![Structural features of a BChl and b Chl pigments as present in bacteria and plants. The methyl group (R = CH3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{3}$$\end{document}) of the Chl-a molecule is replaced by an aldehyde (R = CHO) in the Chl-b molecule. The regions highlighted in green in both panels indicate the structural differences between the BChl and Chl molecules](https://www.researchgate.net/publication/364243944/figure/fig2/AS:11431281136729240@1680574148431/Structural-features-of-a-BChl-and-b-Chl-pigments-as-present-in-bacteria-and-plants-The_Q320.jpg)
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Vol.:(0123456789)
1 3
Photosynthesis Research (2023) 156:147–162
https://doi.org/10.1007/s11120-022-00969-w
REVIEW
Recent progress inatomistic modeling oflight‑harvesting complexes:
amini review
SayanMaity1 · UlrichKleinekathöfer1
Received: 30 June 2022 / Accepted: 22 September 2022 / Published online: 7 October 2022
© The Author(s) 2022
Abstract
In this mini review, we focus on recent advances in the atomistic modeling of biological light-harvesting (LH) complexes.
Because of their size and sophisticated electronic structures, multiscale methods are required to investigate the dynamical
and spectroscopic properties of such complexes. The excitation energies, in this context also known as site energies, excitonic
couplings, and spectral densities are key quantities which usually need to be extracted to be able to determine the exciton
dynamics and spectroscopic properties. The recently developed multiscale approach based on the numerically efficient den-
sity functional tight-binding framework followed by excited state calculations has been shown to be superior to the scheme
based on pure classical molecular dynamics simulations. The enhanced approach, which improves the description of the
internal vibrational dynamics of the pigment molecules, yields spectral densities in good agreement with the experimental
counterparts for various bacterial and plant LH systems. Here, we provide a brief overview of those results and described
the theoretical foundation of the multiscale protocol.
Keywords Multiscale modeling· Light-harvesting complexes· QM/MM simulations· Excited state calculations· Spectral
densities· Exciton dynamics
Introduction
During the photosynthesis process, light-harvesting (LH)
protein–pigment complexes of plants, bacteria, and algae
play a key role in the conversion of solar energy into sus-
tainable forms of chemical energy. Chlorophyll (Chl), bac-
terio-chlorophyll (BChl), and bilin molecules are the major
pigments present in those complexes that absorb sunlight.
The excitation energy is subsequently transferred within the
pigment network via excitation energy transfer processes
(Cogdell etal. 2006; Blankenship etal. 2011; Blankenship
2014). The target of these LH complexes is to transport the
solar energy in the form of excitons to reaction centers where
the electron–hole charge separation takes place for further
processing in photosynthesis.
Photosynthesis can be categorized as oxygenic and
anoxygenic depending on its ability to produce oxygen or
not. Oxygenic photosynthesis is a process in which water
molecules are oxidized into molecular oxygen and is mainly
performed by plants, marine algae such as diatoms, and
cyanobacteria. Green sulfur and purple bacteria are known
to conduct anoxygenic photosynthesis where a terminal
reductant like hydrogen sulfide (H
2
S) is split into a byprod-
uct like sulfur. A general equation of biological photosyn-
thesis (Blankenship 2014) can be given by
where H
2
X is a reducing agent such as
H2O
or
H2S
used to
produce carbohydrates
(CH2O)n
. In both types of photosyn-
thesis, the solar energy is saved as chemical energy in the
form of adenosine triphosphate (ATP) which is then utilized
together with the reduced nicotinamide adenine dinucleotide
phosphate (NADPH) in the Calvin cycle for the production
of carbohydrates (Blankenship 2014). In this process also
carbon fixation takes place, i.e., molecular CO
2
is converted
into carbohydrates.
During the energy transfer processes in LH complexes,
the excitons are being spread over several pigments and a
transfer of the absorbed solar energy in the form of exci-
tons through the pigment network toward a reaction center
(1)
nCO2+2nH2X+hv
→
(CH2O)n+2nX+nH2O,
* Ulrich Kleinekathöfer
u.kleinekathoefer@jacobs-university.de
1 Department ofPhysics andEarth Sciences, Jacobs
University Bremen, Campus Ring 1, 28759Bremen,
Germany
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148 Photosynthesis Research (2023) 156:147–162
1 3
is necessary for further processing of the energy. In the last
decade, an enormous interest in some LH complexes of
bacteria has been triggered especially by experimental find-
ings of long-lived quantum coherence at low temperature in
the Fenna–Mathew–Olson (FMO) complex of green sulfur
bacteria (Engel etal. 2007; Panitchayangkoon etal. 2010).
Later on, a similar kind of quantum coherences was also
reported for conjugated polymers (Collini and Scholes 2009)
and marine algae (Collini etal. 2010) at room temperature.
It was initially proposed that those long-lived coherences
were purely electronic in nature and that they were caused
by correlated fluctuations of the BChl excitation energies
(Lee etal. 2007; Wolynes 2009). Theoretical calculations
based on classical MD simulations did, however, not find
any such correlated fluctuations of the site energies belong-
ing to neighboring pigment molecules (Olbrich etal. 2011a;
Shim etal. 2012). Furthermore, recent experiments based on
two-dimensional electronic spectroscopy raise some ques-
tions concerning the long-lived coherences for LH model
systems (Duan etal. 2017a; Thyrhaug etal. 2018; Cao etal.
2020). By now, it is believed that the long-lived oscillations
are vibronic or vibrational in nature (Duan etal. 2017b)
and are too short-lived to play any significant role in energy
transfer processes in LH systems (Cao etal. 2020).
