Extracting the Excitonic Hamiltonian of the Fenna-Matthews-Olson Complex Using Three-Dimensional Third-Order Electronic Spectroscopy

Abstract

We extend traditional two-dimensional (2D) electronic spectroscopy into a third Fourier dimension without the use of additional optical interactions. By acquiring a set of 2D spectra evenly spaced in waiting time and dividing the area of the spectra into voxels, we can eliminate population dynamics from the data and transform the waiting time dimension into frequency space. The resultant 3D spectrum resolves quantum beating signals arising from excitonic coherences along the waiting frequency dimension, thereby yielding up to 2n-fold redundancy in the set of frequencies necessary to construct a complete set of n excitonic transition energies. Using this technique, we have obtained, to our knowledge, the first fully experimental set of electronic eigenstates for the Fenna-Matthews-Olson (FMO) antenna complex, which can be used to improve theoretical simulations of energy transfer within this protein. Whereas the strong diagonal peaks in the 2D rephasing spectrum of the FMO complex obscure all but one of the crosspeaks at 77 K, extending into the third dimension resolves 19 individual peaks. Analysis of the independently collected nonrephasing data provides the same information, thereby verifying the calculated excitonic transition energies. These results enable one to calculate the Hamiltonian of the FMO complex in the site basis by fitting to the experimental linear absorption spectrum.

Introduction

Photosynthetic antennae absorb sunlight and transfer this energy to the reaction center with near-perfect quantum efficiency (1). The mechanism by which these antenna complexes operate with such high efficiency remains unclear, however, and in recent studies investigators have sought to understand and model energy transfer dynamics in biological systems (2–9). The Fenna-Matthews-Olson (FMO) complex from green sulfur bacteria serves as a model complex for such studies (10–15) because it contains only seven electronically coupled bacteriochlorophyll-a chromophores, a relatively small number of pigments for this class of complexes (1). Although FMO contains identical chromophores, electrostatic interactions with the protein bath introduce variations in the excitation energies of each individual chromophore (16). The site-basis Hamiltonian for this system contains the seven individual excitation energies along the diagonal and the dipole-dipole coupling terms in the off-diagonal positions. Diagonalization of this Hamiltonian yields a set of eigenenergies that correspond to the eigenstates in the exciton basis (17). The energies of these excitons correspond to transitions that are observable in a linear absorption spectrum, but unfortunately only three peaks can be resolved in the linear absorption spectrum of FMO at 77 K due to homogeneous and inhomogeneous broadening. Consequently, the energies of the seven excitons have never been experimentally measured, although a great deal of theoretical work has been done to accurately model this system (2,3,10,16,18–24). Simulations of energy transfer within FMO therefore rely on approximate values obtained by fitting theoretical simulations to spectra obtained experimentally (5–8,14,18,24).

The first attempt to calculate the excitonic Hamiltonian of FMO was made shortly after the structure of the complex was solved by Fenna and Matthews in 1975 (25). Using the extended dipole method with atomic coordinates obtained from the FMO crystallographic data, Pearlstein and Hemenger (16) calculated interchromophoric coupling energies but did not obtain a good qualitative fit to the experimental linear absorption spectrum. They concluded that the local environment of each chromophore must produce a different bathochromic shift, and thus the problem of calculating the excitonic Hamiltonian requires not only the dipole coupling energies but also the individual site-basis transition energies. Gudowska-Nowak et al. (19) obtained a set of site-basis energies from quantum calculations by taking into account the conformational deviations of the chlorin rings observed in the crystal structure, as well as interactions with axial ligands and nonbonded neighboring residues, but the resulting absorption spectrum was significantly broader than the experimental spectrum. Gülen (20) and Vulto and co-workers (22,23) sought to calculate the electronic structure of FMO by fitting simulations to experimental linear absorption, linear and circular dichroism, and triplet-minus-singlet spectra, and Cho and co-workers (2) were able to further refine these calculations by fitting to two-dimensional (2D) photon echo spectra. Subsequently, Adolphs and Renger (10) calculated electrochromic shifts using crystallographic data by considering electrostatic interactions with neighboring residues, and found good agreement between this site-basis Hamiltonian and one obtained by fitting experimental spectra with the use of a genetic algorithm.

Whereas first-order spectroscopic techniques such as linear absorption, linear dichroism, and circular dichroism probe individual excitonic dipole strengths, third-order techniques such as 2D electronic spectroscopy probe both auto- and cross-correlation functions of the transition dipoles (26). In 2D photon echo spectroscopy, cross correlations generate peaks both above and below the diagonal (crosspeaks) at positions corresponding to the frequencies of the two coupled excitons (2). Resolving the linear absorption spectrum into a second dimension can therefore facilitate the assignment of transition energies in congested spectra. As in a linear absorption experiment, however, homogeneous and inhomogeneous broadening often thwarts attempts to resolve individual crosspeaks in the 2D spectra of such systems because nearby peaks overlap.

