Within the conceptual framework of Redfield theory, the optical response function arises from the dynamical evolution of the system’s density operator, where nonunitary relaxation is encoded in the Redfield relaxation superoperator. In the conventional approach, the so-called secular approximation neglects terms that induce transitions between distinct coherences and among coherences and populations. The rationale is that these nonsecular terms are small in comparison to the far more dominant population relaxation and coherence dephasing contributions. Since two-dimensional infrared (2D-IR) spectroscopy has significant contributions arising from population relaxation and transfer pathways, it can be challenging to isolate signatures of the nonsecular relaxation. We report here that in three diiron dithiolate hexacarbonyl complexes that serve as small-molecule models of the [FeFe] hydrogenase H-cluster subsite, a fortuitous vibrational energy structure enables direct and clear signatures of vibrational coherence transfer in alkane solution. This finding holds promise towards developing a molecularly detailed understanding of the mechanism of vibrational coherence transfer processes, thanks to the ease of synthesizing derivatives based on the chemical modularity of these well studied diiron compounds. In addition to the fundamental need to characterize coherence transfer in molecular spectroscopy, we find in this set of molecules a practical utility for the nonsecular dynamics: the ability to determine the frequency of an IR-inactive mode. A coherence generated during the waiting time of the 2D-IR measurement transfers to a coherence involving the single dark CO stretching mode, which modulates some peak amplitudes in the 2D spectrum, revealing its transient excitation.
Coupling and fluctuations drive molecular relaxation processes, and while these dynamics are encoded in linear absorption spectra, the most direct measurements are provided by multidimensional spectroscopy.1–3 Perturbative descriptions of spectroscopic observables are generally successful at reproducing experimental data and offer valuable insight. After two decades, optical two-dimensional (2D) spectroscopy has proven highly effective at measuring energy relaxation and redistribution, energy and charge transfer, as well as solvation dynamics and spectral diffusion. In both electronic and vibrational 2D spectroscopy, excited state absorption is prominently displayed due to its opposite sign relative to the ground state bleach and stimulated emission signals. What unites all of these basic processes is that they can be readily understood within various theoretical frameworks, the most well-known of which is Redfield theory.4 The Redfield equation is the result of a second-order perturbation analysis of the system-bath interaction, which inherently assumes that the interaction is weak,
In Eq. (1), the reduced density matrix element, σjk, evolves according to the instantaneous frequency difference ωjk = ωj − ωk and the relaxation superoperator Γ. The diagonal elements of σ correspond to system state populations, and the off-diagonal elements are coherences between pairs of system states. Population transfer between, for example, states j and k, is mediated by the Γjj,kk and Γkk,jj elements according to detailed balance. The dephasing (i.e., decoherence) of a coherence between states j and k is governed by the Γjk,jk element. Under the so-called secular approximation to Redfield theory, these are the only nonzero elements of the relaxation superoperator. The transfer from one coherence to another, Γjk,lm, is neglected, as is direct transfer between population and coherence (i.e., Γjk,kk). Invoking the secular approximation enables a straightforward interpretation of the relaxation superoperator elements, while greatly simplifying the complexity of computational propagation. Additionally, without the secular approximation, it is possible for the reduced density operator to have negative diagonal elements, which is at odds with the usual view that these elements correspond to system state occupation probabilities.5,6 It is interesting and noteworthy that a new approach, developed by Loring et al., for evaluating semiclassical response functions based on optimized mean-trajectories allows a convenient way to treat coherence transfer without the pathologies of nonsecular Redfield theory.7–9
The cost of making the secular approximation is an artificial diminution of the system-bath interactions and the dynamical independence of coherence and population dynamics.5 Hence, phenomena such as electronic excitation energy transfer and vibrational redistribution and relaxation processes can be incorrectly described. Despite the relatively well-defined nature of both Redfield theory and its secular approximation, the degree to which such an approximation is violated can be challenging to assess. There is, therefore, the possibility for significant insight into first simply identifying violations of secular Redfield and, second, determining the chemical information made available by such a breakdown. As we have previously reported,10,11 and present further evidence here, two-dimensional spectroscopy of the highly coupled carbonyl vibrations in transition metal coordination complexes provides a means both to test and develop the breakdown of secular Redfield theory. Here, we also find a remarkable utility of this breakdown, namely, that we are able to determine the frequency of an infrared (IR) inactive mode by virtue of its participation in coherence transfer among the many CO stretches of several hydrogenase enzyme active site mimics.
