Theory of single molecule line shapes of multichromophoric macromolecules

Absorption and Circular Dichroism Spectra of Molecular Aggregates With the Full Cumulant Expansion

The exciton Hamiltonian of multichromophoric aggregates can be probed by spectroscopic techniques such as linear absorption and circular dichroism. To compare calculated Hamiltonians to experiments, a lineshape theory is needed, which takes into account the coupling of the excitons with inter- and intramolecular vibrations. This coupling is normally introduced in a perturbative way through the cumulant expansion formalism and further approximated by assuming a Markovian exciton dynamics, for example with the modified Redfield theory. Here, we present the implementation of the full cumulant expansion (FCE) formalism (J. Chem. Phys.142, 2015, 09410625747060) to efficiently compute absorption and circular dichroism spectra of molecular aggregates beyond the Markov approximation, without restrictions on the form of exciton–phonon coupling. By employing the LH2 system of purple bacteria as a challenging test case, we compare the FCE lineshapes with the Markovian lineshapes obtained with the modified Redfield theory, showing that the latter presents a less satisfying agreement with experiments. The FCE approach instead accurately describes the lineshapes, especially in the vibronic sideband of the B800 peak. We envision that the FCE approach will become a valuable tool for accurately comparing model exciton Hamiltonians with optical spectroscopy experiments.

Theory for polariton-assisted remote energy transfer

Strong-coupling between light and matter produces hybridized states (polaritons) whose delocalization and electromagnetic character allow for novel modifications in spectroscopy and chemical reactivity of molecular systems. Recent experiments have demonstrated remarkable distance-independent long-range energy transfer between molecules strongly coupled to optical microcavity modes. To shed light on the mechanism of this phenomenon, we present the first comprehensive theory of polariton-assisted remote energy transfer (PARET) based on strong-coupling of donor and/or acceptor chromophores to surface plasmons. Application of our theory demonstrates that PARET up to a micron is indeed possible. In particular, we report two regimes for PARET: in one case, strong-coupling to a single type of chromophore leads to transfer mediated largely by surface plasmons while in the other case, strong-coupling to both types of chromophores creates energy transfer pathways mediated by vibrational relaxation. Importantly, we highlight conditions under which coherence enhances or deteriorates these processes. For instance, while exclusive strong-coupling to donors can enhance transfer to acceptors, the reverse turns out not to be true. However, strong-coupling to acceptors can shift energy levels in a way that transfer from acceptors to donors can occur, thus yielding a chromophore role-reversal or "carnival effect". This theoretical study demonstrates the potential for confined electromagnetic fields to control and mediate PARET, thus opening doors to the design of remote mesoscale interactions between molecular systems.

Role of quantum coherence in shaping the line shape of an exciton interacting with a spatially and temporally correlated bath

Kubo's fluctuation theory of line shape forms the backbone of our understanding of optical and vibrational line shapes, through such concepts as static heterogeneity and motional narrowing. However, the theory does not properly address the effects of quantum coherences on optical line shape, especially in extended systems where a large number of eigenstates are present. In this work, we study the line shape of an exciton in a one-dimensional lattice consisting of regularly placed and equally separated optical two level systems. We consider both linear array and cyclic ring systems of different sizes. Detailed analytical calculations of line shape have been carried out by using Kubo's stochastic Liouville equation (SLE). We make use of the observation that in the site representation, the Hamiltonian of our system with constant off-diagonal coupling J is a tridiagonal Toeplitz matrix (TDTM) whose eigenvalues and eigenfunctions are known analytically. This identification is particularly useful for long chains where the eigenvalues of TDTM help understanding crossover between static and fast modulation limits. We summarize the new results as follows. (i) In the slow modulation limit when the bath correlation time is large, the effects of spatial correlation are not negligible. Here the line shape is broadened and the number of peaks increases beyond the ones obtained from TDTM (constant off-diagonal coupling element J and no fluctuation). (ii) However, in the fast modulation limit when the bath correlation time is small, the spatial correlation is less important. In this limit, the line shape shows motional narrowing with peaks at the values predicted by TDTM (constant J and no fluctuation). (iii) Importantly, we find that the line shape can capture that quantum coherence affects in the two limits differently. (iv) In addition to linear chains of two level systems, we also consider a cyclic tetramer. The cyclic polymers can be designed for experimental verification. (v) We also build a connection between line shape and population transfer dynamics. In the fast modulation limit, both the line shape and the population relaxation, for both correlated and uncorrelated bath, show similar behavior. However, in slow modulation limit, they show profoundly different behavior. (vi) This study explains the unique role of the rate of fluctuation (inverse of the bath correlation time) in the sustenance and propagation of coherence. We also examine the effects of off-diagonal fluctuation in spectral line shape. Finally, we use Tanimura-Kubo formalism to derive a set of coupled equations to include temperature effects (partly neglected in the SLE employed here) and effects of vibrational mode in energy transfer dynamics.