This problem can largely be overcome by resolving a third-order spectrum into another frequency dimension to separate quantum beating signals that arise from interexcitonic zero-quantum coherences. In a four-wave mixing experiment, such as 2D photon echo spectroscopy, three laser pulses interact perturbatively with a sample to generate a signal pulse in a unique phase-matched direction (26). The time delays between pulses 1 and 2 (the coherence time τ) and pulses 2 and 3 (the waiting time T) are controlled experimentally to obtain two time dimensions. The signal pulse, emitted after the rephasing time t, is frequency- and phase-resolved by means of spectral interferometry (27,28). The first time delay is generally incremented in even steps to permit Fourier transformation of the coherence time dimension into frequency space, yielding a set of 2D frequency-frequency (ω_τ-ω_t) spectra with temporal resolution in T. The evolution of spectral features in T provides information on the population and coherence dynamics of the system (11). Interference between response pathways gives rise to the amplitude of a signal peak in these spectra: if the phase of one of these pathways oscillates in T because of the unitary evolution of a zero-quantum coherence, the magnitude of the interference will oscillate, giving rise to a quantum beat (29). This beating therefore provides the spectroscopic signature of a quantum coherence. Accordingly, transformation of the waiting time dimension into frequency space gives a 3D spectrum with peaks corresponding to coherences that appear at positions in waiting frequency equal to the coherence difference frequencies.

This third-order technique was previously reported by Turner et al. (30) in a study of the nondegenerate light-hole and heavy-hole excitons in GaAs quantum wells. The crosspeaks in a 2D spectrum of this confined system arise from response pathways in which the system is in a zero-quantum coherence during the waiting time, as well as pathways in which the system is in the ground state during the waiting time (sequential absorption). By resolving the electronic spectrum into a third dimension, Turner et al. (30) were able to separate the previously overlapping signals from these response pathways into distinct peaks. The authors also anticipated the application of this technique to the study of light-harvesting complexes. Fifth-order 3D spectroscopy techniques have also been developed for use in the infrared (31) and optical (32) regimes. Because these techniques involve five interactions with a laser field, they can access much more information than their corresponding third-order techniques, including higher lying vibrational states and three-body correlations.

A complete set of the frequency coordinates of the 3D third-order peaks for a system with n excitons provides 2n-fold redundancy for constructing an accurate set of exciton transition energies. Calhoun et al. (33) recently used a similar strategy to deconstruct the absorption spectrum of the LHC II antenna complex by transforming points along the diagonal of a set of nonrephasing 2D spectra to acquire a 2D difference frequency versus exciton frequency spectrum. Here, we apply the full transformation to the 2D electronic spectra of FMO to obtain a fully experimental set of electronic eigenstates for the complex and use this information to calculate the Hamiltonian in the site basis.

Theory

The density operator ρ(t) for an ensemble described by a statistical mixture of systems with probabilities P_n and corresponding electronic wave functions |ψ_n(t)〉 written in the excitonic basis |ϕ_j〉 is given by

$$\begin{array}{c}\mathit{\rho }\left(t\right)={\sum }_n{P}_n|{\mathit{\psi }}_n\left(t\right)\mathrm{\rangle \langle }{\mathit{\psi }}_n\left(t\right)|,\\ |{\mathit{\psi }}_n\left(t\right)\mathrm{\rangle }={\sum }_j{e}^{-i{E}_{j}t/h}{c}_{j,n}|{\mathit{\varphi }}_j\mathrm{\rangle }.\end{array}$$ (1)

The diagonal (population) elements of the density matrix ρ_jj(t) are simply given by the statistically weighted sum of the square magnitudes of the respective expansion coefficients:

$${\mathit{\rho }}_{jj}\left(t\right)=\mathrm{\langle }{\mathit{\varphi }}_j\left|\mathit{\rho }\left(t\right)\right|{\mathit{\varphi }}_j\mathrm{\rangle }={\sum }_n{P}_n|c_{j,n}{|}^2.$$ (2)

The off-diagonal (coherence) elements ρ_jk(t), on the other hand, contain both the stationary phases ϕ_jk,n between the basis functions |ϕ_j〉 and |ϕ_k〉 that comprise the superposition states |ψ_n(t) 〉 and the time-dependent phases that arise from the energy dependence of the time propagator:

$${\mathit{\rho }}_{jk}\left(t\right)=\mathrm{\langle }{\mathit{\varphi }}_j\left|\mathit{\rho }\left(t\right)\right|{\mathit{\varphi }}_k\mathrm{\rangle }={\sum }_n{P}_n{c}_{j,n}^{\ast }{c}_{k,n}{e}^{-i\left(E_j-E_k\right)t/h}={\sum }_n{P}_n\left|c_{j,n}\right|c_{k,n}|e^{-i{\mathrm{\omega }}_{jk}t+{\mathit{\varphi }}_{jk,n}}.$$ (3)

The time-dependent phase evolves with a frequency ω_jk equal to the difference between the respective population excitation frequencies, and this evolution results in the spectroscopically observable beating signature of quantum coherences (29). Preparing the ensemble by excitation with a coherent light source fixes the value of ϕ_jk for all n, thereby eliminating destructive interference between individual systems in the ensemble.

