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

Photosynthetic Light-Harvesting Complexes

The light-harvesting antenna systems found in photosynthetic organisms function to collect light and transfer energy in the photon to a reaction center, where electron transfer gives rise to long-term energy storage. The antenna systems found in different types of photosynthetic organisms adapt the organisms to very different photic environments, and almost certainly have been invented multiple times during evolution. The diverse collection of photosynthetic antenna systems is described in terms of their pigment and protein components and their organization in the photosystem. The Förster theory is described as the physical basis of energy transfer in photosynthetic antennas, although in many systems it is not adequate to describe energy transfer in complexes with closely interacting pigments. Regulatory aspects of antennas are described, including the process of non-photochemical quenching.

Quantum chemical description of absorption properties and excited-state processes in photosynthetic systems

The theoretical description of the initial steps in photosynthesis has gained increasing importance over the past few years. This is caused by more and more structural data becoming available for light-harvesting complexes and reaction centers which form the basis for atomistic calculations and by the progress made in the development of first-principles methods for excited electronic states of large molecules. In this Review, we discuss the advantages and pitfalls of theoretical methods applicable to photosynthetic pigments. Besides methodological aspects of excited-state electronic-structure methods, studies on chlorophyll-type and carotenoid-like molecules are discussed. We also address the concepts of exciton coupling and excitation-energy transfer (EET) and compare the different theoretical methods for the calculation of EET coupling constants. Applications to photosynthetic light-harvesting complexes and reaction centers based on such models are also analyzed.

Non-Markovian finite-temperature two-time correlation functions of system operators: beyond the quantum regression theorem

An extremely useful evolution equation that allows systematically calculating the two-time correlation functions (CF's) of system operators for non-Markovian open (dissipative) quantum systems is derived. The derivation is based on perturbative quantum master equation approach, so non-Markovian open quantum system models that are not exactly solvable can use our derived evolution equation to easily obtain their two-time CF's of system operators, valid to second order in the system-environment interaction. Since the form and nature of the Hamiltonian are not specified in our derived evolution equation, our evolution equation is applicable for bosonic and/or fermionic environments and can be applied to a wide range of system-environment models with any factorized (separable) system-environment initial states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equation is valid for both L = L(†) and L ≠ L(†) cases, in contrast to those evolution equations valid only for L = L(†) case in the literature. The derived equation that generalizes the quantum regression theorem (QRT) to the non-Markovian case will have broad applications in many different branches of physics. We then give conditions on which the QRT holds in the weak system-environment coupling case and apply the derived evolution equation to a problem of a two-level system (atom) coupled to the finite-temperature bosonic environment (electromagnetic fields) with L ≠ L(†).

Quantum decoherence in finite size exciton-phonon systems

Based on the operatorial formulation of the perturbation theory, the properties of a confined exciton coupled with phonons in thermal equilibrium is revisited. Within this method, the dynamics is governed by an effective Hamiltonian which accounts for exciton-phonon entanglement. The exciton is dressed by a virtual phonon cloud whereas the phonons are clothed by virtual excitonic transitions. Special attention is thus paid for describing the time evolution of the excitonic coherences at finite temperature. As in an infinite lattice, temperature-enhanced quantum decoherence takes place. However, it is shown that the confinement softens the decoherence. The coherences are very sensitive to the excitonic states so that the closer to the band center the state is located, the slower the coherence decays. In particular, for odd lattice sizes, the coherence between the vacuum state and the one-exciton state exactly located at the band center survives over an extremely long time scale. A superimposition involving the vacuum and this specific one-exciton state behaves as an ideal qubit insensitive to its environment.

The effect of correlated bath fluctuations on exciton transfer

Excitation dynamics of various light harvesting systems have been investigated with many theoretical methods including various non-Markovian descriptions of dissipative quantum dynamics. It is typically assumed that each excited state is coupled to an independent thermal environment, i.e., that fluctuations in different environments are uncorrelated. Here the assumption is dropped and the effect of correlated bath fluctuations on excitation transfer is investigated. Using the hierarchy equations of motion for dissipative quantum dynamics it is shown for models of the B850 bacteriochlorophylls of LH2 that correlated bath fluctuations have a significant effect on the LH2→LH2 excitation transfer rate. It is also demonstrated that inclusion of static disorder is crucial for an accurate description of transfer dynamics.