Justification for the utility of coordination complexes to serve as general model systems
Though central to all optical absorption spectroscopy, coherence has received somewhat renewed interest due to the many observations of oscillatory features in time-resolved two-dimensional spectroscopy, particularly in photosynthetic systems.12–20 Transition metal coordination complexes, despite having no structural or even length-scale similarity to large photosynthetic proteins, nevertheless have the potential to serve as ideal model systems to test fundamental aspects of quantum dynamics and two-dimensional spectroscopy.21 Although proteins are large, most theoretical models consider primarily the chromophore sites.22 In the Fenna-Matthews-Olson complex, for example, there are seven strongly coupled sites, whose excitonic states have been described using semiempirical Hamiltonians. The Hilbert space for such a model is no larger than a similarly excitonic description of seven coupled vibrational modes. Metal carbonyl complexes have CO stretching vibrations near 2000 cm−1, an order of magnitude lower in frequency than the visible transitions in photosynthetic proteins, but the vibrational modes are split by roughly 50–100 cm−1. Hence, relative to the site energies, metal carbonyl couplings are of the same order as those in photosynthetic proteins and other electronic aggregates.
What transition metal carbonyls lack is the inconvenience of Franck-Condon active lower frequency vibrations, which can complicate analysis of the purely excitonic dynamics. Most transient IR and 2D-IR spectra that report coherent oscillations find frequencies that precisely match the known vibrational mode differences easily obtained from FTIR spectra. Interesting cases have been found, such as the hydrogen bonded acetic acid dimer, where wavepacket oscillation results from the coupling of the high-frequency OH stretches to low-frequency interdimer modes.23,24 In these cases, the one-dimensional IR spectrum exhibits characteristic progressions similar to those seen in UV and visible spectra. Excitonic models for multicarbonyl complexes have been shown to be able to reproduce solvent dependent spectral line shapes using the same Gaussian site disorder approaches that have been widely adopted in modeling electronic aggregates.25 In other words, although there may be no purely exitonic electronic systems, these vibrational complexes are effectively purely excitonic and can be used to identify new signatures of nonsecular dynamics in 2D spectroscopy. Moreover, since coordination complexes can be manipulated chemically, it is possible to alter the geometrical and electronic structure in order to modulate the relaxation dynamics.
Previous 2D-IR studies of vibrational coherence transfer
The concept of vibrational coherence transfer has been invoked to interpret multidimensional IR spectral features, primarily in the dicarbonyl complexes Rh(CO)2(acac) (RDC, acac = acetylacetonato)26 and Ir(CO)2(acac) (IDC).27 In the first 2D-IR investigation of coherence transfer, Khalil et al. observed two peaks that are forbidden since each corresponds to excited state absorption to a mode whose transition dipole moment is orthogonal to the initially excited mode (Fig. 1).26 For example, the dicarbonyl modes are well approximated as the symmetric (s) and antisymmetric (a) CO stretching vibrations, with mutually orthogonal transition dipole moments.28,29 One unexpected peak in the 2D spectrum appears at excitation of the fundamental of a (2015 cm−1), followed by emission at the s-as combination band transition (1989 cm−1). The other peak corresponds to excitation at s (2084 cm−1) and emission from the a-2s transition (2058 cm−1) frequency. One origin of this unexpected peak (Fig. 1) is that excitation of the ρ0s coherence (2084 cm−1) transfers to ρ0a (2015 cm−1) during t1. The second field-matter interaction creates the population ρaa. The third field-matter interaction excites the coherence ρ2a,a (2002 cm−1), which undergoes coherence transfer to ρas,a (2058 cm−1) during t3. The net result is a peak at (ωex, ωdet) = (2084 cm−1, 2058 cm−1). The other unexpected peak arises from swapping the roles of a and s, giving a peak at (ωex, ωdet) = (2015 cm−1, 1989 cm−1). These pathways require that the first two field-matter interactions be resonant with both the 0-a and 0-s transitions. The data were modeled using the Redfield equation with parameters extracted from experiment, with the explicit inclusion of coherence transfer processes.