Four-wave mixing experiments probe the third-order polarization P⁽³⁾(t) of the system (26). One could greatly simplify the theoretical calculation of the third-order polarization, which is independent of the bath degrees of freedom, by performing a partial trace over the bath degrees of freedom and using a reduced equation of motion for the system density operator; however, we present no such calculations here. Therefore, the purely illustrative treatment that follows involves the complete system-bath density matrix ρ for the sake of simplicity. The total time-dependent polarization, which is equal to the expectation value of the dipole operator V, can be written in terms of the density operator directly from the definition of ρ:

$$P\left(t\right)=\langle V\left(t\right)\rangle =\text{Tr}\left[V\mathit{\rho }\left(t\right)\right]$$ (4)

Performing a perturbative expansion on both the polarization and the density operator yields an expression for the third-order polarization:

$$P^{\left(3\right)}\left(t\right)=\text{Tr}\left[V{\mathit{\rho }}^{\left(3\right)}\left(t\right)\right].$$ (5)

Finding the third-order density matrix elements necessary to calculate P⁽³⁾ requires an equation of motion for the density operator. Just as the dynamics of a wave function are given by the time-dependent Schrödinger equation, the time evolution of the density operator is governed by the Liouville-von Neumann equation:

$$\dot{\rho }=-\frac{i}{h}\left[\mathit{H}\left(t\right),\mathit{\rho }\left(t\right)\right].$$ (6)

Separating H into a system Hamiltonian H₀ and a perturbative interaction with the radiation field H_int, and again performing an expansion on ρ gives the nth-order perturbation to the Liouville-von Neumann equation:

$$\dot{\rho }{}^{\left(n\right)}\left(t\right)=-\frac{i}{h}\left(\left[{\mathit{H}}_0,\mathit{\rho }{}^{\left(n\right)}\left(t\right)\right]+\left[{\mathit{H}}_{\text{int}}\left(t\right),\mathit{\rho }{}^{\left(n-1\right)}\left(t\right)\right]\right).$$ (7)

An expression for ρ⁽³⁾ can be obtained from this equation by starting from a stationary ρ⁽⁰⁾ and iteratively integrating three times, giving the final expression a total of 48 terms, called Liouville pathways (26).

These terms correspond to the three radiation fields that interact with either the ket or the bra of the density operator in a particular order. By experimentally controlling the order of the interactions, we measure only a small subset of these response pathways. We further restrict which pathways we observe by employing a noncollinear arrangement for the three pulses and measuring only signal in the k_s = –k₁ + k₂ + k₃ phase-matched direction, where k_n is the wave vector of each pulse. The response pathways can be conveniently represented using double-sided Feynman diagrams as shown in Fig. 1. Diagram A shows a response pathway in which pulse 1 interacts before pulse 2 (τ > 0), leaving the system in a single-quantum coherence that evolves phase at the optical frequency ω_ag during τ. After interacting with pulse 2, the system is in a zero-quantum coherence that evolves phase at a frequency ω_ab during T as given by Eq. 3. Finally the third pulse arrives, returning the system to a single-quantum coherence that evolves phase at a frequency ω_gb ≈ –ω_ag during t. Because the phase of the system evolves in opposite directions during τ and t, the ensemble is said to rephase during t in pathways for which τ > 0. Diagram B illustrates another class of response pathways, called nonrephasing pathways, in which pulse 2 interacts with the system before pulse 1 (τ < 0). In these cases, phase evolution occurs at a frequency ω_ga during both τ and t, leading to a free induction decay.

Figure 1.

Third-order response pathways represented graphically using double-sided Feynman diagrams. Time progresses upward along the axis shown to the left on which the intervals τ, T, and t are indicated. The left and right vertical lines represent the ket and bra of the density operator, respectively, and the curved arrows indicate interactions with a radiation field. The ground state of the system is given by |g〉, and different single-exciton states are given by |e_a〉 and |e_b〉. These two diagrams represent pathways that contribute to the signal in the k_s = –k₁ + k₂ + k₃ phase-matching direction. Diagram A depicts a rephasing pathway in which the coherence and rephasing frequencies are not equal, yielding a crosspeak. Diagram B depicts a nonrephasing pathway in which these frequencies are equal, yielding a peak along the main diagonal. In both cases, the system is in a zero-quantum coherence during T, resulting in a periodic beating of the corresponding signal.

In both of the Liouville pathways shown, the system is in a zero-quantum coherence during the waiting time T, and therefore the amplitude of the corresponding peak beats at the difference frequency ω_ab. Because the phase evolution during the coherence and rephasing times occurs at different frequencies in diagram A, this rephasing pathway gives rise to crosspeaks that appear off the main diagonal in 2D spectra. In diagram B, however, the coherence and rephasing frequencies are equal, and these nonrephasing pathways therefore give rise to diagonal peaks (29). By acquiring the signal from rephasing and nonrephasing pathways separately, we can analyze the beating in crosspeaks and diagonal peaks independently.