Lowest electronic states of the CP47 antenna protein complex of photosystem II: simulation of optical spectra and revised structural assignments

In this work, we present simulated steady-state absorption, emission, and nonresonant hole burning (HB) spectra for the CP47 antenna complex of photosystem II (PS II) based on fits to recently refined experimental data (Neupane et al. J. Am. Chem. Soc. 2010, 132, 4214). Excitonic simulations are based on the 2.9 Å resolution structure of the PS II core from cyanobacteria (Guskov et al. Nat. Struct. Mol. Biol. 2009, 16, 334), and allow for preliminary assignment of the chlorophylls (Chls) contributing to the lowest excitonic states. The search for realistic site energies was guided by experimental constraints and aided by simple fitting algorithms. The following experimental constraints were used: (i) the oscillator strength of the lowest-energy state should be approximately ≤0.5 Chl equivalents; (ii) the excitonic structure must explain the experimentally observed red-shifted (∼695 nm) emission maximum; and (iii) the excitonic interactions of all states must properly describe the broad (non-line-narrowed, NLN) HB spectrum (including its antihole) whose shape is extremely sensitive to the excitonic structure of the complex, especially the lowest excitonic states. Importantly, our assignments differ significantly from those previously reported by Raszewski and Renger (J. Am. Chem. Soc. 2008, 130, 4431), due primarily to differences in the experimental data simulated. In particular, we find that the lowest state localized on Chl 526 possesses too high of an oscillator strength to fit low-temperature experimental data. Instead, we suggest that Chl 523 most strongly contributes to the lowest excitonic state, with Chl 526 contributing to the second excitonic state. Since the fits of nonresonant holes are more restrictive (in terms of possible site energies) than those of absorption and emission spectra, we suggest that fits of linear optical spectra along with HB spectra provide more realistic site energies.

Parametric resonance-induced time-convolutionless master equation breakdown in finite size exciton-phonon systems

A detailed analysis is performed to show that the second order time-convolutionless master equation fails to describe the exciton-phonon dynamics in a finite size lattice. To proceed, special attention is paid to characterizing the coherences of the exciton reduced density matrix. These specific elements measure the ability of the exciton to develop superimpositions involving the vacuum and the one-exciton states. It is shown that the coherences behave as wavefunctions whose dynamics is governed by a time-dependent effective Hamiltonian defined in terms of the so-called time-dependent relaxation operator. Due to the confinement, quantum recurrences provide to the relaxation operator an almost periodic nature, so the master equation reduces to a linear system of differential equations with almost periodic coefficients. We show that, in accordance with the Floquet theory, unstable solutions emerge due to parametric resonances involving specific frequencies of the relaxation operator and specific excitonic eigenfrequencies. These resonances give rise to an unphysical exponential growth of the coherences, indicating the breakdown of the second order master equation.

Phonon anharmonicity-induced decoherence slowing down in exciton-phonon systems

Based on a generalized Fröhlich model, a time-convolutionless master equation is established for studying the dynamics of an exciton coupled with anharmonic phonons. Special attention is paid to describing the influence of the phonon anharmonicity on specific elements of the exciton reduced density matrix. These elements, called coherences, characterize the ability of the exciton to develop quantum states that are superimpositions involving the vacuum and the local one-exciton states. Whether the phonons are harmonic or not, it is shown that dephasing limited-coherent motion takes place. The coherences irreversibly decrease with time, the decay rate being the so-called dephasing rate, so that they experience a localization phenomenon and propagate over a finite length scale. However, it is shown that the phonon anharmonicity softens the influence of the phonon bath and reduces the dephasing rate. A slowdown in the decoherence process appears so that the coherences are able to explore a larger region along the lattice. Moreover, the phonon anharmonicity modifies the way the dephasing rate depends on both the adiabaticity and the temperature. In particular, the dephasing rate increases linearly with the temperature in the weak anharmonicity limit whereas it becomes almost temperature-independent in the strong anharmonicity limit. Note that the present formalism is applied to describe amide-I excitons (vibrons) in a lattice of H-bonded peptide units.

