Frequency Fluctuation Correlation Function Research Articles

Measuring the Ultrafast Correlation Dynamics of a Multilevel System Using the Center Line Slope Analysis in Two-Dimensional Electronic Spectroscopy.

In a two-dimensional (2D) optical spectrum of a multilevel system, there are diagonal peaks and off-diagonal cross-peaks that correlate the different levels. The time-dependent properties of these diagonal peaks and cross-peaks contain much information about the dynamics of the multilevel system. The time-dependent diagonal peakshape that depends on the spectral diffusion dynamics of the associated transition and characterized by the frequency-fluctuation correlation function (FFCF) is well studied. However, the time-dependent peakshape of a cross-peak that provides the correlation dynamics between different transitions is much less studied or understood. We derived the third-order nonlinear response functions that describe the cross-peaks in a 2D electronic spectrum of a multilevel system that arise from processes sharing a common ground state and/or from internal conversion and population transfer. We can use the center line slope (CLS) analysis to characterize the cross-peaks in conjunction with the diagonal peaks. This allows us to recover the frequency-fluctuation cross-correlation functions (FXCFs) between two transitions. The FXCF and its subsidiary quantities such as the initial correlation and the initial covariance between different transitions are important for studying the correlation effects between states in complex systems, such as energy-transfer processes. Furthermore, knowledge of how various molecular processes over different timescales affect simultaneously different transitions can also be obtained from the measured FXCF. We validated and tested our derived equations and analysis process by studying, as an example, the 2D electronic spectra of metal-free phthalocyanine in solution. We measured and analyzed the diagonal peaks of the Qx and Qy transitions and the cross-peaks between these two transitions of this multilevel electronic system and obtained the associated FFCFs and FXCFs. In this model system, we measured negative components of FXCF over the tens of picosecond timescale. This suggests that in phthalocyanine, the Qx and Qy transitions coupling with the solvent molecule motion are anticorrelated to each other.

Site-selective dynamics of ligand-free and ligand-bound azidolysozyme.

Azido-modified alanine residues (AlaN3) are environment-sensitive, minimally invasive infrared probes for the site-specific investigation of protein structure and dynamics. Here, the capability of the label is investigated to query whether or not a ligand is bound to the active site of lysozyme and how the spectroscopy and dynamics change upon ligand binding. The results demonstrate specific differences for center frequencies of the asymmetric azide stretch vibration, the longtime decay, and the static offset of the frequency fluctuation correlation function (FFCF)-all of which are experimental observables-between the ligand-free and the ligand-bound N3-labeled protein. The center-frequency shifts range from 1 to 8 cm-1, which is detectable from state-of-the art experiments. Similarly, the nonvanishing static component Δ0 of the FFCF between ligand-free and ligand-bound protein can differ by up to a factor of 2.5. This makes the azide label a versatile and structurally sensitive probe to report on the dynamics of proteins in a variety of environments and for a range of different applications. Ligand-induced differences in the dynamics are also mapped onto changes in the local and through-space coupling between residues by virtue of dynamical cross correlation maps. This demonstrates that the position where the label is placed also influences the local and global protein motions.

Multipolar Force Fields for Amide-I Spectroscopy from Conformational Dynamics of the Alanine Trimer.

The dynamics and spectroscopy of N-methyl-acetamide (NMA) and trialanine in solution are characterized from molecular dynamics simulations using different energy functions, including a conventional point charge (PC)-based force field, one based on a multipolar (MTP) representation of the electrostatics, and a semiempirical DFT method. For the 1D infrared spectra, the frequency splitting between the two amide-I groups is 10 cm-1 from the PC, 13 cm-1 from the MTP, and 47 cm-1 from self-consistent charge density functional tight-binding (SCC-DFTB) simulations, compared with 25 cm-1 from experiment. The frequency trajectory required for the frequency fluctuation correlation function (FFCF) is determined from individual normal mode (INM) and full normal mode (FNM) analyses of the amide-I vibrations. The spectroscopy, time-zero magnitude of the FFCF C(t = 0), and the static component Δ02 from simulations using MTP and analysis based on FNM are all consistent with experiments for (Ala)3. Contrary to this, for the analysis excluding mode-mode coupling (INM), the FFCF decays to zero too rapidly and for simulations with a PC-based force field, the Δ02 is too small by a factor of two compared with experiments. Simulations with SCC-DFTB agree better with experiment for these observables than those from PC-based simulations. The conformational ensemble sampled from simulations using PCs is consistent with the literature (including PII, β, αR, and αL), whereas that covered by the MTP-based simulations is dominated by PII with some contributions from β and αR. This agrees with and confirms recently reported Bayesian-refined populations based on 1D infrared experiments. FNM analysis together with a MTP representation provides a meaningful model to correctly describe the dynamics of hydrated trialanine.

Direct comparison of amplitude and geometric measures of spectral inhomogeneity using phase-cycled 2D-IR spectroscopy.