Apart from the LH complexes of bacteria and algae, the
LH complexes of plants have of course attracted quite some
interest and especially also the photoprotective mecha-
nism. Under excess solar light conditions, LH complexes of
plants release excess energy as heat in order to avoid photo-
damage. This mechanism is known as non-photochemical
quenching (NPQ) of higher plants (Ruban etal. 2007, 2012;
Chmeliov etal. 2016). In the NPQ process, an increase of
the pH gradient across the thylakoid membrane triggers
the switch between the light-harvesting and the quenching
modes of the antenna complexes belonging to photosystem
II (PSII; Tian etal. 2019; Nicol etal. 2019). Apart from the
pH gradient, binding of protein PsbS (PSII subunit S) can
also induce conformational changes in the LH complexes
leading to an activation of the quenching mechanism (Li
etal. 2000; Correa-Galvis etal. 2016; Liguori etal. 2019;
Daskalakis etal. 2019a, b). The molecular details and the
interplay between different processes are presently an active
field of research. In addition to harvest light in a frequency
range different from that of the Chl molecules, carotenoid
molecules are instrumental in regulating the flow of (excess)
energy which eventually can be released as heat (Ruban
2016, 2018; Maity etal. 2019). Based on experimental stud-
ies, it is believed that the major light-harvesting complex
LHCII and the minor antenna CP29 play the most important
role in the PSII complex in order to protect the photosyn-
thetic apparatus from excess solar energy and thus photo-
damage (Dall’Osto etal. 2017; Son and Schlau-Cohen 2019;
Guardini etal. 2020).
In order to understand the energy transfer dynamics in
LH complexes, various exciton transfer models have been
built based on crystal structures of plant, bacteria, and algae
complexes. In experiment, often two-dimensional spectros-
copy has been employed, whereas in most theoretical inves-
tigation, classical molecular dynamics (MD) simulations
followed by quantum chemistry calculations were carried
out. The excitation energies, also known as site energies,
excitonic couplings and spectral densities are key param-
eters which can be extracted in such studies (Olbrich and
Kleinekathöfer 2010; Olbrich etal. 2011b, c; Shim etal.
2012; Gao etal. 2013; Cupellini etal. 2016; Sláma etal.
2020). Subsequently, these properties need be utilized as
input either in density matrix calculations or in ensemble-
average wave-packet dynamics (Aghtar etal. 2012). Spec-
tral densities represent the frequency-dependent system-bath
couplings within the framework of open quantum system
(May and Kühn 2011) and can be determined via the auto-
correlation functions of the site energy fluctuations of the
individual pigment molecules. Various approaches have
applied to determine the energy gap fluctuations. Among
the first ones was the configuration interaction with singles
(CIS) scheme (Damjanović etal. 2002) but more popular
became the semi-empirical ZINDO/S-CIS scheme (Zern-
er’s intermediate neglect of differential orbital method with
spectroscopic parameters together with configuration inter-
action using single excitation) and time-dependent density
functional theory (TDDFT) calculations. All these theo-
ries have to be applied in a quantum mechanics/molecular
mechanics (QM/MM) fashion to account for the environ-
ments of the pigments and the fluctuations thereof. Because
of a high-computational demand of TDDFT calculations
when employed along MD trajectories, recently, the time-
dependent extension of the density functional tight-bind-
ing theory (TD-DFTB) and its long-range corrected (LC)
version (Kranz etal. 2017) became popular which has an
accuracy similar to that of standard long-range corrected
TD-DFT approaches but with a significantly reduced numer-
ical effort (Bold etal. 2020). Moreover, various computa-
tional demanding quantum chemical methods such as the
second-order coupled cluster (CC2) scheme (Suomivuori
etal. 2019), pair natural orbital coupled cluster theory,
i.e., a domain-based local pair natural orbital similarity
transformed equation of motion-coupled cluster singles
and doubles (DLPNO-STEOM-CCSD) (Sirohiwal etal.
2021), the algebraic-diagrammatic construction (ADC)
formalism (Suomivuori etal. 2019; Bold etal. 2020), the
Bethe–Salpeter equation in the GW approximation (GW/
BSE) (Hashemi and Leppert 2021), the multireference con-
figuration interaction-DFT (MRCI-DFT) approach (Maity
etal. 2019; List etal. 2013; Bold etal. 2020), and complete
active-space SCF (CASSCF) in combination with perturba-
tion theory (CASPT2) (Anda etal. 2019) have been utilized
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149Photosynthesis Research (2023) 156:147–162
1 3
in order to calculate and benchmark the site energies mostly
for a geometry-optimized single conformations of pigment
molecules. In the present review, we focus on the quanti-
ties and functions that were extracted and published by our
group for bacterial and plant LH complexes using trajec-
tories based on QM/MM dynamics of the chromophores
(Bold etal. 2020; Maity etal. 2020, 2021a, b; Sarngad-
haran etal. 2022). In the following sections, we will describe
the multiscale protocol that was successfully applied to LH
complexes in order to calculate site energies, excitonic cou-
plings, and most importantly the so-called spectral densi-
ties. Due to the size and the electronic complexity of the
Chl molecules of the LH systems, we have employed in our
multiscale strategy the numerically efficient DFTB method
(Elstner etal. 1998) as the basis of the ground dynamics and
the excited states calculations in a QM/MM fashion.
This mini review (for more complete reviews see, e.g.,
Jang and Mennucci 2018; Segatta etal. 2019; Cignoni etal.
2022), which is certainly incomplete and likely lacks many
references of interest, is organized as follows: first several
structural properties of LH complexes of bacteria such as
LH2 and FMO as well as of plants such as the LHCII and the
CP29 complexes are described and discussed. Thereafter, a
flowchart of the multiscale strategy is presented and most
of the key parameters involved in the scheme are described.
Finally, the results regarding several properties like site ener-
gies and spectral densities are reviewed while brief conclu-
sions are drawn at the end.
LH antenna complexes ofbacteria
andplants
In LH complexes of bacteria, BChl-a molecules are the main
pigments involved in harvesting the solar light and in the
excitation energy transfer processes. Especially within the
last two decades, the FMO and LH2 complexes of bacteria
have been extensively studied model systems(see Fig.1).
In case of the FMO complex, the potential of long-lived
quantum coherence phenomena at low and ambient tem-
peratures created an immense attention (Engel etal. 2007;
Harel and Engel 2012; Duan etal. 2017a; Maiuri etal. 2018;
Thyrhaug etal. 2018; Cao etal. 2020). The FMO complex is
a water-soluble trimeric LH system of green sulfur bacteria
with C
3
symmetry containing eight BChl in each monomer.
Invivo, these proteins are placed in between chlorosomes,
i.e., large antenna complex with thousands of pigment mol-
ecules, and reaction centers. Their main function is to enable
the transport of the excitation energy from the chlorosome
to the respective reaction center and thus acts as a kind of
excitonic wire. The first X-ray structure for the FMO com-
plex was resolved in 1975 by Fenna and Olson which found
seven BChl in each monomer (Fenna and Matthews 1975).