This picture was revisited by Marroux and Orr-Ewing using pulses spectrally narrowed by a pulse shaper to test for the presence of vibrational coherence transfer in RDC.30 For a pump pulse pair with a spectral width limited to one of the two coupled CO modes of RDC, the pathways proposed as being responsible for the unexpected peaks would not be possible. Nevertheless, the forbidden peaks do appear in each “half-pumped” 2D spectrum. The data were interpreted as arising primarily from population transfer (i.e., intramolecular vibrational redistribution, IVR). Indeed, this is the explanation offered by Baiz et al.10 in their earlier theoretical work on coherence transfer in transition metal complexes. When IVR is very rapid, as in the case of RDC, there will be some amplitude for relaxation-induced cross peaks even at very early time. For an exponential with a 3 ps time constant, by 300 fs, 10% of the relaxation will already have taken place. Since the half-bandwidth pulses were at least 300 fs in duration (the sharp spectral edge also creates temporal ringing oscillations), it is indeed likely that rapid IVR can explain the forbidden peaks. In the transform-limited experiments by Khalil et al., a similar argument would hold since the realistic time resolution of 2D-IR spectroscopy limits the minimum waiting time to 100–300 fs for pulses of ∼200 cm−1 bandwidth.
It is also possible that fifth-order ladder climbing can account for the large signal at early waiting time due to the Morse energy level anharmonicity.31 The transition from the doubly excited symmetric stretch (2s) to the triply excited symmetric stretch (3s) has the right frequency to account for the forbidden peak. The situation is the same for the asymmetric stretch. It is difficult to determine if this fifth-order explanation is more likely than the IVR origin, but we do note that our own detailed measurements on RDC show only a very small magnitude for the forbidden peak.32 Our apparatus generally uses lower laser power and looser focusing conditions than other setups reported in the literature, which would make the overall intensity lower and favor the third-order response over the fifth-order contributions. For the present work, although we do not perform a power dependence due to the limits of signal-to-noise ratios, we do note that fifth-order contributions would lead to unexpected peaks in the 2D spectrum and to coherent oscillations at frequencies set by the vibrational anharmonicities.33 As we do not observe either of these characteristics, we do not attribute the features described below to fifth-order contributions. A brief discussion of the fifth-order scenario is given in the supplementary material.
Several years ago, Nee et al. reported the observation of oscillatory cross peaks in the rephasing 2D-IR spectra of dimanganese dodecacarbonyl’s [Mn2(CO)10, DMDC] carbonyl vibrational modes.11 One cross peak exhibits oscillatory features at not one but two frequencies, both of which correspond to differences of vibrational modes known from the FTIR spectrum. Cross peaks in 2D-IR rephasing spectra are expected to oscillate but at only one frequency, and the presence of a second frequency component is anomalous within the usual secular approximation to Redfield theory often used to model 2D-IR spectra. Nee et al. suggested that the additional frequency component could arise from vibrational coherence transfer during one or more of the experimental delay times.11 In a subsequent theoretical study, Baiz et al. compared the data reported by Nee et al. to 2D-IR spectra of DMDC simulated using Redfield theory but including the possibility for nonsecular terms to contribute.10 When coherence-coherence coupling terms were included in the Redfield model, the same cross peak oscillatory features observed experimentally were found in the simulated 2D spectra, supporting the assignment of multifrequency cross peak oscillations as spectral signatures of coherence transfer.
Unfortunately, DMDC also suffers from an overabundance of spectral symmetry. Only four of the ten carbonyl vibrational modes are IR-active, and two of these are degenerate. DMDC’s carbonyl vibrational spectrum thus consists of only three bands and two difference frequencies. This relative sparsity of IR-active vibrations was sufficient to establish the presence of anomalous cross peak frequency components but inadequate to provide any significant insight into the dynamical phenomena associated with the vibrational coherence transfer. By contrast, the diiron hexacarbonyl systems that we report here are quite different (Fig. 2), having five of their six carbonyl vibrations IR-active and none (in most cases) degenerate. The larger number of vibrational bands and difference frequencies in these systems affords a wealth of experimental information that would be inaccessible in many of the 2D-IR benchmark metal carbonyl complexes. We report the observation of clear patterns of intracarbonyl vibrational coherence transfer, and the first observation of vibrational coherence transfer between bright and dark vibrational modes. This latter finding displays the practical utility of coherence transfer to provide spectral information that is invisible in FTIR spectra or in 2D-IR spectra within the secular approximation to Redfield theory.