Materials and Methods

The FMO sample was isolated from Chlorobium tepidum as described previously (34) and solubilized in 800 mM tris/HCl buffer (pH 8.0) with 0.1% laurydimethylamine oxide detergent. This solution was mixed 35:65 v/v with glycerol and loaded into quartz cell with a path length of 200 μm (Starna, Atascadero, CA). The cell was cooled in a cryostat (Oxford Instruments, Abingdon, UK) to 77 K. The optical density of the sample at 809 nm was measured to be 0.32.

The output of a self mode-locking Ti:Sapphire oscillator (Coherent Micra, Santa Clara, CA) was regeneratively amplified (Coherent Legend Elite) to produce 38 fs pulses with a spectral bandwidth of 35 nm centered at 806 nm at a repetition rate of 5.0 kHz. The beam was split with the use of a 50:50 beam splitter (CVI Melles Griot, Albuquerque, NM), and a retroreflector (PLX, Deer Park, NY) mounted on a translation stage (Aerotech, Pittsburgh, PA) was used to introduce a time delay (the waiting time T) between the beams. The two beams were focused onto a diffractive optic (Holoeye Photonics, Berlin, Germany) to generate two pairs of phase-locked pulses arranged in a box geometry. Beams 1 and 2 were directed through different pairs of 1° fused silica wedges (Almaz Optics, Marlton, NJ) mounted on translation stages (Aerotech) to introduce another time delay (the coherence time τ). The local oscillator beam was attenuated by means of absorptive neutral density filters with a total optical density of 3.1 at 809 nm. The four beams were focused to a spot size of ∼70 μm diameter in the sample, and the total power incident on the sample was 4.8 nJ (1.6 nJ per pulse). The local oscillator beam was aligned in a 0.3 m spectrometer (Andor Shamrock, Belfast, Northern Ireland) and focused on a 1600 × 5 pixel region centered on a thermoelectrically cooled, back-illuminated CCD (Andor Newton). Calibration of the delay stages by spectral interferometry was performed as described previously (28,35).

Spectra were acquired by stepping the coherence time from −500 to 500 fs in steps of 4 fs for each fixed waiting time. The waiting time was stepped from 0 to 1800 fs in steps of 20 fs, and multiple spectra taken at T = 0 were acquired throughout the data acquisition to monitor sample integrity. Although the pump-probe signal was also acquired at each waiting time to permit separation of the 2D signal into real (absorptive) and imaginary (dispersive) parts, here we used the absolute value of the signal to avoid introducing beating artifacts as a result of phase errors. Scatter subtraction, Fourier windowing, and transformation to frequency-frequency space were performed as described elsewhere (28).

Because the signal at any point in the 2D spectrum decays during the waiting time due to population relaxation, this decay must be removed before the frequency-time-frequency data are transformed. The coherence and rephasing frequency axes of each 2D spectrum were divided into 120 regions between 11,900 cm⁻¹ and 12,800 cm⁻¹, giving a total of 14,400 voxels. The signal amplitude inside each voxel was integrated at each waiting time, and the decay of the signal in T was fit to a sum of three exponential terms. The fit was subtracted at each point to obtain a residual 3D data set. Because the signal decays most rapidly within the first few hundred femtoseconds, removal of the exponential fit often introduces artificial beating signals at early waiting times. Accordingly, only the residual data from T = 500–1500 fs were used to generate the 3D spectrum. Fortunately, the protein provides significant protection against decoherence, and the unusually long lifetimes of the zero-quantum coherences permit this analysis strategy (4,12). The cropped 3D data set was zero-padded on either side to yield a total of 700 T points, and the data points were apodized with a Blackman window before each voxel was transformed into frequency space to obtain the final frequency-frequency-frequency spectrum. This process was performed separately on both the rephasing (τ > 0) and nonrephasing (τ < 0) data.

Results

The 3D rephasing spectrum of FMO is shown in Fig. 2. This representation facilitates the identification and isolation of peaks from which a set of (ω_τ,ω_T,ω_t) coordinates can be deduced. Because the amplitudes of the crosspeaks beat in the rephasing signal, this spectrum could theoretically provide sevenfold redundancy in this set of coordinates: although only a complete set of the crosspeaks associated with one particular exciton are necessary to construct a complete set of transition energies from difference frequencies and a single absolute transition frequency, here we have the crosspeaks associated with all seven excitons. Of course, sevenfold redundancy represents a theoretical upper bound, and we do not observe all 42 crosspeaks in our experimental spectrum. We do, however, resolve 19 crosspeaks, specifically the 1-2, 2-4, 2-5, 3-6, 4-2, 4-3, 4-7, 5-2, 5-3, 5-6, 5-7, 6-2, 6-4, 6-5, 6-7, 7-3, 7-4, and 7-6 crosspeaks (where the first and second indices represent the excitons corresponding to the observed rephasing and coherence frequencies, respectively). This is a marked improvement over the frequency-time-frequency data, in which we can only resolve a single crosspeak that in fact is the sum of two overlapping crosspeaks (13).