Dynamics of coherence, localization and excitation transfer in disordered nanorings

Self-assembled supramolecular aggregates are excellent candidates for the design of efficient excitation transport devices. Both artificially prepared and natural photosynthetic aggregates in plants and bacteria present an important degree of disorder that is supposed to hinder excitation transport. Besides, molecular excitations couple to nuclear motion affecting excitation transport in a variety of ways. We present an exhaustive study of exciton dynamics in disordered nanorings with long-range interactions under the influence of a phonon bath taking the LH2 system of purple bacteria as a model. Nuclear motion is explicitly taken into account by employing the Davydov ansatz description of the polaron and quantum dynamics are obtained using a time-dependent variational method. We reveal an optimal exciton-phonon coupling that suppresses disorder-induced localization and facilitate excitation de-trapping. This excitation transfer enhancement, mediated by environmental phonons, is attributed to energy relaxation toward extended, low-energy excitons provided by the precise LH2 geometry with anti-parallel dipoles and long-range interactions. An analysis of localization and spectral statistics is followed by dynamic measures of coherence and localization, transfer efficiency and superradiance. Linear absorption, 2D photon-echo spectra and diffusion measures of the exciton are examined to monitor the diffusive behavior as a function of the strengths of disorder and exciton-phonon coupling.

Contribution of low-temperature single-molecule techniques to structural issues of pigment-protein complexes from photosynthetic purple bacteria

As the electronic energies of the chromophores in a pigment-protein complex are imposed by the geometrical structure of the protein, this allows the spectral information obtained to be compared with predictions derived from structural models. Thereby, the single-molecule approach is particularly suited for the elucidation of specific, distinctive spectral features that are key for a particular model structure, and that would not be observable in ensemble-averaged spectra due to the heterogeneity of the biological objects. In this concise review, we illustrate with the example of the light-harvesting complexes from photosynthetic purple bacteria how results from low-temperature single-molecule spectroscopy can be used to discriminate between different structural models. Thereby the low-temperature approach provides two advantages: (i) owing to the negligible photobleaching, very long observation times become possible, and more importantly, (ii) at cryogenic temperatures, vibrational degrees of freedom are frozen out, leading to sharper spectral features and in turn to better resolved spectra.

Optimal Energy Transfer in Light-Harvesting Systems

Photosynthesis is one of the most essential biological processes in which specialized pigment-protein complexes absorb solar photons, and with a remarkably high efficiency, guide the photo-induced excitation energy toward the reaction center to subsequently trigger its conversion to chemical energy. In this work, we review the principles of optimal energy transfer in various natural and artificial light harvesting systems. We begin by presenting the guiding principles for optimizing the energy transfer efficiency in systems connected to dissipative environments, with particular attention paid to the potential role of quantum coherence in light harvesting systems. We will comment briefly on photo-protective mechanisms in natural systems that ensure optimal functionality under varying ambient conditions. For completeness, we will also present an overview of the charge separation and electron transfer pathways in reaction centers. Finally, recent theoretical and experimental progress on excitation energy transfer, charge separation, and charge transport in artificial light harvesting systems is delineated, with organic solar cells taken as prime examples.

Generalized master equation with non-Markovian multichromophoric Förster resonance energy transfer for modular exciton densities

A generalized master equation (GME) governing quantum evolution of modular exciton density (MED) is derived for large scale light harvesting systems composed of weakly interacting modules of multiple chromophores. The GME-MED offers a practical framework to incorporate real time coherent quantum dynamics calculations of small length scales into dynamics over large length scales, and also provides a non-Markovian generalization and rigorous derivation of the Pauli master equation employing multichromophoric Förster resonance energy transfer rates. A test of the GME-MED for four sites of the Fenna-Matthews-Olson complex demonstrates how coherent dynamics of excitonic populations over coupled chromophores can be accurately described by transitions between subgroups (modules) of delocalized excitons. Application of the GME-MED to the exciton dynamics between a pair of light harvesting complexes in purple bacteria demonstrates its promise as a computationally efficient tool to investigate large scale exciton dynamics in complex environments.