Effective-mode representation of non-Markovian dynamics: a hierarchical approximation of the spectral density. II. Application to environment-induced nonadiabatic dynamics

The non-Markovian approach developed in the companion paper [Hughes et al., J. Chem. Phys. 131, 024109 (2009)], which employs a hierarchical series of approximate spectral densities, is extended to the treatment of nonadiabatic dynamics of coupled electronic states. We focus on a spin-boson-type Hamiltonian including a subset of system vibrational modes which are treated without any approximation, while a set of bath modes is transformed to a chain of effective modes and treated in a reduced-dimensional space. Only the first member of the chain is coupled to the electronic subsystem. The chain construction can be truncated at successive orders and is terminated by a Markovian closure acting on the end of the chain. From this Mori-type construction, a hierarchy of approximate spectral densities is obtained which approach the true bath spectral density with increasing accuracy. Applications are presented for the dynamics of a vibronic subsystem comprising a high-frequency mode and interacting with a low-frequency bath. The bath is shown to have a striking effect on the nonadiabatic dynamics, which can be rationalized in the effective-mode picture. A reduced two-dimensional subspace is constructed which accounts for the essential features of the nonadiabatic process induced by the effective environmental mode. Electronic coherence is found to be preserved on the shortest time scale determined by the effective mode, while decoherence sets in on a longer time scale. Numerical simulations are carried out using either an explicit wave function representation of the system and overall bath or else an explicit representation of the system and effective-mode part in conjunction with a Caldeira-Leggett master equation.

Effective-mode representation of non-Markovian dynamics: a hierarchical approximation of the spectral density. I. Application to single surface dynamics

An approach to non-Markovian system-environment dynamics is described which is based on the construction of a hierarchy of coupled effective environmental modes that is terminated by coupling the final member of the hierarchy to a Markovian bath. For an arbitrary environment, which is linearly coupled to the subsystem, the discretized spectral density is replaced by a series of approximate spectral densities involving an increasing number of effective modes. This series of approximants, which are constructed analytically in this paper, guarantees the accurate representation of the overall system-plus-bath dynamics up to increasing times. The hierarchical structure is manifested in the approximate spectral densities in the form of the imaginary part of a continued fraction similar to Mori theory. The results are described for cases where the hierarchy is truncated at the first-, second-, and third-order level. It is demonstrated that the results generated from a reduced density matrix equation of motion and large dimensional system-plus-bath wavepacket calculations are in excellent agreement. For the reduced density matrix calculations, the system and hierarchy of effective modes are treated explicitly and the effects of the bath on the final member of the hierarchy are described by the Caldeira-Leggett equation and its generalization to zero temperature.

Dynamics of light harvesting in photosynthesis

We review recent theoretical and experimental advances in the elucidation of the dynamics of light harvesting in photosynthesis, focusing on recent theoretical developments in structure-based modeling of electronic excitations in photosynthetic complexes and critically examining theoretical models for excitation energy transfer. We then briefly describe two-dimensional electronic spectroscopy and its application to the study of photosynthetic complexes, in particular the Fenna-Matthews-Olson complex from green sulfur bacteria. This review emphasizes recent experimental observations of long-lasting quantum coherence in photosynthetic systems and the implications of quantum coherence in natural photosynthesis.

Narrow band exciton coupled with acoustical anharmonic phonons: application to the vibrational energy flow in a lattice of H-bonded peptide units

A time-convolutionless master equation is established for describing the transport properties of amide-I vibrons coupled with acoustic phonons in a lattice of H-bonded peptide units. Within the non-adiabatic weak coupling limit, it is shown that the vibron dynamics strongly depends on the nature of the phonons and two distinct mechanisms have been identified. Harmonic phonons, which support spatial correlations over an infinite length scale, induce a fast dephasing-rephasing mechanism in the short time limit. Consequently, the vibron keeps its wavelike nature and a coherent vibrational energy flow takes place whatever the temperature. By contrast, anharmonic phonons carry spatial correlations over a finite length scale, only. As a result, the rephasing process no longer compensates the dephasing mechanism so that dephasing-limited band motion occurs. It gives rise to the incoherent diffusion of the vibron characterized by a diffusion coefficient whose temperature dependence scales as 1/T(α). In the weak anharmonicity limit, the exponent α is about 2. It becomes smaller than unity in the strong anharmonicity limit, indicating that the diffusion coefficient behaves as a slowly decaying function of the temperature.

