ArticlePublisher preview available

Self-consistent approach to the dynamics of excitation energy transfer in multichromophoric systems

AIP Publishing
The Journal of Chemical Physics
Authors:
Veljko Janković at Institute of Physics Belgrade
Veljko Janković
Tomas Mancal at Charles University in Prague
Tomas Mancal
Read publisher preview
Download citation
To read the full-text of this research, you can request a copy directly from the authors.

Abstract and Figures

Computationally tractable and reliable, albeit approximate, methods for studying exciton transport in molecular aggregates immersed in structured bosonic environments have been actively developed. Going beyond the lowest-order (Born) approximation for the memory kernel of the quantum master equation typically results in complicated and possibly divergent expressions. Starting from the memory kernel in the Born approximation, and recognizing the quantum master equation as the Dyson equation of Green’s functions theory, we formulate the self-consistent Born approximation to resum the memory-kernel perturbation series in powers of the exciton–environment interaction. Our formulation is in the Liouville space and frequency domain and handles arbitrary exciton–environment spectral densities. In a molecular dimer coupled to an overdamped oscillator environment, we conclude that the self-consistent cycle significantly improves the Born-approximation energy-transfer dynamics. The dynamics in the self-consistent Born approximation agree well with the solutions of hierarchical equations of motion over a wide range of parameters, including the most challenging regimes of strong exciton–environment interactions, slow environments, and low temperatures. This is rationalized by the analytical considerations of coherence-dephasing dynamics in the pure-dephasing model. We find that the self-consistent Born approximation is good (poor) at describing energy transfer modulated by an underdamped vibration resonant (off-resonant) with the exciton energy gap. Nevertheless, it reasonably describes exciton dynamics in the seven-site model of the Fenna–Matthews–Olson complex in a realistic environment comprising both an overdamped continuum and underdamped vibrations.
Diagrammatic representation of (a) the interacting Green’s superoperator and (b) the free-exciton Green’s superoperator. (c) In an arbitrary diagram, the environmental assistance starting at sj, ending at sl (t ≥ sl ≥ sj ≥ 0), and involving the environment that surrounds chromophore njl is represented by a dashed circumference connecting sj and sl. The connections of the circumference with free-exciton lines, represented by full dots, correspond to the superoperators in Eqs. (17) and (18). Diagrammatic representation of (d) the perturbation expansion for G(t) up to the fourth order in the exciton–environment interaction, (e) the Dyson equation [Eq. (19)], and (f) the self-energy superoperator up to the fourth order in the exciton–environment interaction.
Diagrammatic representation of (a) the interacting Green’s superoperator and (b) the free-exciton Green’s superoperator. (c) In an arbitrary diagram, the environmental assistance starting at sj, ending at sl (t ≥ sl ≥ sj ≥ 0), and involving the environment that surrounds chromophore njl is represented by a dashed circumference connecting sj and sl. The connections of the circumference with free-exciton lines, represented by full dots, correspond to the superoperators in Eqs. (17) and (18). Diagrammatic representation of (d) the perturbation expansion for G(t) up to the fourth order in the exciton–environment interaction, (e) the Dyson equation [Eq. (19)], and (f) the self-energy superoperator up to the fourth order in the exciton–environment interaction.
… 
Diagrammatic representation of (a1) the BA self-energy superoperator [Eq. (26) or Eq. (27)], (b1) the SCBA self-energy superoperator [Eq. (33)], (a2) Green’s superoperator in the BA, and (b2) Green’s superoperator in the SCBA.
Diagrammatic representation of (a1) the BA self-energy superoperator [Eq. (26) or Eq. (27)], (b1) the SCBA self-energy superoperator [Eq. (33)], (a2) Green’s superoperator in the BA, and (b2) Green’s superoperator in the SCBA.