Later in 2009, an eighth pigment was found to be present in
each monomer which seems to be stably bound only in the
trimeric arrangement and can apparently easily be lost in
purification processes (Tronrud etal. 2009; Olbrich etal.
2011a). In the year 2007, a wave-like energy transfer in the
FMO complex was reported by Engel etal. based on two-
dimensional electronic spectroscopy and the coherence time
was measured to be up to 1.4 ps at a temperature of 77 K
(Engel etal. 2007). An extensive amount of experimental
as well as theoretical studies were carried out on this and
similar LH model systems with the aim to identify the origin
of this long-lived coherence phenomenon (Engel etal. 2007;
Harel and Engel 2012; Duan etal. 2017a; Maiuri etal. 2018;
Thyrhaug etal. 2018; Cao etal. 2020). It was proposed that
correlated site energy fluctuations of the BChl pigments
could be reason behind the electronic nature of this phenom-
enon. Later however, theoretical studies based on classical
MD simulations ruled out this assumption (Olbrich etal.
2011a; Shim etal. 2012). Moreover, recent experimental
studies lead to the conclusion that the long-lived oscillations
are of vibronic or vibrational nature rather than being purely
coherences (Duan etal. 2017b; Thyrhaug etal. 2018; Cao
etal. 2020).
Other complexes investigated using two-dimensional
electronic spectroscopy to look for long-lived coherences
were LH2 complexes of purple bacteria (Harel and Engel
2012). They also have been investigated theoretically quite
intensively (Hu etal. 2002; Damjanović etal. 2002; Olbrich
and Kleinekathöfer 2010; Cupellini etal. 2016). LH2 com-
plexes from purple bacteria are ring-shaped complexes and
based on the organism can contain a different number of
chromophores, e.g., Rhodopseudomonas acidophila and
Rhodospirillum molischianum include 27 and 24 BChl-a
pigment molecules, respectively. Besides the Chl molecules,
these LH2 systems also contain carotenoids which act as
accessory pigments and are much less studied in experiment
and theory partially due to their complex electronic struc-
ture. The LH2 systems are split into two rings, one B800
and one B850 ring. These rings are named according to their
major absorption wave length in units of nm. The B800 ring
contains 9 or 8 BChl pigments, whereas the B850 ring holds
18 or 16 BChl pigments for R. acidophila (McDermott etal.
1995) and R. molischianum (Koepke etal. 1996), respec-
tively. As can already be seen from these numbers, these
systems have either a 8 or 9-fold symmetry depending on
the organism. Within the B850 ring, each symmetry unit
contains two BChl molecules with a slightly different protein
environment leading also to two types of couplings (Alden
etal. 1997; Hu etal. 1997). These couplings do, of course,
have an influence on the exciton transfer rates between the
pigments in the B850 ring (Mirkovic etal. 2017). Since the
pigments in the B850 rings are much more closely packed
than in the B800 rings, the couplings in the B850 rings are
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150 Photosynthesis Research (2023) 156:147–162
1 3
considerably higher than in the B800 rings leading to exciton
delocalization in the former rings (Damjanović etal. 2002).
For some bacteria, the LH2 complexes together with LH1
systems and F
1
ATPase proteins as well as other proteins
are assembled in an organelle termed chromatophore (Sin-
gharoy etal. 2019). In such a chromatophore, several LH2
rings surround an LH1 complex, and at the center of these
LH1 complexes, a reaction center is present. Recently, an
atomistic model of such a chromatophore was constructed
containing more than 130 millions atoms (Singharoy etal.
2019). In principle, this model system contains the complete
molecular machinery from photon absorption over exciton
transfer to the conversion into chemical energy in form of
ATP.
In case of plants, Chl-a molecules are the main actors in
the transfer of the excitation energy to the reaction center,
whereas Chl-b pigments act as accessory pigments and
absorb high-energy sunlight which is subsequently trans-
ferred to the Chl-a pool. Besides harvesting solar light and
producing a significant amount of the oxygen on earth, the
LH complexes of higher plants have to protect themselves
against too much sunlight. Excess solar energy can lead to an
increased lifetime of singlet excited Chls which in turn can
trigger a process causing oxidative damage to the pigments.
To prevent this damage to happen, an increased pH gradient
across the thylakoid membrane is build up in the presence of
excess solar energy. Moreover, the binding of PsbS proteins
can trigger conformational changes of the protein matrix of
the LH complexes in order to release the excess energy as
heat. For this NPQ process (Bergantino etal. 2003; Ruban
etal. 2007, 2012; Chmeliov etal. 2016; Daskalakis 2018;
Liguori etal. 2019), the major antenna complex LHCII and
the minor antenna CP29 of PSII have been experimentally
identified to play a major role (Dall’Osto etal. 2017; Son and
Schlau-Cohen 2019; Guardini etal. 2020). The LHCII com-
plex is present in the periphery of the PSII complex, whereas
CP29 acts as a bridge between LHCII and the PSII-core com-
plex. In the presence of excess sunlight, these two complexes
are believed to be part of an elegant feedback mechanisms to
harmlessly dissipate excess energy (Ruban 2018), while also
other proposals for energy quenching exist which are based
on fluctuating antenna systems and excitation transfer being
modeled in terms of random walks (Chmeliov etal. 2014;
Amarnath etal. 2016; Bennett etal. 2018).
Fig. 1 The upper panels depict
the bacterial antenna complexes
FMO and LH2, whereas the
lower panels show the plant
antenna complexes LHCII and
CP29 with the protein backbone
in green cartoon representation.
The different types of Chl pig-
ment molecules are highlighted
in red and blue
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151Photosynthesis Research (2023) 156:147–162
1 3
The crystal structure of the LHCII system shows that it
is a trimeric complex and each monomer contains 14 Chl
molecules, i.e., 8 Chl-a and 6 Chl-b pigments. Moreover,
two luteins (Lut), one neoxanthin (Neo), and one violaxan-
thin (Vio) carotenoid are present in each monomer (Liu etal.