Amide-I lifetime-limited vibrational energy flow in a one-dimensional lattice of hydrogen-bonded peptide units

A time-convolutionless master equation is established for describing the amide-I vibrational energy flow in a lattice of H-bonded peptide units. The dynamics is addressed within the small polaron formalism to account for the strong coupling between the amide-I vibron and the phonons describing the H-bond vibrations. Therefore, special attention is paid to characterize the influence of the amide-I relaxation on the polaron transport properties. This relaxation is modeled by assuming that each amide-I mode interacts with a bath of intramolecular normal modes whose displacements are strongly localized on the C=O groups. It has been shown that the energy relaxation occurs over a very short time scale which prevents any significant delocalization of the polaron. At biological temperature, the polaron explores a finite region around the excited site whose size is about one or two lattice parameters. However, two regimes occur depending on whether the vibron-phonon coupling is weak or strong. For a weak coupling, the energy propagates coherently along the lattice until the polaron disappears. By contrast, for a strong coupling, a diffusive regime occurs so that the polaron explores a finite size region incoherently. In both cases, the finite polaron lifetime favors the localization of the vibron density whose amplitude decreases exponentially.

Photoinduced Vibrational Coherence Transfer in Molecular Dimers

At short times that are faster than dephasing, photoinduced evolution of the vibrational subsystem in an electron-phonon molecular structure depends strongly on the electronic evolution. As the electronic population shifts between the donor and acceptor states, in the diabatic description the state with the largest population determines the equilibrium positions and frequencies of the vibrational modes, which oscillate continuously and without loss of coherence. The vibrational coherence transfer between the electronic states detected recently in a number of systems is described theoretically by application of the quantized Hamiltonian dynamics (QHD) formalism [J. Chem. Phys. 2000, 113, 6557] to the coupled electronic and vibrational degrees of freedom of a model heterodimer. The observed coherent modulation of the frequency of the probe signal is represented with simple analytic and numeric QHD models.

On the effects of PufX on the absorption properties of the light-harvesting complexes of Rhodobacter sphaeroides

Some species of purple bacteria as, e.g., Rhodobacter sphaeroides contain the protein PufX. Concurrently, the light harvesting complexes 1 (LH1) form dimers of open rings. In mutants without PufX, the LH1s are closed rings and photosynthesis breaks down, because the ubiquinone exchange at the reaction center is blocked. However, the main purpose of the LH1 is light harvesting. We therefore investigate the effects that the PufX-induced dimerization has on the absorption properties of the core complexes. Calculations with a dipole model, which compare the photosynthetic efficiency of various configurations of monomeric and dimeric core complexes, show that the dimer can absorb photons directly into the reaction centers more efficiently, but that the performance of the more sophisticated dimeric LH1 antenna degrades faster with structural perturbations. The calculations predict an optimal orientation of the reaction centers relative to the LH1 dimer, which agrees well with the experimentally found configuration. Based on experimental observations indicating that the dimeric core complexes are indeed rather rigid, we hypothesize that in PufX(+) species the association between the LH1 and the reaction centers is enhanced. This mechanical stabilization of the core complexes would lead to the observed quinone blockage, when PufX is missing.

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.

Two-site realization of the Davydov model in a finite size lattice: a time-convolutionless master equation approach

A two-site realization of the Davydov model is introduced to study the energy flow between two amide-I modes embedded in a finite size lattice of hydrogen-bounded peptide units. The non-Markovian nature of the energy transfer is addressed by using a time convolutionless master equation for the population difference of quanta between the two sites of the dimer. It is shown that both the lattice size and the dimer position discriminate between two dynamical regimes. For specific values of these parameters, the population difference shows damped oscillations. It decreases exponentially and rapidly vanishes, which indicates that the equilibrium corresponds to a uniform energy distribution over the two sites of the dimer. By contrast, for other specific values of the lattice size and dimer position, a slowdown of the decoherence takes place. The population difference does not decay exponentially but evolves by steps during which the damping of the oscillations is very small. In addition, the occurrence of revivals characterizing an amplification of the coherence over a finite time scale is observed. Nevertheless, both the decoherence slowdown and the revivals are limited by pure dephasing so that the population difference finally vanishes, but after a rather large coherent time.

Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory

Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.