To confirm the significance of the waiting time (t2) dependent oscillations, the Fourier transform of each relevant spectral peak was compared to coherence maps. This comparison allows clear identification of frequency components localized at specific peaks, and of frequency components arising from spectral overlap with other peaks, or from those due to trivial experimental noise.
2D-IR pulse sequence and response functions
To investigate the ultrafast coherent dynamics in our compounds, we employ 2D-IR. Two-dimensional infrared spectroscopy in the background-free, noncolinear geometry1,34–36 is carried out using three fields with wavevectors k1, k2, and k3, respectively, giving rise to two signals, rephasing (kR = −k1 + k2 + k3) and nonrephasing (kN = +k1 − k2 + k3). The rephasing and nonrephasing sequences can be performed with the same beam alignment and signal direction but with a swapped timing of k1 and k2. The delay between the first two pulses, t1, known as the “coherence time,” is Fourier transformed to create an excitation frequency axis, and the time delay between the second and third pulses, known as the “waiting time” or t2, provides experimental time resolution. The detection axis is directly read from the spectrometer, with the field being measured by spectral interferometry37 with a time delayed local oscillator reference field derived from the same pulse used for k3. The response of the system to the sequence of experimental field-matter interactions may be written as a sum of four-point correlation functions of the system’s dipole moment μ as in the following equation:35
where the indices i, j, k, and l represent the laboratory-frame polarizations of the respective field and τn represents the time period between field-matter interactions n and n + 1. Assuming that the vibrational and orientational responses of the system are separable, the total response of the system may be written as a product of the two, as shown in the following equation:28,29
where Rνκλχ is the four-point correlation function of the system’s vibrational transition dipole moments ν, κ, λ, and χ with the four field-matter interactions, respectively, and is the orientational response function of that correlation function, which accounts for the effect of orientational diffusion on the response of vibrational transition dipoles ν, κ, λ, and χ to field-matter interactions of i, j, k, and l polarizations, respectively. Note that the measured response of the system will include only pathways with nonvanishing orientational response functions.
By invoking the rotating wave approximation, the vibrational response function defined in Eq. (2) may be expressed as 16 individual four-point correlation functions, six of which are responses to the rephasing pulse sequence and six of which are responses to the nonrephasing pulse sequence.36 These four-point correlation functions are often represented as Feynman diagrams to illustrate the time-dependent evolution of the system’s density matrix, and this sequence of states is often referred to as a Liouville pathway. The Liouville pathways associated with rephasing and nonrephasing pulse sequences are shown in Fig. 3. Liouville pathways are read upwards from the bottom, with the arrows representing field-matter interactions and the state of the density matrix during each waiting written inside the diagram.35 Solid horizontal lines represent field-matter interaction which delineates time intervals, and dashed horizontal lines may be used to represent the occurrence of a coherence transfer during a time interval.
In 2D spectra, the amplitude of peaks that receive contributions from a Liouville pathway with a t2 coherence will oscillate at the difference frequency of the two superposed states. The positions of such peaks vary depending upon the experimental pulse sequence.34 In a nonrephasing sequence, the pathways involving t2 coherences contribute to peaks on the diagonal of the spectrum, and thus, in a time-resolved nonrephasing experiment, each diagonal peak may oscillate at frequencies corresponding to the difference between the excited mode and each of the other modes with which it shares a ground state. In a rephasing sequence, the pathways involving t2 coherences contribute to off-diagonal cross peaks, each of which beats at the difference between the excitation and detection frequencies. Hence, it is not unusual for nonrephasing diagonal peaks to exhibit multiple frequency oscillations in multilevel systems.