… 
Dependence of the quantity δΣ(k) [Eq. (37)] monitoring the convergence of the self-consistent algorithm on the iteration number k in the examples analyzed in (a) Figs. 7(a)–7(d) and (b) Figs. 12(b1) and 12(b2). Note the logarithmic scale on both axes.
Dependence of the quantity δΣ(k) [Eq. (37)] monitoring the convergence of the self-consistent algorithm on the iteration number k in the examples analyzed in (a) Figs. 7(a)–7(d) and (b) Figs. 12(b1) and 12(b2). Note the logarithmic scale on both axes.
… 
(a) and (b) Heat maps of the maximum trace distance D((SC)BA|HEOM) [Eq. (43)] between the numerically exact and BA [(a)] or SCBA [(b)] excitonic RDM over the interval [0, tmax]. (c) Heat map of the ratio D(BA|HEOM)D(SCBA|HEOM) between the performance metrics used in (a) and (b). All quantities are computed for different values of the resonance coupling J and the site-energy gap Δɛ, the remaining parameters assume their default values listed in Table II, the initial condition is specified in Eq. (38), while tmax = 2 ps.
(a) and (b) Heat maps of the maximum trace distance D((SC)BA|HEOM) [Eq. (43)] between the numerically exact and BA [(a)] or SCBA [(b)] excitonic RDM over the interval [0, tmax]. (c) Heat map of the ratio D(BA|HEOM)D(SCBA|HEOM) between the performance metrics used in (a) and (b). All quantities are computed for different values of the resonance coupling J and the site-energy gap Δɛ, the remaining parameters assume their default values listed in Table II, the initial condition is specified in Eq. (38), while tmax = 2 ps.
… 
(a) Time dependence of the real part of the coherence in a symmetric (Δɛ = 0) pure-dephasing dimer. (b) Frequency profile of the imaginary part of the matrix element of the self-energy superoperator describing coherence dephasing in a symmetric pure-dephasing dimer. We compare the results of the SCBA (solid lines), BA (dashed lines), and Redfield theory (dotted lines) to the exact result (double dashed-dotted lines). The lines in (a) display Fourier-transformed frequency-domain results using the self-energies given in Eq. (45) (label “exact”), Eq. (48) (label “SCBA”), Eq. (46) (label “BA”), and Eq. (47) (label “Red”). Full symbols in (a) display the analytical results given by Eq. (49) (circles), Eq. (50) (down-triangles), and Eq. (51) (up-triangles). For visual clarity, the BA and Redfield self-energies in (b) are scaled down by a factor of 4.
+8
(a) Time dependence of the real part of the coherence in a symmetric (Δɛ = 0) pure-dephasing dimer. (b) Frequency profile of the imaginary part of the matrix element of the self-energy superoperator describing coherence dephasing in a symmetric pure-dephasing dimer. We compare the results of the SCBA (solid lines), BA (dashed lines), and Redfield theory (dotted lines) to the exact result (double dashed-dotted lines). The lines in (a) display Fourier-transformed frequency-domain results using the self-energies given in Eq. (45) (label “exact”), Eq. (48) (label “SCBA”), Eq. (46) (label “BA”), and Eq. (47) (label “Red”). Full symbols in (a) display the analytical results given by Eq. (49) (circles), Eq. (50) (down-triangles), and Eq. (51) (up-triangles). For visual clarity, the BA and Redfield self-energies in (b) are scaled down by a factor of 4.
… 
Figures - available from: The Journal of Chemical Physics
This content is subject to copyright. Terms and conditions apply.
A preview of this full-text is provided by AIP Publishing.
Learn more
Content available from The Journal of Chemical Physics 
This content is subject to copyright. Terms and conditions apply.
