Theoretical Studies of Energy Transport in Complex Systems
Abstract
Photosynthesis involves the absorption of photons by lightharvesting pigments and the subsequent transfer of excitation from the absorption centre to the reaction centre. This highly eﬃcient phenomenon of excitation transfer has traditionally been explained by the Forster mechanism of incoherent hopping of excitation from one chromophore to another. Recently 2D electronic spectroscopic evidences were gathered by Fleming and coworkers on the photosynthetic FennaMatthewsOlson (FMO) complex in green sulfur bacteria [1]. Subsequent simulation studies by the same group [2] led to the proposition of a quantummechanical, coherent, wavelike transfer of excitation among the chromophores. However, Fleming's conclusions regarding retention of coherence appeared surprising because, the complex would interact with the numerous degrees of freedom of the protein scaﬀold surrounding it, leading to decoherence, which is expected to be rapid. Thus, we were interested in proposing an analytical treatment to rationalize the excitation transfer.
Traditional approaches employed for studying excitation energy transfer involve the master equation techniques where the systembath coupling is perturbative and is truncated after a few orders. It is important to note that the systembath coupling causes both decoherence and population relaxation. Such a perturbative approximation is diﬃcult to justify for the photosystem, as the systembath coupling and the interchromophoric electronic coupling have comparable values. Also, these treatments are largely numerical studies and demand involved calculations. Thus, exact calculations for such a system (7level) are very diﬃcult. Consequently, we were interested in developing an analytical approach where the coupling is treated as nonperturbative. We devised a novel analytical treatment which employs a unitary transformation analogous to the one used for the theory of nonadiabatic eﬀects in chemical reactions [3]. Our treatment rests on an adiabatic basis which are eigenstates calculated at each nuclear position (i.e. at each configuration of the bath) bearing a parametric dependence in Qi, where Qi denotes the shift of the exciton at site `i' due to the environment. The treatment is justified because in the case of coherent transfer, the excitation would travel mostly amongst the adiabatic states and the eﬀects of nonadiabaticity are small.
We observed that the systembath coupling, after the unitary transformation, could be decoupled at the lowest order into two parts: a) an adiabatic contribution, which accounts solely for decoherence (this is evaluated almost exactly in our approach) and b) a nonadiabatic contribution which accounts for population relaxation from one adiabatic state to another (treated by a Markovian master equation). When we applied our technique to the FMO complex, our prediction for population evolution at the chromophores showed excellent correspondence with those obtained by Nalbach and coworkers using pathintegral calculations [4], which are exact. These were calculations where the environment was modelled using a Drude spectral density. Our method allowed the calculations to be readily performed for diﬀerent temperatures as well. It should be specifically emphasized that, unlike the involved and cumbersome pathintegral calculations by Nalbach and coworkers [4] or the hierarchical equation calculations by Ishizaki et al. [2], our method is simple, easy to apply and computationally expedient. Further it became evident that the ultraeﬃciency of energy transfer in photosynthetic complexes is not completely captured by coherence alone but is the result of an interplay of coherence and the dissipative influence of the environment (also known as ENAQT or Environment Assisted Quantum Transport [5]).
An added advantage of our analytical treatment was the flexibility it oﬀered. Thus, we could use our formalism to perform expedient analyses on the behavior of the system under various conditions. For example, we may wish to evaluate the consequences of introducing correlations among the bath degrees of freedom on the eﬃciency of transfer to the reaction centre. To this end, we applied our formalism by introducing correlations among the bath degrees of freedom and then by introducing anticorrelations among the bath degrees of freedom. The conclusions were interesting, for they suggested that the eﬃciency of transfer to the reaction centre was enhanced by the presence of anticorrelations, when compared with an uncorrelated bath. Uncorrelated baths, in turn, had a higher eﬃciency of energy transfer than correlated baths [6]. Thus, the population evolution is fastest for the anticorrelated bath, followed by the uncorrelated bath and is slowest for the correlated bath. Similar conclusions have been reached at by Tiwari et al. [7].
We could also extend the formalism for studying the system under diﬀerent spectral densities for the environment, apart from just the Drude spectral density which is popularly used in literature associated with FMO calculations. For instance, the FMO system could be analyzed for the AdolphsRenger spectral density [3, 8]. Once again our results showed excellent agreement with those reported by Nalbach. We also analyzed the FMO system under the spectral density proposed by Kleinekathofer and coworkers [9]. It was found that these latter spectral densities had more profound participation from the environment, therefore coherences were destroyed more eﬀectively and population relaxation was faster. The excitation transfer to the final site (site closest to the reaction centre in the FMO complex) was found to be faster for the Adolphs and Renger spectral density and the spectral density proposed by Kleinekathofer and coworkers, when compared to the Drude spectral density. Also, the excitation transfer was fastest when we modelled the environment using the Kleinekathofer spectral density. This reinforced the previous conclusions that the dissipative eﬀects of the environment promote a faster energy transport.