All details of experimental and computational methods are standard and are reported in the supplementary material. A breakthrough was made when maps of coherence amplitudes were used to highlight the regions of the 2D spectrum oscillating at a given frequency.38 Borrowing ideas from kinetics maps,39 these coherence maps are cuts through the 3D volume that contains the full third-order response function, Fourier transformed along the waiting time (t2) axis. The specific explanation of the construction of coherence maps is given in the supplementary material. The three different diiron hexacarbonyl complexes used in this study are shown in Fig. 2(a). All three have a diiron hexacarbonyl core and differ in the bridging dithiolate group; two are bridged by organic groups, an ethanedithiolate (edt) and an o-xylyldithiolate (xyl), respectively, while in the third complex, the sulfides are not bridged by any organic group. All data referenced in this study were collected using hexadecane as the solvent. Our previous work has investigated the spectral diffusion dynamics of this class of complexes, finding evidence for intramolecular flexibility in nonpolar solvents.40 The low-frequency skeletal fluctuations of the molecule induce transient spectral inhomogeneity, which undergoes spectral diffusion on a 5–10 ps time scale. Thus, there is a coupling between the high frequency CO stretches and the low frequency modes, of which there are many [∼15 modes in the edt complex below 200 cm−1 according to density functional theory (DFT) calculations]. From a microscopic perspective, coherence transfer is enabled by the mutual coupling of two high-frequency modes to a common bath degree of freedom, where, in this case, a “bath” mode can be any mode not directly excited by the laser fields.
The metal carbonyl vibrational spectrum for each compound is shown in Fig. 2(b). We reference the vibrational modes in descending order as modes 1–5. The vibrational spectra are similar among the three complexes with two exceptions: the vibrational frequencies of the disulfide are blue-shifted several wavenumbers relative to the frequencies of the edt- and o-xylyl- bridged compounds and, in the disulfide, modes 3 and 4 are degenerate.
Sample coherence maps of the edt and xyl compounds are shown in Fig. 3. We refer to vibrational peaks in our spectra by their excitation and detection frequencies according to the mode numbering scheme introduced in Fig. 2. For example, the cross peak at (ω1 = mode 1, ω3 = mode 2) will be denoted as (1, 2). Figure 4(a) shows the 71–73 cm−1 frequency range in the rephasing spectrum of xyl. It has amplitude at the (1, 4) cross peak and its conjugate (4, 1), as expected. Figure 4(b) shows the 36–39 cm−1 frequency range in the rephasing spectrum of xyl. That frequency is expected for the (1, 2) cross peak and its conjugate but is not expected for the (1, 4) cross peak and its conjugate. Figure 4(c) shows the 80–83 cm−1 frequency range components of the nonrephasing edt spectrum. The 80–83 cm−1 frequency is localized on (1, 1) and (4, 4) diagonal peaks, as expected. Figure 4(d) shows the 28–31 cm−1 oscillatory components of the nonrephasing edt spectrum. This frequency is expected for the (2, 2) and (3, 3) diagonal peaks but not for the (1, 4) cross peak. Many additional data sets are provided in the supplementary material; we focus on the rationale, interpretation, and significance of these nonsecular dynamics in the following.