Photon emission statistics and photon tracking in single-molecule spectroscopy of molecular aggregates: dimers and trimers

Based on the generating function formalism, we investigate broadband photon statistics of emission for single dimers and trimers driven by a continuous monochromatic laser field. In particular, we study the first and second moments of the emission statistics, which are the fluorescence excitation line shape and Mandel's Q parameter. Numerical results for this line shape and the Q parameter versus laser frequency in the limit of long measurement times are obtained. We show that in the limit of small Rabi frequencies and laser frequencies close to resonance with one of the one-exciton states, the results for the line shape and Q parameter reduce to those of a two-level monomer. For laser frequencies halfway the transition frequency of a two-exciton state, the photon bunching effect associated with two-photon absorption processes is observed. This super-Poissonian peak is characterized in terms of the ratio between the two-photon absorption line shape and the underlying two-level monomer line shapes. Upon increasing the Rabi frequency, the Q parameter shows a transition from super- to sub- to super-Poissonian statistics. Results of broadband photon statistics are also discussed in the context of a transition (frequency) resolved photon detection scheme, photon tracking, which provides a greater insight in the different physical processes that occur in the multi-level systems.

The electronically excited states of LH2 complexes from Rhodopseudomonas acidophila strain 10050 studied by time-resolved spectroscopy and dynamic Monte Carlo simulations. I. Isolated, non-interacting LH2 complexes

We have employed time-resolved spectroscopy on the picosecond time scale in combination with dynamic Monte Carlo simulations to investigate the photophysical properties of light-harvesting 2 (LH2) complexes from the purple photosynthetic bacterium Rhodopseudomonas acidophila. The variations of the fluorescence transients were studied as a function of the excitation fluence, the repetition rate of the excitation and the sample preparation conditions. Here we present the results obtained on detergent solubilized LH2 complexes, i.e., avoiding intercomplex interactions, and show that a simple four-state model is sufficient to grasp the experimental observations quantitatively without the need for any free parameters. This approach allows us to obtain a quantitative measure for the singlet-triplet annihilation rate in isolated, noninteracting LH2 complexes.

Spectral diffusion and electron-phonon coupling of the B800 BChl a molecules in LH2 complexes from three different species of purple bacteria

We have investigated the spectral diffusion and the electron-phonon coupling of B800 bacteriochlorophyll a molecules in the peripheral light-harvesting complex LH2 for three different species of purple bacteria, Rhodobacter sphaeroides, Rhodospirillum molischianum, and Rhodopseudomonas acidophila. We come to the conclusion that B800 binding pockets for Rhodobacter sphaeroides and Rhodopseudomonas acidophila are rather similar with respect to the polarity of the protein environment but that the packaging of the alphabeta-polypeptides seems to be less tight in Rb. sphaeroides with respect to the other two species.

Single-molecule spectroscopy on a ladder-type conjugated polymer: electron-phonon coupling and spectral diffusion

We employ low-temperature single-molecule spectroscopy combined with pattern recognition techniques for data analysis on a methyl-substituted ladder-type poly(para-phenylene) (MeLPPP) to investigate the electron-phonon coupling to low-energy vibrational modes as well as the origin of the strong spectral diffusion processes observed for this conjugated polymer. The results indicate weak electron-phonon coupling to low-frequency vibrations of the surrounding matrix of the chromophores, and that low-energy intrachain vibrations of the conjugated backbone do not couple to the electronic transitions of MeLPPP at low temperatures. Furthermore, these findings suggest that the main line-broadening mechanism of the zero-phonon lines of MeLPPP is fast, unresolved spectral diffusion, which arises from conformational fluctuations of the side groups attached to the MeLPPP backbone as well as of the surrounding host material.