Figure 2.

(a) 3D third-order electronic spectrum of FMO. Features in the spectrum are presented in both color and opacity scales to permit visualization of the entire data set. (b) 2D spectrum extracted from the 3D spectrum by cutting through the solid at a fixed waiting frequency. As expected, the spectrum shows features broadened along the diagonal that appear above and below the diagonal. These features correspond to positions in the spectrum where the absolute value of the difference between the coherence frequency (x axis) and the rephasing frequency (y axis) is equal to the fixed waiting frequency. (c) 2D spectrum taken at a fixed rephasing frequency. A series of spectra of the latter type corresponding to each exciton are shown in Fig. 3.

The 2D spectra extracted from the 3D spectrum in Fig. 2 represent cuts through the full data set taken at particular values of ω_T and ω_t. A set of ω_τ versus ω_T spectra taken at values of ω_t equal to the energies of the seven excitons deduced from the set of crosspeaks is shown in panels a–g of Fig. 3. The lines overlaid on the spectra are not statistical fits; rather, they are guidelines corresponding to positions at which the absolute value of the difference between the coherence frequency (y axis) and the fixed rephasing frequency is equal to the waiting frequency (x axis). Most of the major peaks clearly fall along these lines, as expected. The peak that appears off the line in the spectrum corresponding to exciton 1 can be traced upward through the rephasing frequency dimension, showing that it is simply the tail of the strong 2-5 peak visible in the exciton 2 spectrum. We attribute the stray peaks in the exciton 7 spectrum to noise arising from the weakness of the signal at the blue edge of the excitation pulse bandwidth. The peaks in the spectra of excitons 2, 3, and 4 for which ω_τ = ω_t and ω_T = 160 cm⁻¹ likely include some contribution from the tail of the 1-2 peak, but we believe the large amplitudes of these ostensibly anomalous features may indicate strong coupling between coherence and population elements of the density matrix as predicted by Palmieri and co-workers (36).

Figure 3.

(a–g) 2D spectra taken from the 3D spectrum at fixed values of rephasing frequency corresponding to each of the seven excitons. Each spectrum is normalized to its respective maximum. The guidelines overlaid on the spectra correspond to positions at which the absolute value of the difference between the coherence frequency (y axis) and the fixed rephasing frequency is equal to the waiting frequency (x axis). A total of 19 crosspeaks appear along these lines. In panel h, points along the main diagonal of the nonrephasing data set are plotted against waiting frequency. The dotted lines correspond to the diagonal frequencies of the seven excitons. The peaks in this spectrum provide additional redundancy for calculating the transition energies of the system. The same slices taken from a simple 3D Gaussian dressed stick model are presented in panels i–p for comparison.

Additional redundancy can be obtained to reduce the error associated with each transition energy by analyzing the nonrephasing data. Because the amplitudes of the diagonal peaks beat in the nonrephasing signal, only the peaks along the main diagonal must be considered. Panel h of Fig. 3 shows a 2D spectrum obtained by cutting through the 3D nonrephasing spectrum along the diagonal (ω_τ = ω_t). Each diagonal peak should contain beating contributions from all six zero-quantum coherences associated with that particular exciton, so this 2D slice also theoretically provides sevenfold redundancy. The coordinates obtained from this spectrum were used along with those obtained from the rephasing 3D spectrum to construct the set of seven exciton transition energies for FMO consistent with the full set of difference frequencies. These values are given in Table 1. Our results are consistent with those of Cho et al. (2) but significantly red-shifted from those reported by Adolphs and Renger (10).

Table 1.

Experimental exciton transition energies for FMO (in cm⁻¹)

Exciton	Energy ± SD
1	12,121 ± 15
2	12,274 ± 20
3	12,350 ± 16
4	12,415 ± 12
5	12,454 ± 17
6	12,520 ± 19
7	12,606 ± 25