2004). The CP29 complex contains 9 Chl-a and 4 Chl-b
pigments as well as 1 Lut, 1 Neo, and 1 Vio carotenoid (Pan
etal. 2011)(see Fig.1).
Let us have a brief view on the different Chl types in
plants and bacteria since this becomes of interest further
below. Chl and BChl molecules are structurally very similar
and both pigment have phytyl chain. Moreover, in case of
Chl molecules, the Mg-porphyrin ring contains three unsatu-
rated and one saturated pyrrole ring, whereas in BChl, two
saturated and two unsaturated pyrrole rings are present in the
Mg-porphyrin ring as shown in Fig.2. Furthermore, in BChl
molecules, an acetyl (COCH
3
) group is connected to the
Mg-porphyrin ring instead of an ethylene group (CH=CH
2
)
in Chl molecules. As a consequence, Chl molecules absorb
at a slightly higher energy and are less flexible than BChl
molecules which has consequences on the excitation energy
transfer processes (Maity etal. 2021b).
Multiscale description oftheLH systems
To study the excitation energy transfer dynamics in LH
complexes of bacteria, plants or algae several models
have developed within the framework of open quantum
systems (Renger etal. 2001; May and Kühn 2011). In this
context, mixed quantum-classical ground state dynamics
simulations have been performed together with excited
state calculations along the trajectory in order to deter-
mine the excitonic population dynamics within the pig-
ment network. Over the last years, we mainly followed two
strategies to investigate the dynamical properties within
LH antenna complexes. On the one hand, a time-average
Hamiltonian together with the spectral densities of each
pigment can be extracted and utilized as input in density
matrix calculations. On the other hand, the time-dependent
Hamiltonian can be employed directly to perform ensem-
ble-average wave-packet dynamics. In previous studies
especially in our own research group, classical MD simu-
lations were used for the ground state dynamics (Olbrich
and Kleinekathöfer 2010; Olbrich etal. 2011c; Aghtar
etal. 2014; Aghtar and Kleinekathöfer 2016). It became
evident, however, that the force fields are not good enough
to describe the internal vibrational modes of the pigment
molecules and that these active parts need to be treated
on a quantum level. Because of the size and electronic
complexity of the pigment molecules, we choose DFTB
as the basis of our multiscale scheme for the ground and
excited state simulations. A schematic representation of
the present multiscale protocol is given in Fig.3.
The aim of the atomistic simulation is to determine
the parameters for a reduced model Hamiltonian. Usually
these tight-binding Hamiltonians are restricted to the one-
exciton manifold. With this restriction, the Hamiltonian of
the Frenkel exciton model is given by
where
Em
denotes the excitation energy, i.e., site energy, of
pigment m and
Vmm
the excitonic coupling between pigments
m and n. The excitonic state
�
𝛼
⟩
can be expressed in terms of
the site-local states
�m⟩
as
and can be obtained by diagonalizing the Hamiltonian. In
the following subsections, we briefly explain some of the
key ingredients and features of the multiscale scheme and
the recent progress in rather accurately describing the exci-
ton dynamics and potentially spectroscopic properties of LH
complexes.
(2)
H
S=
m
Em
m
m
+
n≠m
Vmn
n
m
,
(3)
𝛼
=
m
c𝛼
m
m
,
Fig. 2 Structural features of a BChl and b Chl pigments as present in
bacteria and plants. The methyl group (R = CH
3
) of the Chl-a mol-
ecule is replaced by an aldehyde (R = CHO) in the Chl-b molecule.
The regions highlighted in green in both panels indicate the structural
differences between the BChl and Chl molecules
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152 Photosynthesis Research (2023) 156:147–162
1 3
Site energies
The diagonal elements of the Frenkel Hamiltonian as given
in Eq.2 are the excitation energies, i.e., site energies, of
the individual pigment molecules. As mentioned above, one
way to obtain those site energies is to determine them using
quantum chemical approaches along a classical MD trajec-
tory. To this end, mainly the ZINDO/S and TD-DFT (often
in a long-range corrected version) methods were utilized in
a QM/MM setting and then averaged over time when neces-
sary (Olbrich and Kleinekathöfer 2010; Olbrich etal. 2011b,
c; Shim etal. 2012; Gao etal. 2013; Claridge and Troisi
2018; Claridge etal. 2018; Cupellini etal. 2016; Cignoni
etal. 2022). TD-DFT schemes are computationally more
demanding than the semi-empirical ZINDO/S formalism,
and despite some deficiencies, they have been validated and
extensively tested for excited state energies of pigment mole-
cules. At the same time, numerous higher-level QM methods
for excited state calculations exist but due to their numerical
cost, they are usually limited to benchmark calculations for
single conformations of pigment molecules. These methods
include DFT/MRCI (List etal. 2013; Bold etal. 2020), CAS-
SCF/CASPT2 (Anda etal. 2019), CC2 (Suomivuori etal.
2019), DLPNO-STEOM-CCSD (Sirohiwal etal. 2021),
and ADC (Suomivuori etal. 2019; Bold etal. 2020). The
recently developed time-depended extension of the long-
range corrected DFTB (TD-LC-DFTB) method leads to a
numerical formalism which can be employed for many con-
formations along a trajectory with an accuracy close to that
of standard long-range corrected TD-LC-DFT approaches
(Bold etal. 2020). For this reason, we have included the
TD-LC-DFTB method as the scheme for the excited state
calculations into our multiscale strategy as shown in Fig.3.
Since using purely classical MD simulations for the ground
state dynamics had clear drawbacks, e.g., in the vibrational
dynamics of the pigment molecules, in our recent version
of the multiscale scheme, we replaced this classical MD by
a DFTB-based MD for the ground state dynamics of pig-
ment molecules in a QM/MM fashion in order to improve
the accuracy of the scheme considerably (Maity etal. 2020,
2021a). This change in the multiscale scheme to DFTB/MM
MD for the chromophores improved the site energies but
also the spectral densities considerably.