Quantum chemical calculations predict an additional dark vibrational mode appearing in frequency between modes 4 and 5 in all three compounds. In (μ-S)2, modes 3 and 4 are degenerate. The vibrational frequencies of (μ-S)2 are blue shifted by several wavenumbers from the corresponding vibrational frequencies in the other two compounds, and we attribute this shift to the electron-donating nature of the bridging alkyl and aryl groups. Carbon monoxide ligands bond to transition metals in two modes, but the most spectroscopically significant bonding mode is π-backbonding, where the metal atom’s d-orbitals donate electron density into the carbon monoxide’s π*-antibonding molecular orbital, causing an increase in the carbon-oxygen bond length and a decrease in the vibrational frequency.41,42 Alkyl and aryl groups are electron-donating, and the red shift in carbonyl vibrational frequency in the two compounds with bridging ligands is due to increased electron density in the diiron core of the complex. The computed atomic displacements of each carbonyl vibrational mode are provided in the supplementary material and are highly delocalized, allowing the carbonyl vibrations to be viewed as “vibrational excitons,” as has previously been done for DMDC.25
The main features of the 2D-IR spectra are unremarkable and reflect the expected coupling among the IR bands readily measured using FTIR. Somewhat more surprising is the significant degree of spectral diffusion that we have previously reported. This spectral diffusion indicates significant coupling between the high-frequency CO stretches and lower-frequency skeletal distortions, such as the torsional turnstile motion.40 The principal observation of this work that we analyze in detail is the coherent modulation of the cross peaks in the 2D spectra. Although cross peaks in 2D-IR spectra are expected to beat at single frequencies in rephasing spectra and not to beat at all in nonrephasing spectra, our data show clear frequency components on some of the nonrephasing cross peaks and multiple frequency components on some of the rephasing cross peaks. The anomalous coherences in each of our compounds are distinct and unique from each of the others—for example, as shown in Fig. S7, the (2, 5) cross peak in edt has an anomalous frequency component of ∼29 cm−1, corresponding to the (2, 3) difference frequency. This is unique to edt; we do not observe this anomalous frequency component in either xyldt or (μ-S)2. However, we do observe specific anomalous frequencies that are consistently present in all three compounds, and we devote the remainder of this work to the discussion of these consistently anomalous cross peak frequencies. The spectral locations of cross peaks with the same anomalous frequency components in all three compounds are shown in Fig. 5(a) for both the rephasing and nonrephasing spectra. All three of the compounds show multiple frequencies on the (1, 4) cross peaks and their conjugates in the rephasing spectra, and all three show an oscillatory frequency on the (1, 2), (2, 3), and (2, 4) cross peaks and their conjugates in the nonrephasing spectra.
The most prominent unexpected oscillatory features in the rephasing spectra are on the (1, 4) cross peak and its conjugate. In all three of our compounds, the (1, 4) cross peak shows an additional oscillatory frequency that matches the (2, 4) difference frequency, and in the xyldt, an additional frequency component corresponds to the (1, 5) difference frequency.
In both xyldt and (μ-S)2, an ∼87 cm−1 frequency component is apparent on the (4, 1) cross peak and its conjugate which does not correspond to the difference frequency of any two bright vibrational modes in the system. The 87 cm−1 frequency component in the spectrum of xyldt is shown in Figs. 5(b) and 5(c). This is the only frequency we have observed in our data set that does not correspond to a difference frequency of any two bright vibrational modes, and we consider below several potential sources for this feature.
One potential source is a modulation of the (1, 4) coherence by a low frequency molecular vibrational mode. Our DFT calculations do indicate that there are several vibrational modes in our compounds with ∼90 cm−1 frequencies. However, it is unclear to us why only one metal carbonyl coherence, the (1, 4) coherence, would be noticeably modulated by a low-frequency structural vibration. Another possibility is that this anomalous frequency component arises from an interaction with the dark carbonyl vibrational mode. A pronounced IR-Raman noncoincidence effect43 inhibits an identification of the dark mode’s frequency using Raman spectroscopy, but quantum chemistry calculations place it between modes 4 and 5, which have difference frequencies of ∼78 cm−1 and ∼93 cm−1, respectively. This assignment corresponds well with the observed frequency of ∼87 cm−1 and is consistent with our observation that all of the other anomalous frequency components we observe correspond directly with the difference frequencies of other carbonyl vibrational modes. We thus propose that the frequency of ∼87 cm−1 which we observe on the (1, 4) cross peak and its conjugate in xyldt and (μ-S)2 arises from vibrational coherence transfer between mode 4 and the dark mode during one or more of our experimental waiting times. To the best of our knowledge, this is the first use of coherence transfer to identify and investigate otherwise inaccessible dark modes using 2D-IR spectroscopy.
The most consistent anomaly in our rephasing data is a frequency of ∼40 cm−1 on the (4, 1) cross peak and its conjugate. In xyldt, the (1, 2) and (2, 4) difference frequencies are indistinguishable and both match the anomalous frequency component on the (4, 1) cross peak, but in edt and (μ-S)2, the (2, 4) difference frequency most closely matches the anomalous frequency component of the (4, 1) cross peak and its conjugate. In the two bridged compounds we examine, edt and xyldt, the frequency difference between the excited state absorption and ground state bleach of the (4, 1) cross peak is too small to allow unambiguous assignment, but in (μ-S)2 the frequency difference is great enough to assign the ∼37 cm−1 feature to the excited state absorption of the cross peak. The consistent presence and significant amplitude of the (2, 4) frequency on the (4, 1) cross peak and its conjugate suggest that a specific set of coherence transfer pathways are active. Several potential Liouville pathways—described below—may be written to account for this feature, including coherence-coherence and population-coherence transfer. In the case of the nonrephasing spectra, the cross peaks between modes 1 and 2 oscillate at the (1, 2) frequency in all three data sets. The same features are observed on the cross peaks between modes 2 and 3 and modes 2 and 4.