The Journal
of Chemical Physics ARTICLE pubs.aip.org/aip/jcp
Self-consistent approach to the dynamics
of excitation energy transfer
in multichromophoric systems
Cite as: J. Chem. Phys. 161, 204108 (2024); doi: 10.1063/5.0237483
Submitted: 5 September 2024 •Accepted: 12 November 2024 •
Published Online: 26 November 2024
Veljko Jankovi´
c1,a) and Tomáˇ
s Manˇ
cal2
AFFILIATIONS
1Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
2Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Prague 2, Czech Republic
a)Author to whom correspondence should be addressed: veljko.jankovic@ipb.ac.rs
ABSTRACT
Computationally tractable and reliable, albeit approximate, methods for studying exciton transport in molecular aggregates immersed in
structured bosonic environments have been actively developed. Going beyond the lowest-order (Born) approximation for the memory
kernel of the quantum master equation typically results in complicated and possibly divergent expressions. Starting from the memory
kernel in the Born approximation, and recognizing the quantum master equation as the Dyson equation of Green’s functions theory, we
formulate the self-consistent Born approximation to resum the memory-kernel perturbation series in powers of the exciton–environment
interaction. Our formulation is in the Liouville space and frequency domain and handles arbitrary exciton–environment spectral densities.
In a molecular dimer coupled to an overdamped oscillator environment, we conclude that the self-consistent cycle significantly improves
the Born-approximation energy-transfer dynamics. The dynamics in the self-consistent Born approximation agree well with the solutions of
hierarchical equations of motion over a wide range of parameters, including the most challenging regimes of strong exciton–environment
interactions, slow environments, and low temperatures. This is rationalized by the analytical considerations of coherence-dephasing dynam-
ics in the pure-dephasing model. We find that the self-consistent Born approximation is good (poor) at describing energy transfer modulated
by an underdamped vibration resonant (off-resonant) with the exciton energy gap. Nevertheless, it reasonably describes exciton dynamics
in the seven-site model of the Fenna–Matthews–Olson complex in a realistic environment comprising both an overdamped continuum and
underdamped vibrations.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0237483
I. INTRODUCTION
The light absorption and thus initiated excitation energy trans-
fer (EET) in molecular aggregates constitute the first steps of
solar-energy conversion in both natural1,2 and artificial3–6 systems.
The EET takes place in a complex dynamic spatiotemporal land-
scape stemming from the competition of interactions promoting
exciton delocalization (resonance coupling between molecules) and
localization (static and dynamic disorder).7As the energy scales
of these counteracting interactions are typically comparable to
one another,8,9 theoretical descriptions of EET dynamics are quite
challenging.
When approached from the perspective of the theory of open
quantum systems,10,11 the challenge transforms into describing
non-Markovian quantum dynamics of excitons interacting with
their environment. As standard theories (such as Redfield12 and
Förster13 theories) do not meet this challenge,14 various numerically
exact methods have been developed. Two most common founda-
tions of these are (i) the Feynman–Vernon influence functional
theory15 and (ii) the Nakajima–Zwanzig [time-convolution (TC)]
quantum master equation (QME).16–18 The approaches rooted in
(i) include the hierarchical equations of motion (HEOM)19–21 and
a host of path-integral and process-tensor-based methods.22–29 Each
of these approaches develops a different representation of the so-
called exact reduced evolution superoperator (or the dynamical
map) U(t), which becomes its central object. Meanwhile, the main
aim of the approaches originating from (ii) is to evaluate the exact
memory-kernel superoperator K(t).30–32 All the above-referenced
J. Chem. Phys. 161, 204108 (2024); doi: 10.1063/5.0237483 161, 204108-1
Published under an exclusive license by AIP Publishing
ResearchGate has not been able to resolve any citations for this publication.
Self-consistent dynamical maps for open quantum systems
Article
Full-text available
  • Jan 2024
In several cases, open quantum systems can be successfully described using master equations relying on Born-Markov approximations, but going beyond these approaches has become often necessary. In this work, we introduce the NCA and NCA-Markov dynamical maps for open quantum systems, which go beyond these master equations replacing the Born approximation with a self-consistent approximation, known as non-crossing approximation (NCA). These maps are formally similar to master equations, but allow to capture non-perturbative effects of the environment at a moderate extra numerical cost. To demonstrate their capabilities, we apply them to the spin-boson model at zero temperature for both a Ohmic and a sub-Ohmic environment, showing that they can both qualitatively capture its strong-coupling behaviour, and be quantitatively correct beyond standard master equations.