With the exception of homodyne peaks (ω_τ = ω_t, ω_T = 0) and peaks in the rephasing spectrum attributed to population-coherence coupling (ω_τ = ω_t, ω_T ≠ 0), all fully resolved peaks (local maxima) in both the 3D rephasing spectrum and the diagonal slice of the 3D nonrephasing spectrum with amplitudes > 5% of the respective global maxima were used in the calculation of exciton transition energies. Because the position of peaks in the coherence and rephasing frequency dimensions can shift significantly due to interference from neighboring peaks, the waiting frequency was taken to be the most accurate coordinate for each peak. This dimension also has the finest frequency resolution, a consequence of the binning required in the coherence and rephasing dimensions before the transformation into waiting frequency space. We generated one set of frequencies from the set of rephasing peaks by taking the coherence frequencies to be equal to the values given by the spectral coordinates and adding (ω_τ < ω_t) or subtracting (ω_τ > ω_t) the waiting frequencies, and similarly generated a second set by taking the rephasing frequencies from the spectral coordinates. We then generated a final set from the nonrephasing data by adding and subtracting the waiting frequencies to the diagonal frequencies and discarding values < 12,100 cm⁻¹ or > 12,700 cm⁻¹ (in each of these cases, the other value obtained from that peak fell within these bounds). These frequencies were arranged numerically and separated into seven groups according to the largest gaps between successive frequencies. The weighted mean of each group was calculated with the amplitudes of the peaks used as statistical weights. These calculated frequencies were then used to generate a new set of frequencies as before by adding or subtracting the appropriate waiting frequencies to these values. Again, a set of mean frequencies was calculated, and the process was iterated until a self-consistent set of transition energies was obtained.

To verify the origin of the observed crosspeaks, we generated very simple rephasing and nonrephasing 3D spectra analogous to Gaussian dressed stick spectra from the experimental transition energies. Orientationally averaged integrated peak amplitudes were calculated as

$$A_{jk}=|{\mathit{\mu }}_j{|}^2\mathrm{\cdot |}{\mathit{\mu }}_{\mathit{\kappa }}{|}^2\mathrm{\cdot }\frac{1}{\mathrm{30}}\left(2+4{\cos }^2{\mathit{\theta }}_{jk}\right),$$ (8)

where θ_jk is the angle between the dipoles of excitons j and k, and the orientational factor is taken from Hochstrasser for parallel laser fields (37). The frequency-time-frequency rephasing spectrum was calculated according to

$$\begin{array}{l}C\left({\mathit{\omega }}_{\mathit{\tau }},T,{\mathit{\omega }}_t\right)=\sum _{j=1}^7\sum _{k=1}^7{A}_{jk}\mathrm{\cdot }\sqrt{S\left({\mathit{\omega }}_{\mathit{\tau }}\right)}\mathrm{\cdot }\sqrt{S\left({\mathit{\omega }}_t\right)}\mathrm{\cdot }\sqrt{S\left({\mathit{\omega }}_{\mathit{\tau }}\right)}\mathrm{\cdot }\\ \exp \left(\frac{-\left[{\left({\mathit{\omega }}_{\mathit{\tau }}-{\mathit{\omega }}_j\right)}^2+{\left({\mathit{\omega }}_t-{\mathit{\omega }}_k\right)}^2\right]}{2{\mathit{\sigma }}^{\mathit{2}}}-2\mathit{\pi }i\mathrm{\cdot }\left[\left({\mathit{\omega }}_j-{\mathit{\omega }}_k\right)\mathrm{\cdot }T+\mathit{\pi }\right]-\frac{T}{1000}\right),\end{array}$$ (9)

where S(ω) is the laser power spectrum and σ is the variance of the lineshape, chosen to be 60 cm⁻¹ for all peaks. The penultimate term causes the peak amplitudes to beat at the appropriate difference frequency with the arbitrary initial phase of π, whereas the final term sets a uniform dephasing rate of 33 cm⁻¹. This spectrum was then cropped, windowed, zero-padded, and transformed to frequency space as described above for the residual experimental spectrum. The function for generating the nonrephasing spectrum was similarly calculated according to

$$\begin{array}{l}C\left(\mathit{\omega ,}T\right)=\sum _{j=1}^7\sum _{k=1}^7{A}_{jk}\mathrm{\cdot }\sqrt{S\left(\mathit{\omega }\right)}\mathrm{\cdot }\sqrt{S\left({\mathit{\omega }}_k\right)}\mathrm{\cdot }\\ \exp \left(\frac{-{\left(\mathit{\omega }-{\mathit{\omega }}_j\right)}^2}{2{\mathit{\sigma }}^{\mathit{2}}}-2\mathit{\pi }i\mathrm{\cdot }\left[\left({\mathit{\omega }}_j-{\mathit{\omega }}_k\right)\mathrm{\cdot }T+\mathit{\pi }\right]-\frac{T}{1000}\right)\end{array}$$ (10)

Here we assign the frequency of the second and third interactions to the exciton transition energy. The 2D slices from these model spectra are shown in Fig. 3, i–p. The data and model spectra are clearly in good qualitative agreement, although one clear inconsistency is the strong 1-5 peak in the model exciton 1 rephasing spectrum, which does not appear in the data. The model employs a single dephasing rate for all coherences, however, and therefore may overrepresent some crosspeaks.