An example of site energies calculated for the FMO com-
plex based on the TD-LC-DFTB method along a DFTB/MM
MD trajectory is shown in Fig.4. Details of these simula-
tions can be found in Maity etal. (2020). These results also
show that longer ground state trajectories are necessary to
obtain reasonably converged estimates of the average site
energies of the individual pigments. We need to mention
here that the fluctuations in the MD simulations shown in
Fig.4 are smaller than those in the QM/MM results since
in those classical MD simulations bond constraints were
employed which are not present in the QM region of the
QM/MM simulations (Maity etal. 2020). Moreover, please
note that all these site energies are higher than the experi-
mental findings since almost all excited state calculations
based on DFT schemes are known to overestimate the exci-
tation energies and band gaps. Within the present QM/MM
calculations, only an electrostatic embedding was consid-
ered which seems to be a reasonable and an often employed
approximation. Moreover, the QM region of the BChl and
Fig. 3 A schematic representation of the multiscale approach
employed here to study LH complexes
1234567
Pigment
1.65
1.8
1.95
2.1
2.25
E0->Qy
[eV]
MD (1 ns) : TD-LC-DFTB
MD (100 ns) : TD-LC-DFTB
QM/MM MD (40 ps) : TD-LC-DFTB
QM/MM MD (1 ns) : TD-LC-DFTB
8A 8B
Fig. 4 An example of average site energies with the respective fluctu-
ations as error bars for the FMO complex showing excitation energies
based on the TD-LC-DFTB method along MD and DFTB/MM MD
trajectories of different lengths as published in Maity etal. (2020)
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153Photosynthesis Research (2023) 156:147–162
1 3
Chl molecules has been truncated at the C
1
–C
2
bond of the
phytyl tail since the phytyl chain does not significantly con-
tribute to the excitation energies and the truncation of the
QM part considerably reduces the computational cost. A
more sophisticated QM/MM embedding scheme is often
used by the Mennucci group using a polarizable formal-
ism, i.e., the so-called QM/MMPol approach, in which the
MM region is described by a polarizable force field (Juri-
novich etal. 2015b; Aghtar etal. 2017; Segatta etal. 2019;
Cignoni etal. 2022). In this case, the QM and MM regions
can mutually polarize each other which is lacking in our
electrostatic embedding where only the MM environment
can polarize the QM region. Such a polarizable embedding
can certainly account more accurate effects of the environ-
ment but is computationally more expensive than the more
common electrostatic embedding using non-polarizable
force fields. Nevertheless, this improved scheme has been
applied to biomolecular systems as well (Viani etal. 2014;
Loco etal. 2017; Curutchet and Mennucci 2017; Bondanza
etal. 2020; Macaluso etal. 2021).
Excitonic couplings
The second type of ingredients for the system Hamiltonian is
the excitonic couplings between pigment molecules, i.e., the
off-diagonal terms in the site basis. The excitonic couplings
can be seen to consist of two parts: the long-range Cou-
lomb and the short-range exchange contribution. The later
term is based on the orbital overlap between the pigment
molecules and thus exponentially decays with the distance
between the pigments. Due to the pigment distances in LH
complexes, the Coulomb term which decays only algebrai-
cally with the inter-pigment distance is by far dominating.
Thus, the exchange contribution is usually small and often
neglected. In a supramolecular approach, however, the full
excitonic coupling including exchange and Coulomb parts
are included. In such a scheme and for identical donor and
acceptor molecules (in the same conformation), the coupling
between pigments m and n is given by Kenny and Kassal
(2016) and Bold etal. (2020)
where
E1
and
E2
denote the two lowest excitonic energies of
the dimer molecule.
Without doubt the simplest scheme to determine the Cou-
lomb coupling between two molecules is the point-dipole
approximation (PDA) but its range of validity is limited to
rather large inter-pigment distances. The coupling between
pigments m and n in the PDA is given by
where
𝛍𝐦
and
𝛍𝐧
denote the transition dipole moments of
the respective donor and acceptor molecules. A schematic
representation of the PDA is shown in Fig.5. The next sim-
plest scheme is that of extended dipoles in which two tran-
sition charges are used per molecule, so that the Coulomb
interaction becomes more accurate at intermediate distances
(Kenny and Kassal 2016). Using one transition charge per
(heavy) atom leads to a scheme called transition charges
from electrostatic potential (TrESP) developed by Renger
and co-workers (Madjet etal. 2006; Renger etal. 2013). This
scheme is basically as accurate as the transition density cube
method (Krueger etal. 1998), where the transition charges
are located on a Cartesian grid, but the former is numeri-
cally much more efficient. Thus, we employed the TrESP
formalism in all results shown here. The coupling is given by
where the atomistic transition charges
qT
I
and
qT
J
are pre-
sent at atoms I and J of pigments m and n, respectively (see
also Fig.5). The transition charges can be determined by
(4)
V
mn =
1
2
(E2−E1)
,
(5)
V
PDA
mn =1
4𝜋𝜖0
[
𝜇m⋅𝜇n
R3
mn
−
3
(𝜇m⋅Rmn)(𝜇n⋅Rmn )
R5
mn ],
(6)
V
TrESP
mn =f
4𝜋𝜖0
m,n
∑
I,J
q
T
l
⋅q
T
J
∣rI
m
−rJ
n
∣
,
Fig. 5 A schematic representa-
tion of a the TrESP and b the
point-dipole approximation for
a pigment dimer
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154 Photosynthesis Research (2023) 156:147–162
1 3
the so-called CHELPG (charges from electrostatic poten-
tials using a grid) fitting of the transition density such
that the later can be given by a sum of transition charges
∑qT
𝛿
(r−R)
. Since TD-DFT-based transition charges usu-
ally overestimate the transition dipole moment of pigment
molecules with respect to the experimental values, the
transition charges are normally rescaled to reproduce the
experimental transition dipole values before using them in
coupling calculations.
As the space between the pigment molecules is not
empty in biological systems, the excitonic couplings are
screened. Determining this screening accurately is far from
trivial. Mennucci etal. (Curutchet etal. 2009; Curutchet
and Mennucci 2017; Aghtar etal. 2017; Cignoni etal. 2022)
employed the QM/MMPol approach to determine the envi-
ronmental effects on the coupling values. So far, this scheme
seems the best approach for this purpose but is numerically
quite expensive and thus not easily applicable for a large
number of pigment pairs along trajectories. Scholes etal.