View
Exciton dynamics from the mapping approach to surface hopping: Comparison with Förster and Redfield theories
Article
Full-text available
  • Jan 2024
We compare the recently introduced multi-state mapping approach to surface hopping (MASH) with the Förster and Redfield theories of excitation energy transfer. Whereas Förster theory relies on weak coupling between...
View
A mapping approach to surface hopping
Article
Full-text available
  • Feb 2023
We present a nonadiabatic classical-trajectory approach that offers the best of both worlds between fewest-switches surface hopping (FSSH) and quasiclassical mapping dynamics. This mapping approach to surface hopping (MASH) propagates the nuclei on the active adiabatic potential-energy surface, like in FSSH. However, unlike in FSSH, transitions between active surfaces are deterministic and occur when the electronic mapping variables evolve between specified regions of the electronic phase space. This guarantees internal consistency between the active surface and the electronic degrees of freedom throughout the dynamics. MASH is rigorously derivable from exact quantum mechanics as a limit of the quantum-classical Liouville equation (QCLE), leading to a unique prescription for momentum rescaling and frustrated hops. Hence, a quantum-jump procedure can in principle be used to systematically converge the accuracy of the results to that of the QCLE. This jump procedure also provides a rigorous framework for deriving approximate decoherence corrections similar to those proposed for FSSH. We apply MASH to simulate the nonadiabatic dynamics in various model systems and show that it consistently produces more accurate results than FSSH at a comparable computational cost.
View
Exact Dynamics of Nonadditive Environments in Non-Markovian Open Quantum Systems
Article
Full-text available
  • Feb 2022
When a quantum system couples strongly to multiple baths, then it is generally no longer possible to describe the resulting system dynamics by simply adding the individual effects of each bath. However, capturing such multibath system dynamics typically requires approximations that can obscure some of the nonadditive effects. Here we present a numerically exact and efficient technique for tackling this problem that builds on the time-evolving matrix product operator (TEMPO) representation. We test the method by applying it to a simple model system that exhibits nonadditive behavior: a two-level dipole coupled to both a vibrational and an optical bath. Although not directly coupled, there is an effective interaction between the baths mediated by the system that can lead to population inversion in the matter system when the vibrational coupling is strong. We benchmark and validate multibath TEMPO against two approximate methods—one based on a polaron transformation, the other on an identification of a reaction coordinate—before exploring the regime of simultaneously strong vibrational and optical coupling where the approximate techniques break down. Here we uncover a new regime where the quantum Zeno effect leads to a fully mixed state of the electronic system.
View
Recent progress in atomistic modeling of light-harvesting complexes: a mini review
Article
Full-text available
  • Oct 2022
  • PHOTOSYNTH RES
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 density 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.
View
Exact simulation of pigment-protein complexes unveils vibronic renormalization of electronic parameters in ultrafast spectroscopy
Article
Full-text available
  • May 2022
The primary steps of photosynthesis rely on the generation, transport, and trapping of excitons in pigment-protein complexes (PPCs). Generically, PPCs possess highly structured vibrational spectra, combining many discrete intra-pigment modes and a quasi-continuous of protein modes, with vibrational and electronic couplings of comparable strength. The intricacy of the resulting vibronic dynamics poses significant challenges in establishing a quantitative connection between spectroscopic data and underlying microscopic models. Here we show how to address this challenge using numerically exact simulation methods by considering two model systems, namely the water-soluble chlorophyll-binding protein of cauliflower and the special pair of bacterial reaction centers. We demonstrate that the inclusion of the full multi-mode vibronic dynamics in numerical calculations of linear spectra leads to systematic and quantitatively significant corrections to electronic parameter estimation. These multi-mode vibronic effects are shown to be relevant in the longstanding discussion regarding the origin of long-lived oscillations in multidimensional nonlinear spectra. Multimode vibronic mixing in model photosynthetic systems revealed by numerically exact simulations is shown to strongly modify linear and non-linear optical responses and facilitate the persistence of coherent dynamics.