Refinement of a Structural Model of a Pigment−Protein Complex by Accurate Optical Line Shape Theory and Experiments

Time-local and time-nonlocal theories are used in combination with optical spectroscopy to characterize the water-soluble chlorophyll binding protein complex (WSCP) from cauliflower. The recombinant cauliflower WSCP complexes reconstituted with either chlorophyll b (Chl b) or Chl a/Chl b mixtures are characterized by absorption spectroscopy at 77 and 298 K and circular dichroism at 298 K. On the basis of the analysis of these spectra and spectra reported for recombinant WSCP reconstituted with Chl a only (Hughes, J. L.; Razeghifard, R.; Logue, M.; Oakley, A.; Wydrzynski, T.; Krausz, E. J. Am. Chem. Soc. U.S.A. 2006, 128, 3649), the "open-sandwich" model proposed for the structure of the pigment dimer is refined. Our calculations show that, for a reasonable description of the data, a reduction of the angle between pigment planes from 60 degrees of the original model to about 30 degrees is required when exciton relaxation-induced lifetime broadening is included in the analysis of optical spectra. The temperature dependence of the absorption spectrum is found to provide a unique test for the two non-Markovian theories of optical spectra. Based on our data and the 1.7 K spectra of Hughes et al. (2006), the time-local partial ordering prescription theory is shown to describe the experimental results over the whole temperature range between 1.7 K and room temperature, whereas the alternative time-nonlocal chronological ordering prescription theory fails at high temperatures. Modified-Redfield theory predicts sub-100 fs exciton relaxation times for the homodimers and a 450 fs time constant in the heterodimers. Whereas the simpler Redfield theory gives a similar time constant for the homodimers, the one for the heterodimers deviates strongly in the two theories. The difference is explained by multivibrational quanta transitions in the protein which are neglected in Redfield theory.

Multichromophoric Förster Resonance Energy Transfer from B800 to B850 in the Light Harvesting Complex 2: Evidence for Subtle Energetic Optimization by Purple Bacteria

This work provides a detailed account of the application of our multichromophoric Förster resonance energy transfer (MC-FRET) theory (Phys. Rev. Lett. 2004, 92, 218301) for the calculation of the energy transfer rate from the B800 unit to the B850 unit in the light harvesting complex 2 (LH2) of purple bacteria. The model Hamiltonian consists of the B800 unit represented by a single bacteriochlorophyll (BChl), the B850 unit represented by its entire set of BChls, the electronic coupling between the two units, and the bath terms representing all environmental degrees of freedom. The model parameters are determined, independent of the rate calculation, from the literature data and by a fitting to an ensemble line shape. Comparing our theoretical rate and a low-temperature experimental rate, we estimate the magnitude of the BChl-Qy transition dipole to be in the range of 6.5-7.5 D, assuming that the optical dielectric constant of the medium is in the range of 1.5-2. We examine how the bias of the average excitation energy of the B800-BChl relative to that of the B850-BChl affects the energy transfer time by calculating the transfer rates based on both our MC-FRET theory and the original FRET theory, varying the value of the bias. Within our model, we find that the value of bias 260 cm-1, which we determine from the fitting to an ensemble line shape, is very close to the value at which the ratio between MC-FRET and FRET rates is a maximum. This provides evidence that the bacterial system utilizes the quantum mechanical coherence among the multiple chromophores within the B850 in a constructive way so as to achieve efficient energy transfer from B800 to B850.

Electronic excitation energy transfer between two single molecules embedded in a polymer host

Unidirectional electronic excitation energy transfer from a photoexcited donor chromophore to a ground state acceptor chromophore - both linked by a rigid bridge - has been investigated by low temperature high-resolution single molecule spectroscopy. Our approach allows for accurately accessing static disorder in the donor and acceptor electronic transitions and to calculate the spectral overlap for each couple. By plotting the experimentally determined transfer rates against the spectral overlap, we can distinguish and quantify Förster- and non-Förster-type contributions to the energy transfer.

Super- and sub-Poissonian photon statistics for single molecule spectroscopy

We investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave laser field. The spectral diffusion is modeled based on a stochastic approach, in the spirit of the Anderson-Kubo line shape theory. Using a generating function formalism we solve the generalized optical Bloch equations and obtain an exact analytical formula for the line shape and Mandel's Q parameter. The line shape exhibits well-known behaviors, including motional narrowing when the stochastic modulation is fast and power broadening. The Mandel parameter, describing the line shape fluctuations, exhibits a transition from a quantum sub-Poissonian behavior in the fast modulation limit to a classical super-Poissonian behavior found in the slow modulation limit. Our result is applicable for weak and strong laser fields, namely, for arbitrary Rabi frequency. We show how to choose the Rabi frequency in such a way so that the quantum sub-Poissonian nature of the emission process becomes strongest. A lower bound on Q is found and simple limiting behaviors are investigated. A nontrivial behavior is obtained in the intermediate modulation limit, when the time scales for spectral diffusion and the lifetime of the excited state become similar. A comparison is made between our results and previous ones derived, based on the semiclassical generalized Wiener-Khintchine formula.