(2007) tried to develop an exponential screening function by
fitting a large number of data points. The resultant screening
function f is given by
where A, B, and
f0
are fitting parameters obtained by com-
paring to experimental data for several biosystems and have
the numerical values 2.68, 0.27, and 0.54, respectively. In
case of inter-pigment distances larger than 20 Å, the value
of the screening converges to a constant value of 0.54. Alter-
natively, a constant screening factor as recommended by the
Poisson-TrESP approach (Renger and Müh 2012) can be
applied in order to capture the environmental effect on the
coupling fluctuations. More work in this direction is cer-
tainly needed to better include the environmental effects on
the coupling values. For example, a screening function has
been developed that is not only dependent on the distance of
the pigment centers but on all involved atom pairs (Megow
etal. 2014).
Spectral densities
Spectral densities describe the frequency-dependent cou-
pling of a primary system and its environment, also called
thermal bath. It has to be clear that in the present case not
the whole pigment molecules are included in the primary
system but only one mode representing the vertical excita-
tion from ground to the
Qy
excited state. Thus, large parts of
the spectral densities discussed below will be due to inter-
nal vibrational modes of the pigment molecules themselves.
Denoting the dimensionless coupling parameter for pigment
m and mode k by
cm,k
, the spectral density is given by Weiss
(1999), May and Kühn (2011), and Aghtar etal. (2012)
(7)
f(Rmn)=Aexp(−BRmn +f0),
On the experimental side, spectral densities can be extracted
from the fluorescence line-narrowing and spectral hole-
burning measurements, whereas in our multiscale protocol,
we determine it as a half-sided Fourier transformation of the
autocorrelation function belonging to the site energy fluc-
tuations (Damjanović etal. 2002; Valleau etal. 2012; Maity
etal. 2020) multiplied by a thermal prefactor
In this expression,
𝛽
denotes the inverse temperature and
C
m(t)
the site energy autocorrelation function for the corre-
sponding pigment molecule m. This autocorrelation function
can be determined as
where
ΔEm
describes the difference between the site energy
Em
at a certain time point and its time-averaged value, i.e.,
ΔEm(t)=Em(t)−⟨Em⟩
. Moreover, N denotes the number of
snapshots present in the respective part of the trajectory. To
improve the sampling of the correlation functions and the
associate spectral densities along the ground state dynam-
ics, we have employed a windowing technique (Maity etal.
2020). The abrupt start and end of the time series can lead
to well-known problems when performing the Fourier trans-
form and can lead to negative peaks in the spectral density.
These artifacts can partially be removed by convoluting the
time series by a damping function (Cupellini etal. 2018),
which however can lead to new issues (Valleau etal. 2012).
Thus, we refrained from any damping functions for the
results shown here. In the present approach, the negative
peaks have simply set to zero since they are unphysical and
might lead to erroneous results in the exciton dynamics.
We have, however, taken care by tuning the length of the
trajectory pieces and the sampling that these problems are
reduced to a minimum. Moreover, we have applied the con-
cept of zero padding by extending the correlation functions
in order to increase the resolution of the spectral densities.
A schematic representation of a spectral density calculation
is shown in Fig.6. Details of a similar scheme are given in
Cignoni etal. (2022).
Previous strategies of spectral density calculations based
on classical MD trajectories suffered from poor quality
ground state geometries used in excited state calculations.
This is commonly known as geometry-mismatch problem
i.e., the ground state structures based on force fields are
(8)
J
m(𝜔)= 1
2∑
k
c
2
m,k
mk𝜔k
𝛿(𝜔−𝜔k)
.
(9)
J
m(𝜔)= 𝛽𝜔
𝜋
∞
∫
0
dtC
m(t)cos(𝜔t)
.
(10)
C
m(tl)= 1
N−l
N−l
∑
k=1
ΔEm(tl+tk)ΔEm(tk)
,
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155Photosynthesis Research (2023) 156:147–162
1 3
not consistent with the excited state quantum calculations
(Renger and Müh 2013; Jurinovich etal. 2015a; Padula
etal. 2017). Moreover, the internal vibrational dynamics of
the Chl and BChl molecules is not well represented by the
existing force fields. Attempts like re-parametrizing classi-
cal force fields have been made to overcome this problem
partially by considering some quantum effects in ground
state simulations (Do and Troisi 2015; Claridge etal. 2019).
Unfortunately, such approaches still failed to represent the
internal vibrational modes accurately leading to inaccu-
rate high-frequency parts of the spectral densities. For this
reason, multiscale schemes based on semi-empirical meth-
ods (Rosnik and Curutchet 2015) or computationally quite
expensive DFT-based approaches (Blau etal. 2018) were
employed to overcome this issue. An alternative approach
was proposed by Coker and co-workers to calculate the
intermolecular and intramolecular contributions of a spectral
density separately where the later part is computed using a
normal model analysis. In this scheme, the intermolecular
part is calculated from pure electrostatic interactions with
the environment. Although, this approach has shown quite
a good agreement of the resulting spectral density with the
experimental counterpart, individual normal mode analy-
ses followed by Huang–Rhys factor calculations within the
vertical gradient approximation make this scheme compu-
tationally quite demanding (Lee and Coker 2016; Lee etal.
2016; Cignoni etal. 2022). In another approach, Rhee and
co-workers constructed a potential energy surface within a
QM/MM framework to perform long-time nuclear dynamics
and subsequently produced rather accurate ground state con-
formations in order to use in the excited state calculations
(Kim and Rhee 2016; Kim etal. 2018, 2015). Again, how-
ever, the construction of such surfaces is computationally
expensive for Chl-type molecules since many high-level
quantum chemistry calculations are involved. Although this
approach provides a good agreement of the spectral density
with the experiment finding, some peaks in the high fre-
quency are still problematic (Jang and Mennucci 2018). In
addition, completely different approaches exist, e.g., based
on normal mode analysis (Renger etal. 2012; Klinger etal.
2020) which have their own advantages but will not be fur-
ther discussed in this mini review.