Being an almost analytical approach, our technique could be applied to systems with larger number of levels as well. A good example of such a case is the MEHPPV polymer. 2D electronicspectroscopic experiments performed on this polymer in solution speculate that the excitation energy transfer might be coherent even at physiological temperatures [10]. A prototype for studying this system might be a conjugated polymer with around 80100 chromophores.
Linewidths and Lineshapes in the vicinity of Graphene
It has been reported that a vibrating dipole may deexcite by transferring energy nonradiatively to a neighboring metal surface [11]. It is also understood that due to its delocalized picloud, graphene has a continuum of energy states and can behave like a metal sheet and accept energies. Thus, we proposed that if a vibrationally excited dipole deexcites in the vicinity of a graphene sheet, graphene may get electronically excited and thus serve as an eﬀective quencher for such vibrational excitations. Depending on the distance of the dipole from the graphene sheet, the transfer might be intense enough to be spectroscopically probed. We have investigated the rate of such an energy transfer.
We use the Dirac cone approximation for graphene, as this enables us to obtain analytical results. The Fermi Golden rule was used to evaluate the rate of energy transfer from the excited dipole to the graphene sheet [12]. The calculations were performed for both the instances: a) energy transfer from a dipole to undoped graphene and, b) energy transfer from a dipole to doped graphene. For undoped graphene, the carrier (electron) charge density in the conduction band is zero and we would only have transitions from the valence band to the conduction band. As a consequence of absence of carrier charge density in CB (conduction band), the screening of Coulombic interactions in the graphene plane is ineﬀective. Thus, one could use the noninteracting polarizability for undoped graphene in the rate expression [13]. However, when we consider the case of doped graphene where EF is shifted upwards into CB, the conduction band electrons will contribute to screening. In this case, we have two sets of transitions: a) from ki in VB (valence band) to kf in CB and b) ki in CB to kf in CB, where ki and kf are the wavevectors which correspond to the initial and final electronic states in graphene. So we have used the polarizability propagator in the random phase approximation [14] to calculate the rate following the approach of [13].
It is also known that the imaginary part of the frequency domain dipoledipole correlation function is a measure of the lineshape [15]. We were, thus, interested in evaluating the lineshape for these transitions. For evaluating the correlation function, we used the partitioning technique developed by L•owdin [16] and subsequently extracted the lineshape from its imaginary part. Using this method, we calculated lineshape for the vibrational excitation of CO molecule in the vicinity of an undoped graphene lattice. The linewidth for this system also was obtained. It could be seen that the vibrational linewidth for 1 CO in the vicinity (5 A) of undoped graphene (EF = 0:00eV ) is small (0:012 cm ) but could be observed experimentally. The lineshape calculations were also extended to cases where it is possible to have atomic transitions by placing an electronically excited atom in the vicinity of the graphene sheet. We considered the following two cases: a) 3p ! 2s transition in hydrogen atom, at a distance of 12 A from the graphene sheet and, b) 4p ! 3s transition in hydrogen atom, at a distance of 20 A from the graphene sheet. The linewidths for atomic transitions could be easily probed in these cases ( 55 cm 1 for 3p ! 2s and 56 cm 1 for 4p ! 3s). In the preceding calculations, the transition dipoles were considered perpendicular to the graphene surface. It is worthwhile to note that if the transition dipoles are considered parallel to the graphene surface, the respective linewidths would be half of those obtained for the case where the transition dipoles are perpendicular. Another interesting possibility would be to consider a lanthanide metal complex placed within a few nanometers from graphene. Lanthanides are known to have sharp ff transitions [17] and consequently, one could easily observe the eﬀects of broadening due to energy transfer to the electronic system of graphene.
Energy Eigenmodes for arrays of Metal Nanoparticles
In the final part of the thesis we consider organized assemblies of metal nanoparticles, specifically helical and cylindrical assemblies and investigate the plasmonic excitation transfer across these assemblies. These were motivated by recent studies which reported growth of chiral asymmetric assemblies of nanoparticles on D and L isomers of diphenylalanine peptide nanotubes [18]. The plasmons in the helical/cylindrical assemblies are expected to couple with each other via electromagnetic interactions. We construct the Hamiltonian for such systems and evaluate the eigenmodes and energies pertaining to these modes in the wave vector space. We also perform calculations for the group velocity for each eigenmode as this gives us an idea of which eigenmode transports excitation the fastest.
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