The site-basis Hamiltonian and linear absorption spectrum were obtained indirectly from the experimentally determined excitonic Hamiltonian (H_ex) and a complete set of dipole-dipole coupling terms. These coupling terms were calculated within the point dipole approximation from crystallographic data (38) according to the familiar equation:

$$V_{j\mathrm{\ne }k}=s\left(R_{jk}\right)\frac{1}{R_{jk}^2}\left[{\stackrel{\rightharpoonup }{\mu }}_j\mathrm{\cdot }{\stackrel{\rightharpoonup }{\mu }}_{\mathit{\kappa }}-3\left({\stackrel{\rightharpoonup }{\mu }}_j\mathrm{\cdot }{\widehat{\text{e}}}_{jk}\right)\left({\stackrel{\rightharpoonup }{\mu }}_k\mathrm{\cdot }{\widehat{\text{e}}}_{jk}\right)\right],$$ (11)

where s(R_jk) is a scaling factor that corrects for the screening of the Coulomb interaction by the dielectric of the protein environment, R_jk is the distance between the centers of dipoles j and k, and ê_jk is the unit vector between the centers of the dipoles. Here we use the exponential form of s(R_jk) calculated by Scholes et al. (39) for electronically coupled chromophores embedded in a protein dielectric. We assume that the Q_y transition of bacteriochlorophyll-a has a vacuum dipole strength of 37.1 D² (10,40) and is oriented between the nitrogen atoms of rings I and III but rotated slightly toward ring V (Fischer nomenclature). Although Adolphs and Renger (10) found the best agreement between their calculated and experimental circular dichroism spectra when the dipoles were rotated by 7°, they stated that the exact value of this angle is uncertain. Georgakopoulou et al. (41,42) reported rotation angles between −10° and 13° for the bacteriochlorophyll-a chromophores in the different rings of the LH1 (41) and LH2 (42) complexes, demonstrating that the orientation of the transition dipole is strongly dependent on the local environment. Here we use a value of 9°, which gives the best agreement between our calculated and experimental linear and 3D spectra. We constructed an initial guess for the site-basis Hamiltonian $\left(H_{\text{site}}^{\left(0\right)}\right)$ from these coupling terms and the set of theoretical site energies given by Adolphs and Renger (10). Diagonalization of this matrix yielded the transformation matrix U⁽⁰⁾ and a set of exciton transition energies inconsistent with H_ex:

$$U^{\left(n\right)-1}{H}_{\text{site}}^{\left(n\right)}{U}^{\left(n\right)}=H_{\text{ex}}^{\left(n\right)\prime }\mathrm{\ne }{H}_{\text{ex}}.$$ (12)

By transforming H_ex to the site basis using this U matrix, we obtained a set of site energies that we then used to construct a new guess for H_site, again using the calculated off-diagonal terms:

$$U^{\left(n\right)}{H}_{\text{ex}}{U}^{\left(n\right)-1}=H_{\text{site}}^{\left(n+1\right)\prime }$$ (13)

$$H_{\text{site}}^{\left(n+1\right)}=\text{diag}\left(H_{\text{site}}^{\left(n+1\right)\prime }\right)+V.$$ (14)

This process was iterated until $H_{\text{ex}}^{\left(n\right)\prime }$ converged to a set of energies equal to those given by H_ex. The final site-basis Hamiltonian is given in Table 2.

Table 2.

Site-basis Hamiltonian for FMO determined by genetic fit (in cm⁻¹)

	Site 1	Site 2	Site 3	Site 4	Site 5	Site 6	Site 7
Site 1	12,468	−53	5	−4	4	−6	−5
Site 2	−53	12,466	17	6	1	6	5
Site 3	5	17	12,129	−38	−3	−7	25
Site 4	−4	6	−38	12,410	−60	−8	−48
Site 5	4	1	−3	−60	12,320	33	−8
Site 6	−6	6	−7	−8	33	12,593	38
Site 7	−5	5	25	−48	−8	38	12,353

In the case that H_ex and V are precisely known, this problem is equivalent to finding the roots of a seventh-order polynomial and therefore has a single analytic solution. Due to the uncertainty in the fixed parameters, however, this system of equations does not have an exact solution, and the site energies obtained by this method are exquisitely sensitive to the initial guess for H_site. To address this problem, we employed a genetic algorithm to optimize the initial guesses for the seven site energies by optimizing the fit of the linear absorption spectrum calculated from the final H_site to the experimentally measured spectrum. The only condition enforced in the optimization was restricting the lowest-energy site to be site 3 in accordance with previous theoretical and, most recently, experimental results (43). We took the initial site energies from Adolphs and Renger (10) and generated 50 children from this parent by randomly selecting each site energy from a normal distribution centered about each parent value with a variance of 100 cm⁻¹. The 20 best children were selected as parents for the next generation, and this process was repeated until no further improvements were observed for five consecutive generations (15 generations total). We spawned the next generation using a normal distribution with a variance of 30 cm⁻¹, and repeated the process until no further improvements were observed (10 generations). The values were further refined with variances of 15 cm⁻¹, 5 cm⁻¹, and 2 cm⁻¹. The absorption spectrum, shown in Fig. 4, was calculated as the sum of Gaussian peaks with transition dipoles strengths and line widths obtained from the transformation matrix U as described previously (2).