Transition Event Statistics in Genetics and Disordered Kinetics. Theoretical Approaches for Extracting Rate Distributions from Experimental Data

We study the analogies between the theory of rate processes in disordered systems and the overdispersed molecular clocks in evolutionary biology. A biological "molecular clock" expresses the statistics of the number of amino acid or nucleotide substitutions during evolution. Random variations of the evolution rates lead to statistical (overdispersed) molecular clocks which are described by random point processes with random substitution rates. We find that the models for overdispersed molecular clocks are equivalent to those of the random-rate or random channel models used in disordered kinetics. The number of transport (reaction) events in disordered kinetics plays the same role as the number of substitution events in molecular biology. We study the connections between the (observed) statistics of the transition events and the statistics of random rate coefficients and random channels; a unified approach is developed which is valid both in molecular biology and in disordered kinetics. We develop methods for extracting statistical information about the variations of rate coefficients from experimental or observed data regarding the fluctuations of the numbers of substitution, reaction, or transport events. For systems with static disorder, the observed statistics of the number of reaction events, expressed in terms of probabilities at a given time or by the cumulants of the number of transition events at a given time, contains the information necessary for evaluating the cumulants or the probability density of the rate coefficients or the density of states for random channel kinetics. For dynamic disorder this is not possible; further information about multitime probability distributions of the reaction events is needed.

Calculation of absorption spectra for light-harvesting systems using non-Markovian approaches as well as modified Redfield theory

For an ensemble of B850 rings of the light-harvesting system LH2 of purple bacteria the linear absorption spectrum is calculated. Using different Markovian and non-Markovian, time-dependent and time-independent methods based on second-order perturbation theory in the coupling between the excitonic system and its surrounding environment as well as the modified Redfield theory, the influence of the shape of the spectral density on the linear absorption spectrum is demonstrated for single samples and in the ensemble average. For long bath correlation times non-Markovian effects clearly show up in the static absorption line shapes. Among the different spectral densities studied is one of the purple bacterium Rhodospirillum molischianum obtained by a molecular-dynamics simulation earlier. The effect of static disorder on its line shapes in the ensemble average is analyzed and the results of the present calculations are compared to experimental data.

A reduced density-matrix theory of absorption line shape of molecular aggregate

A theory for the absorption line shape of molecular aggregates in condensed phase is formulated based on a reduced density-matrix approach. Intermolecular couplings in the aggregates are assumed to be weak (Förster type of energy transfer mechanism). The spin-Boson model is employed to include the effect of electron-phonon coupling. Using the projection operator technique, we derive kinetic equations for the reduced electronic density matrix associated with the absorption spectrum. General expressions of time-dependent rate constants in the kinetic equations are derived by using the cumulant expansion technique. The resulting time-dependent kinetic equations are solved numerically. We illustrate the applicability of the present theory by calculating the line shape of a dimer (a pair of donor and acceptor of energy transfer). For a J-aggregate type of molecular pair (with excitonic redshift), a tail appears on the blue side of the absorption spectrum due to the existence of inhomogeneity in electronic state mixing which is originated from the electron-phonon coupling.

Extraction of Kinetic Information from Single-Molecule Experiments

We discuss two possible approaches for extracting kinetic information from single-molecule experiments. The first approach is based on computing correlation functions from measured fluorescence signals, and the second on studying the statistics of on and off times of the same fluorescence signal. We show that in both cases it is possible to extract kinetic information about the nature of intramolecular fluctuations of the single molecule. We show that for single-molecule kinetics the intramolecular fluctuations produce stochastic memory effects which lead to new dynamic features that do not exist in traditional chemical kinetics. In particular, we investigate a new type of chemical oscillations in correlation functions observed experimentally by Edman and Rigler (Proc. Natl. Acad. Sci. USA 2000, 97, 8266).