Various spectral densities extracted within the present
DFTB-based multiscale scheme are shown in Fig.3. Espe-
cially the agreement of theory and experiment for the LHCII
complex is remarkable (Maity etal. 2021a), but the same
is true for CP29 (Maity etal. 2021b) and CP43 (Sarngad-
haran etal. 2022) (data not shown). The more surprising
is that for the FMO complex, the theoretical results show
a main line in the middle of the frequency range in Fig.7a
which is not present in the experimental findings. This fact
will need further investigations taking into account that the
properties which are compared here are not exactly identi-
cal. As can also be seen in Fig.7a, the results based on the
QM/MM ground state dynamics are far superior to those
based on a pure MD simulation. Furthermore, we have also
compared our results with the spectral densities computed
by Lee and Coker (2016) as well as the Rhee group (Kim
etal. 2018) in Fig.7c to get a feeling how the other recent
approaches perform. Finally, in Fig.7d, a spectral density of
a BChl-containing complex, here from the FMO complex, is
contrasted with that of a system including Chl-a molecules,
Fig. 6 A schematic represen-
tation of a spectral density
calculation from a autocorrela-
tion function of the site energy
fluctuations via the autocorrela-
tion function (ACF)
0 400 800 1200 1600 2000
Time [fs]
1.4
1.6
1.8
2
2.2
E
0->Q
y
[eV]
Site Energy
0500 1000 1500
2000
Time [fs]
0
1
2
3
4
C(t) [10-3 eV2]
ACF
0400 800 1200 1600
h
_
[cm
-1
]
0
400
800
1200
1600
2000
J( ) [cm
-1
]
SPD
00.05 0.1 0.15 0.2
h
_
[eV]
0
0.05
0.1
0.15
0.2
0.25
J( ) [eV]
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156 Photosynthesis Research (2023) 156:147–162
1 3
i.e., LHCII in this case, within the same DFTB-based mul-
tiscale approach. As can be seen clearly, the rather small
structural variations between BChl-a and Chl-a molecules
highlighted in Fig.2 lead to quite some differences in their
internal vibrational dynamics resulting in markedly different
spectral densities especially in the high-frequency region.
Population dynamics
The time-dependent system Hamiltonian can be directly
employed to calculate exciton dynamics of the system. In
its time-averaged version, it can be employed as in den-
sity matrix calculations where one needs to provide spec-
tral densities as additional input functions. Density matrix
approaches like the hierarchical equation of motion (HEOM)
method are numerically more accurate but computationally
demanding for a complex-structured spectral density of LH
antenna complexes as shown in Fig.7. Alternatively, the
time-dependent Schrödinger equation needs to be solved
using an ensemble-averaged wave-packet-based approach
which uses the system Hamiltonian in its time-dependent
form. This wave-packet-based approach which is also known
as NISE (numerical integration of the Schrödinger equa-
tion) is numerically efficient. At the same time, it does not
approach the proper thermodynamic limit in general since
temperature is not included in these equations but this can be
fixed to some extent by introducing correction functions for
thermalization (Aghtar etal. 2012; Jansen 2018). Dephas-
ing is, however, treated accurately in this approach. One can
indeed show that in certain parameter regimes, converged
density matrix results and ensemble-averaged wave-packet
outcomes do agree (Aghtar etal. 2012). In the wave-packet
calculations, the population of an exciton at pigment site m
is given by
(11)
P
m(t)=
|∑
𝛼
c𝛼
mc𝛼(t)
|
2
,
0 400 800 1200 1600
h
_
[cm
-1
]
0
400
800
1200
1600
2000
J( ) [cm
-1
]
QM/MM MD : TD-LC-DFTB
MD : TD-LC-DFTB
Experiment (FLN)
00.05 0.1 0.15 0.2
h
_
[eV]
0
0.05
0.1
0.15
0.2
0.25
J( ) [eV]
FMO
0400 800 1200 1600
h
_
[cm
-1
]
0
400
800
1200
1600
2000
J( ) [cm
-1
]
QM/MM MD : TD-LC-DFTB
Lee et al.
Kim et al.
00.05 0.1 0.15 0.2
0
0.05
0.1
0.15
0.2
0.25
J( ) [eV]
FMO : BChl 3
0 400 800 1200 1600
h
_
[cm
-1
]
0
1200
2400
3600
4800
J( ) [cm
-1
]
Chl-a Pool (LHCII)
LHCII (Experiment)
00.05 0.1 0.15 0.2
h
_
[eV]
0
0.15
0.3
0.45
0.6
J( ) [eV]
0400 800 1200 1600
h
_
[cm
-1
]
0
1200
2400
3600
4800
J( ) [cm
-1
]
LHCII
FMO
00.05 0.1 0.15 0.2
h
_
[eV]
h
_
[eV]
0
0.15
0.3
0.45
0.6
J( ) [eV]
(a) (b)
(c) (d)
Fig. 7 a Comparison of average spectral densities for the FMO
complex. One spectral density is based on a pure MD ground state
dynamics, while the other on a DFTB-based QM/MM MD dynam-
ics. In both cases, the excited states were determined using TD-LC-
DFTB. b Comparison of a calculated and experimental spectral
density for the LHCII complex. c For BChl 3 of FMO, the present
spectral density is compared to those of Lee and Coker (2016) and
Kim et al. (2018). d Average spectral densities of the LHCII and
FMO complexes which contain Chl and BChl molecules, respectively
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
157Photosynthesis Research (2023) 156:147–162
1 3
where
c𝛼(t)
is the time-dependent coefficient of exciton wave
function in the excitonic basis. In the framework of ensem-
ble averaging, also the population
Pm(t)
has to be averaged
over several realizations to obtain a meaningful result.
As an example, we show in Fig.8 the exciton dynam-
ics in an FMO monomer. In these calculations, BChl 1
is initially excited and the exciton energy redistribution
is monitored up to 1.25 ps. This figure shows that the
exciton leaves the initially excited pigment in a roughly
exponential manner and is first transferred to BChl 2 and
subsequently to the rest of the network. Interestingly, the
transfer to the eighth Chl of the neighboring monomer,
termed BChl 8B, is rather quick, a point which has been
realized already earlier (Olbrich etal. 2011a).