Figure 4.

Experimental (solid blue line) and calculated (dashed red line) linear absorption spectra of FMO.

Discussion

We resolved 19 crosspeaks in this 3D rephasing data set, improving on the single crosspeak observed in the 2D spectrum of FMO. The 3D spectrum provides sufficient information to construct an accurate equilibrium set of electronic eigenstates. A single experimental improvement, however, could provide a much more complete set of crosspeaks. The ultrashort pulses used in this experiment had a bandwidth of 35 nm (FWHM) centered at 806 nm, meaning that the power at 824 nm was less than half that at 806 nm. Consequently, most of the crosspeaks involving the lowest-energy exciton could not be resolved. Increasing the bandwidth of the pulses or red-shifting the center frequency would solve this problem, and future experiments will address this issue. The exciton 1 crosspeaks are especially valuable because they are well separated from the main diagonal and thus should not contain significant contributions from the tails of neighboring peaks.

Factors besides pulse bandwidth, however, must account for the other unresolved crosspeaks. The simplest explanation is that particular pairs of excitons are not strongly electronically coupled. This argument is supported by the fact that most of the crosspeaks observed in the 3D spectrum involve pairs of excitons with small energy gaps. A comparison of the experimental and model 3D spectra (Fig. 2 and Supporting Material, respectively) shows that features in the upper third of the model spectrum are not observed in the data. Because the excitons are spatially delocalized across the complex, the strength of electronic coupling between excitons is largely determined by energetic (rather than spatial) overlap. Finally, it is possible that some coherences simply dephase too quickly to contribute a sufficiently long beating signal to their respective crosspeaks. We previously measured the lifetimes of the 1-2 and 1-3 coherences at 77 K to be 1100 and 700 fs, respectively (13), but localized fluctuations within the complex could cause other coherences to dephase more rapidly.

Although both the rephasing 3D spectrum and nonrephasing 2D diagonal spectrum theoretically contain the same information, analyzing the nonrephasing data is a less robust method. Each diagonal peak in the nonrephasing data contains the sum of six beating signals convolved with some amount of experimental noise. The rephasing beating signals, on the other hand, are spatially separated in the 2D spectra, and each is actually convolved with less experimental noise because the photon echo signal obtained from the rephasing pathways is significantly stronger than the free induction decay emitted from the nonrephasing pathways. Furthermore, the nonrephasing data are necessarily incomplete for a system with evenly spaced excitons. In FMO, for example, the energy gaps between excitons 2 and 3 and excitons 3 and 4 are approximately equal, and the corresponding peaks will—at best—overlap in waiting frequency. At worst, the two signals will beat out of phase and no peak will be observed. By performing both rephasing and nonrephasing analyses, one can obtain a significantly more complete set of energetic coordinates that will greatly reduce the error associated with each transition energy.

Although the electronic eigenstates reported here were obtained experimentally, the site-basis Hamiltonian was obtained by fitting to the linear absorption spectrum of FMO in the spirit of previous theoretical work. This analysis depends on dipole-dipole coupling terms calculated within the point dipole approximation using atomic coordinates obtained from crystallographic data, which may not be representative of the structure of the complex in solution. This calculation also assumes a homogeneous dielectric environment within the protein and identical transition dipole moments for each chromophore. Nevertheless, the calculated Hamiltonian is consistent with the orientation of the FMO complex within the photosynthetic machinery of C. tepidum determined recently by mass spectrometry (43). The primary antenna complex in green sulfur bacteria is the chlorosome, which absorbs sunlight and transfers excitations through FMO to the reaction center. Wen et al. (43) found that chromophores 1 and 6 neighbor the chlorosome, whereas chromophore 3 abuts the reaction center, suggesting that sites 1 and 6 participate in the highest-energy excitons and site 3 primarily participates in exciton 1. Our results agree with this energetically downhill arrangement.

Conclusions

By resolving a set of 2D electronic rephasing spectra into a third Fourier dimension, we were able to identify a suite of crosspeaks corresponding to interexcitonic coherences within the FMO complex that do not appear in the frequency-time-frequency data. We used the frequency coordinates of the maxima of these crosspeaks to assign transition energies to the seven excitons of the complex, thereby obtaining, to our knowledge, the first fully experimental set of electronic eigenstates for the FMO protein. We further refined these values by analyzing the nonrephasing signal along the main diagonal. We then used the exciton transition energies to find the Hamiltonian for FMO in the site basis by fitting to the linear absorption spectrum. These Hamiltonians can be used to refine previous simulations of the energy transfer process in FMO that have provided valuable information on transfer pathways and transport mechanisms in photosynthetic antennae. The technique reported here, however, provides only a time-averaged Hamiltonian. Although the off-diagonal elements in the excitonic basis must average to zero, the magnitude of the fluctuations of these elements about zero has not been quantified. Furthermore, the need to develop an experimental technique to measure the site-basis Hamiltonian of FMO directly remains a challenge.