Our data present both a challenge and opportunity for the theoretical investigation of excitonic coherence transfer. To determine the potential pathways for coherence transfer within our system, we evaluate the constraint imposed on the system’s response function by the orientational contribution. The nonlinear response of a system to the pulse sequence used in 2D spectroscopy is modulated by the orientational response of the system; only specific sequences of field-matter interactions will contribute, as shown in Eq. (3). The sequences of field-matter interactions which contribute to the nonlinear response in a 2D-IR experiment are well known: for two orthogonal transition dipole moments a and b, the sequences of field-matter interactions aaaa, aabb, abab, and abba have a nonvanishing orientational response.29 Previous investigations of vibrational coherence transfer in 2D spectroscopy have focused primarily upon systems with two orthogonal transition dipole moments. This is not the case in the diiron hexacarbonyl complexes we study here, which have five allowed metal carbonyl transition dipole moments, with 1∥4 ⊥ 2∥5 ⊥ 3. To determine the orientational response of our system, we evaluated the orientational response function defined by Golonzka et al. (details are presented in the supplementary material).29 We find that only sequences of field-matter interactions with an even number of terms for each transition dipole moment contribute to the measured response of the system. Thus, Liouville pathways with the 1223 sequence of field-matter interactions will not contribute to the response but the Liouville pathways with the 1224 sequence of field-matter interactions may because modes 1 and 4 have parallel transition dipole moments.
With the constraints of a nonvanishing orientational response and having no more than one coherence transfer event during each time interval, Liouville pathways may be written that correspond to the unexpected frequency components we observe. Typically, two coherence transfer events are necessary to write a Liouville pathway with a nonvanishing orientational response function which could give rise to our observations. This scenario is well exemplified in the case of the 87 cm−1 frequency which we attribute to coherence transfer with the dark mode. Since the mode is dark, its coherence must be introduced and removed from the system’s density matrix without invoking the field-matter interactions of the experimental pulse sequence. We provide example pathways for the consistent anomalous frequencies we observe in our data in Fig. 6. It is clear that further work is necessary to elucidate the dynamics of coherence transfer we observe and why we observe it only on specific peaks and at specific frequencies. Ideally, we would be able to take advantage of anharmonic frequency calculations to explain the propensities for coherence transfer. Although we have performed VPT2 calculations for the edt and related pdt complexes, it is simply not apparent how to link those results to our experimental observations, largely due to the high degree of coupling among all of the modes.
Like coherence-coherence transfer, coherence-population coupling terms are typically neglected in interpreted 2D-IR spectra within the secular Redfield framework. However, like coherence-coherence transfer, coherence-population transfer dynamics have the potential to significantly alter spectra and neglecting coherence-population couplings may impact the fitting of spectral features.10,26,44,45 In addition to being a fundamental aspect of spectroscopy, coherence-population transfer is hypothesized to contribute noticeably to the light-harvesting mechanism in the fragment molecular orbital (FMO) complex and other biological light-harvesting assemblies.12,19 In the context of biological light-harvesting, excitonic coherence-population transfer is sometimes referred to as “quantum transport.”46 When comparing their simulated DMDC 2D-IR spectra to the experimental data, Baiz et al. concluded that coherence-population coupling contributes negligibly to the anomalous cross peak frequency in DMDC.10 However, many of the anomalous frequencies that we observe in our data arise from Liouville pathways which include coherence-population coupling.