Absorption spectra
Based on the above-described properties, there are sev-
eral ways how to determine linear absorption spectra
(Schröder etal. 2006; Dinh and Renger 2015; Zuehlsdorff
etal. 2019). Here, we employ the often used Redfield-
like scheme which makes use of the time-averaged system
Hamiltonian together with the site-specific spectral den-
sity. Within this approximation, the absorption is given
by Novoderezhkin and van Grondelle (2010) and Renger
and Müh (2013)
where
𝜇𝛼
=∑mc𝛼
m𝜇m
denotes the excitonic transition dipole
moments determined from the site basis transition dipole
moments
𝜇m
. The
𝜏𝛼
denotes the lifetime of the excitonic
state
𝛼
and can be derived using a Redfield-like rate equa-
tion. Moreover,
g𝛼
denotes the excitonic line-shape which
can be written as
where the site-dependent line-shape functions
gm
are deter-
mined by the site-dependent spectral densities
Jm
using
An example of a linear absorption spectrum modeled
using the present multiscale approach is shown in Fig.9.
The spectrum belongs to the plant antenna CP29 complex
and was already reported earlier (Maity etal. 2021b) . The
site energies and transition dipole moments were calculated
based on TD-LC-DFTB method along a DFTB/MM MD
trajectory. Subsequently, the site-specific spectral densities
(12)
I
(𝜔)∝𝜔∑
𝛼
∣𝜇𝛼∣2
∞
∫
−∞
e−i(𝜔𝛼−𝜔)t−g𝛼(t)−∣t∣∕𝜏𝛼dt
,
(13)
g
𝛼(t)=
∑
m
∣c𝛼
m∣4gm(t)
,
(14)
g
m(t)=
∞
∫
0
d𝜔
ℏ𝜔2Jm(𝜔)
[
(1−cos(𝜔t)) coth
(
ℏ𝜔
2kBT
)
+i(sin(𝜔t)−𝜔t)
]
.
0200 400600 80010001200
Time [fs]
0
0.2
0.4
0.6
0.8
1
Population
BChl 1
BChl 2
BChl 3
BChl 4
BChl 7
BChl 8A
BChl 8B
Fig. 8 Exciton dynamics in a FMO monomer unit as published in
Maity etal. (2020). The right panel shows the BChl network of an
FMO monomer together with closely coupled BChl 8 pigment from a
neighboring monomer. The left panel shows the exciton dynamics to
all other pigments with only BChl 1 being initially excited
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158 Photosynthesis Research (2023) 156:147–162
1 3
were extracted from the autocorrelation functions of site
energy fluctuations. Moreover, the excitonic couplings were
calculated using the TrESP formalism based on a classi-
cal MD trajectory. The main peak and the high-frequency
vibrational side-band of the calculated spectrum are in good
agreement with the experimental counterpart, while we have
to mention that the calculated spectrum was shifted such
that the peak position agrees. The additional low-frequency
shoulder of the calculated spectrum is possibly due to prob-
lems with the DFTB calculations of the site energies and/or
transition dipole moments. Due to its perturbative nature,
the Redfield-like approximation as employed for absorption
spectra here can, however, be problematic as was recently
shown for the CP43 antenna system (Sarngadharan etal.
2022). In addition, no further broadening due to additional
static disorder has been considered during the computation
of absorption spectra which would certainly broaden the
spectrum to some extent (Cignoni etal. 2022). Neverthe-
less, the shape of the calculated spectrum is very reasonable,
and it is rewarding to see the agreement for the high-energy
shoulder which is due to the high-frequency peaks in the
spectral densities (Sarngadharan etal. 2022).
Conclusions andoutlook
In this mini review, which is certainly incomplete, biased
toward the work of our own research group and likely
neglecting many relevant references, we have tried to give
a general overview of the challenges which one encounters
during the modeling of LH protein–pigment complexes on a
molecular level. To this end, we have presented a multiscale
strategy that combines DFTB-based ground state MD simu-
lations with TD-LC-DFTB-based excited state calculations
and has shown to yield very reasonable results. The key
components of this scheme are the site energies, excitonic
couplings, and spectral densities which were extracted
for various LH antenna complex of bacteria and plants as
reported earlier (Maity etal. 2020, 2021a, b; Sarngadharan
etal. 2022). Moreover, we have highlighted the problems
and the improvements over the previous method based on
classical MD ground state dynamics that was unable to
describe the high-frequency part of the spectral density
accurately. The ingredients that are determined based on
the multiscale scheme can be further employed as an input
to model the exciton dynamics and spectroscopic proper-
ties using many different techniques (Nalbach etal. 2011;
Mühlbacher and Kleinekathöfer 2012; Jansen 2021; Varvelo
etal. 2021; Kundu and Makri 2022; Bose and Walters 2022).
Despite large progress in the field and despite the remark-
able accuracy of the present multiscale scheme, there is
still quite some room for improvement from a computa-
tional point of view. For example, although DFTB-based
ground state and excited state calculations are computation-
ally more efficient than DFT-based approaches, machine
learning models can potentially still reduce the numerical
cost while increasing the accuracy at the same time. First
studies in this direction have been performed already, e.g.,
in Zaspel etal. (2019), Krämer etal. (2020), Chen etal.
(2020), and Westermayr and Marquetand (2020). Moreover,
machine learning-based approaches can be further applied
for the exciton dynamics calculations which are numeri-
cally demanding either in density matrix e.g., HEOM or
temperature corrected NISE calculations (Häse etal. 2017).
Another improvement could be the “on-the-fly” non-adiaba-
tic dynamics of the exciton dynamics instead of constructing
system Hamiltonians as done in the present multiscale proto-
col. Moreover, there is still the question of how to determine
the exciton dynamics in a whole PSII complex or chromato-
phore. In addition, interesting algae systems exist which so
far have investigated far less.
Acknowledgements Financial support by the Deutsche Forschungsge-
meinschaft (DFG) through Grants KL-1299/18-1 and KL-1299/24-1
is gratefully acknowledged. Moreover, we thank current and former
group members for valuable input and discussions on the topics of
this mini review.
Funding Open Access funding enabled and organized by Projekt
DEAL.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
1.75 1.8 1.85 1.9 1.95 2
h
_
[eV]
0
0.2
0.4
0.6
0.8
1
Normalized Intensity [a.u.]
Calculated
Experiment
14000 14500 15000 15500 16000
h
_
[cm
-1
]
CP29
Fig. 9 Absorption spectra at 300 K for the CP29 minor antenna com-
plex compared to the experimental findings. The data have been taken
from Maity etal. (2021b)
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159Photosynthesis Research (2023) 156:147–162
1 3
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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