Lazzarotto, D. Markovitsi, G. Scalmani, M. J. Frisch, V. Barone, and R. Improta, J. Am. Chem. Soc. 128, 607 (2006).
  * Perdew  _et al._ (1996) J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996).
  * Grimme  _et al._ (2010) S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys. 132, 154104 (2010).
  * Rappoport and Furche (2010) D. Rappoport and F. Furche, J. Chem. Phys. 133, 134105 (2010).
  * Macak  _et al._ (2000) P. Macak, Y. Luo, and H. Ågren, Chem. Phys. Lett. 330, 447 (2000).
  * Manthe  _et al._ (1992) U. Manthe, H. D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 97, 3199 (1992).
  * Beck  _et al._ (2000) M. H. Beck, A. Jäckle, G. A. Worth, and H.-D. Meyer, Phys. Rep. 324, 1 (2000).
  * Beyere  _et al._ (2004) L. Beyere, P. Arboleda, V. Monga, and G. Loppnow, Can. J. Chem. 82, 1092 (2004).
  * Hruska  _et al._ (2022) E. Hruska, A. Gale, X. Huang, and F. Liu, J. Chem. Phys. 156, 124801 (2022).
  * Dapprich  _et al._ (1999) S. Dapprich, I. Komáromi, K. S. Byun, K. Morokuma, and M. J. Frisch, J Mol Struct Theochem 461, 1 (1999).
  * Chung  _et al._ (2015) L. W. Chung, W. Sameera, R. Ramozzi, A. J. Page, M. Hatanaka, G. P. Petrova, T. V. Harris, X. Li, Z. Ke, F. Liu, _et al._ , Chem. Rev. 115, 5678 (2015).
  * Frisch  _et al._ (2016) M. Frisch, G. Trucks, H. Schlegel, G. Scuseria, M. Robb, J. Cheeseman, G. Scalmani, V. Barone, G. Petersson, H. Nakatsuji, _et al._ , Inc., Wallingford CT 3 (2016).
  * Yan and Mukamel (1987) Y. J. Yan and S. Mukamel, J. Chem. Phys. 86, 6085 (1987).
  * Merrick  _et al._ (2007) J. P. Merrick, D. Moran, and L. Radom, J. Phys. Chem. A 111, 11683 (2007).