The Fourier spectra of the coherent oscillations observed here contain information that we do not explicitly extract in this initial report. Either in the time or frequency domains, comparing the dephasing of the intraband coherences can reveal more information about the nature of the “secular” dephasing as well as the nonsecular coherence transfer processes. There is some hint in our data that the coherence transfer-induced oscillations exhibit faster dephasing based on comparing the widths of the Fourier spectrum features. Both in the present case and in the previous work on DMDC,11 as well as in numerous other transition metal carbonyl systems we have investigated,21 we do typically find that the intraband coherent oscillations dephase on similar time scales to the fundamental transitions, determined by comparing Fourier transformed waiting time oscillations to the widths in the FTIR spectra. With this diiron system, however, we have many more bands with which to observe coherence transfer, so future work will analyze the comparative dephasing in detail. We do note, in keeping with the spirit of multidimensional spectroscopy, that our coherence Fourier spectra are purely one-dimensional and thus cannot distinguish homogeneous and inhomogeneous contributions to the intraband coherences. Such information would only be obtainable from three-dimensional infrared spectroscopy of the coherences.47–49
Why do intraband coherences show a greater propensity to reveal coherence transfer?
Our earlier work on DMDC showed, albeit much more subtly than in the present examples, that waiting-time coherences expose coherence transfer.10,11 If coherence transfer is as prevalent as our data would indicate, why are signatures of coherence transfer not more readily identified in other 2D-IR spectra? Or phrased differently, we can ask why the intraband coherences seem to be needed to identify evidence of coherence transfer. In alternative versions of multidimensional spectroscopy developed by the Wright group,50,51 specifically those spectra that arise from fully coherent pathways, there are many clear indications of coherence transfer. 2D-IR has both kinds of pathways, depending on the molecular vibrational structure and the pulses used in the experiment. In Fourier transform 2D-IR spectroscopy, using pulses that are broad enough in frequency to simultaneously excite multiple vibrations produces intraband coherences among those modes during the waiting time. That oscillatory signal corresponds to a pathway that does not evolve as a population during the waiting time (Fig. 3). The double-resonance, pump-probe implementation of 2D-IR,52 on the other hand, generally does not produce such pathways because the pump bandwidth is narrow enough not to excite multiple modes. By isolating and analyzing separately the coherent response, we are effectively filtering out the population-visiting pathways, which evidently exposes the coherence transfer. A similar strategy has effectively been employed by Chuntonov and Ma in applying the methods of quantum process tomography (QPT) in a coupled diketone molecule using two different polarization conditions to isolate the coherence transfer contribution.45 The QPT approach is diagnostic, but it is not clear whether it facilitates a molecular interpretation for the origin of the coherence transfer.
We report what appear to be marked signatures of vibrational coherence transfer in three diiron hexacarbonyl organometallic complexes. The signatures of vibrational coherence transfer are the anomalous frequency oscillations of specific but ordinary peaks in the 2D-IR spectrum. Different and complementary signatures of coherence transfer are observed in the rephasing and nonrephasing 2D-IR spectra, including what appears to be coherence transfer between bright and dark vibrational modes. This finding offers a practical utility of coherence transfer in 2D-IR to probe spectroscopically dark modes. Our data represent a new challenge for detailed theoretical understanding, and further work is necessary to elucidate the set of conditions that predispose these compounds to coherence transfer dynamics and that appear to facilitate specific coherence transfer events. Future experimental studies on additional diiron derivatives as well as dicobalt hexacarbonyl structural analogs are currently underway. Since these complexes are important components of catalysis, electrocatalysis, and bio-orthogonal chemistry, understanding the detailed vibrational dynamics will enhance our ability to track energy flow and structural dynamics during chemical transformations. The ability to make chemical modifications in order to explore their influence on coherence transfer promises a new route to testing the widely employed secular approximation to Redfield theory.
Online supplementary material contains the following: (1) Details of experimental methods, including synthesis of the compounds studied, a description of the 2D-IR spectrometer, and the method used to produce the coherence maps; (2) vibrational mode assignments; (3) a discussion of the orientational response function appropriate for the symmetry and transition dipole directions of the modes in the three complexes; (4) various additional coherence maps for the three different complexes studied; (5) tables of the difference frequencies (among the carbonyl vibrational modes); and (6) a discussion of potential fifth-order contributions for rhodium dicarbonyl.
This work was supported by the National Science Foundation (Grant No. CHE-1565795).