View
The atomistic modeling of light-harvesting complexes from the physical models to the computational protocol
Article
Full-text available
The function of light-harvesting complexes is determined by a complex network of dynamic interactions among all the different components: the aggregate of pigments, the protein, and the surrounding environment. Complete and reliable predictions on these types of composite systems can be only achieved with an atomistic description. In the last few decades, there have been important advances in the atomistic modeling of light-harvesting complexes. These advances have involved both the completeness of the physical models and the accuracy and effectiveness of the computational protocols. In this Perspective, we present an overview of the main theoretical and computational breakthroughs attained so far in the field, with particular focus on the important role played by the protein and its dynamics. We then discuss the open problems in their accurate modeling that still need to be addressed. To illustrate an effective computational workflow for the modeling of light harvesting complexes, we take as an example the plant antenna complex CP29 and its H111N mutant.
View
View
Accuracy of approximate methods for the calculation of absorption-type linear spectra with a complex system-bath coupling
Article
  • Aug 2022
The accuracy of approximate methods for calculating linear optical spectra depends on many variables. In this study, we fix most of these parameters to typical values found in photosynthetic light-harvesting complexes of plants and determine the accuracy of approximate spectra with respect to exact calculation as a function of the energy gap and interpigment coupling in a pigment dimer. We use a spectral density with the first eight intramolecular modes of chlorophyll a and include inhomogeneous disorder for the calculation of spectra. We compare the accuracy of absorption, linear dichroism, and circular dichroism spectra calculated using the Full Cumulant Expansion (FCE), coherent time-dependent Redfield (ctR), and time-independent Redfield and modified Redfield methods. As a reference we use spectra calculated with the Exact Stochastic Path Integral Evaluation method. We find the FCE method to be the most accurate for the calculation of all spectra. The ctR method performs well for the qualitative calculation of absorption and linear dichroism spectra when pigments are moderately coupled (∼15 cm ⁻¹ ), but ctR spectra may differ significantly from exact spectra when strong interpigment coupling (>100 cm ⁻¹ ) is present. The dependence of the quality of Redfield and modified Redfield spectra on molecular parameters is similar, and these methods almost always perform worse than ctR, especially when the interpigment coupling is strong or the excitonic energy gap is small (for a given coupling). The accuracy of approximate spectra is not affected by resonance with intramolecular modes for typical system-bath coupling and disorder values found in plant light-harvesting complexes.
View
Partially polaron-transformed quantum master equation for exciton and charge transport dynamics
Article
  • Jul 2022
Polaron-transformed quantum master equation (PQME) offers a unified framework to describe the dynamics of quantum systems in both limits of weak and strong couplings to environmental degrees of freedom. Thus, PQME serves as an efficient method to describe charge and exciton transfer/transport dynamics for a broad range of parameters in condensed or complex environments. However, in some cases, the polaron transformation (PT) being employed in the formulation invokes an over-relaxation of slow modes and results in premature suppression of important coherence terms. A formal framework to address this issue is developed in the present work by employing a partial PT that has smaller weights for low frequency bath modes. It is shown here that a closed form expression of a 2nd order time-local PQME including all the inhomogeneous terms can be derived for a general form of partial PT, although more complicated than that for the full PT. All the expressions needed for numerical calculation are derived in detail. Applications to a model of two-level system coupled to a bath of harmonic oscillators, with test calculations focused on those due to homogeneous relaxation terms, demonstrate the feasibility and the utility of the present approach.
View