Two-dimensional infrared (2D-IR) spectroscopy provides access to equilibrium dynamics with the extraction of the frequency-fluctuation correlation function (FFCF) from the measured spectra. Several different methods of obtaining the FFCF from experimental spectra, such as the center line slope (CLS), ellipticity, phase slope, and nodal line slope, all depend on the geometrical nature of the 2D line shape and necessarily require spectral extent in order to achieve a measure of the FFCF. Amplitude measures, on the other hand, such as the inhomogeneity index, rely only on signal amplitudes and can, in principle, be computed using just a single point in a 2D spectrum. With a pulse shaper-based 2D-IR spectrometer, in conjunction with phase cycling, we separate the rephasing and nonrephasing signals used to determine the inhomogeneity index. The same measured data provide the absorptive spectrum, needed for the CLS. Both methods are applied to two model molecular systems: tungsten hexacarbonyl (WCO6) and methylcyclopentadienyl manganese tricarbonyl [Cp'Mn(CO)3, MCMT]. The three degenerate IR modes of W(CO)6 lack coherent modulation or noticeable intramolecular vibrational redistribution (IVR) and are used to establish a baseline comparison. The two bands of the MCMT tripod complex include intraband coherences and IVR as well as likely internal torsional motion on a few-picosecond time scale. We find essentially identical spectral diffusion, but faster, non-equilibrium dynamics lead to differences in the FFCFs extracted with the two methods. The inhomogeneity index offers an advantage in cases where spectra are complex and energy transfer can mimic line shape changes due to frequency fluctuations.

Site-selective dynamics of azidolysozyme.

The spectroscopic response of and structural dynamics around all azido-modified alanine residues (AlaN3) in lysozyme are characterized. It is found that AlaN3 is a positionally sensitive probe for the local dynamics, covering a frequency range of ∼15 cm-1 for the center frequency of the line shape. This is consistent with findings from selective replacements of amino acids in PDZ2, which reported a frequency span of ∼10 cm-1 for replacements of Val, Ala, or Glu by azidohomoalanine. For the frequency fluctuation correlation functions, the long-time decay constants τ2 range from ∼1 to ∼10 ps, which compares with experimentally measured correlation times of 3 ps. Attaching azide to alanine residues can yield dynamics that decays to zero on the few ps time scale (i.e., static component Δ0 ∼ 0 ps-1) or to a remaining, static contribution of ∼0.5 ps-1 (corresponding to 2.5 cm-1), depending on the local environment on the 10 ps time scale. The magnitude of the static component correlates qualitatively with the degree of hydration of the spectroscopic probe. Although attaching azide to alanine residues is found to be structurally minimally invasive with respect to the overall protein structure, analysis of the local hydrophobicity indicates that the hydration around the modification site differs for modified and unmodified alanine residues, respectively.

Anomalous Temperature Dependence of Exciton Spectral Diffusion in Tetracene Thin Film.

In this work, an ultrafast spectral diffusion of the lowest exciton in a tetracene ultrathin film is studied by two-dimensional electronic spectroscopy. From the analysis of the nodal line slope, the frequency-fluctuation correlation function (FFCF) of the exciton band is extracted. The FFCF contains two components with decay times of 400 and 80 fs; while the former can be understood by a linear exciton-phonon coupling model, the latter shows an order of magnitude increase in its amplitude from 96 to 186 K that cannot be explained by the same model. A novel scheme of the energy-gap fluctuations is examined, in which an intramolecular high-frequency mode causes the spectral diffusion that is enhanced through an anharmonic coupling to low-frequency phonon modes. This finding provides a valuable input for future theoretical predictions on the ultrafast nonadiabatic dynamics of the molecular exciton.

Measuring Ultrafast Spectral Diffusion and Correlation Dynamics by Two-Dimensional Electronic Spectroscopy.

The frequency fluctuation correlation function (FFCF) measures the spectral diffusion of a state's transition while the frequency fluctuation cross-correlation function (FXCF) measures the correlation dynamics between the transitions of two separate states. These quantities contain a wealth of information on how the chromophores or excitonic states interact and couple with its environment and with each other. We summarize the experimental implementations and theoretical considerations of using two-dimensional electronic spectroscopy to characterize FFCFs and FXCFs. Applications can be found in systems such as the chlorophyll pigment molecules in light-harvesting complexes and CdSe nanomaterials.

Studying the spectral diffusion dynamics of chlorophyll a and chlorophyll b using two-dimensional electronic spectroscopy

Chlorophyll a (Chl a) and Chlorophyll b (Chl b) are major pigments in the photosynthetic machinery in green plants. Despite their similarities in chemical structures, the relaxation dynamics as observed in the ultrafast methods, such as pump probe and multidimensional spectroscopy, demonstrate a distinct kinetics. Here, we employ the ultrafast two-dimensional electronic spectroscopy (2DES) method to characterize the frequency fluctuation correlation function (FFCF) for Chl a and Chl b in various solvent environments, through the Centre Line Slope (CLS) method. We observe that the FFCF decay for both Chl a and Chl b in the methanol solvent environment has a ~40 ps component that is absent in aprotic solvents, tetrahydrofuran and diethyl ether. We observe also that Chl b exhibit higher values of inhomogeneous broadening than Chl a.

Probing the Differential Dynamics of the Monomeric and Dimeric Insulin from Amide-I IR Spectroscopy.

The monomer-dimer equilibrium for insulin is one of the essential steps in forming the receptor-binding competent monomeric form of the hormone. Despite this importance, the thermodynamic stability, in particular for modified insulins, is quite poorly understood, in part, due to experimental difficulties. This work explores one- and two-dimensional infrared spectroscopy in the range of the amide-I band for the hydrated monomeric and dimeric wild-type hormone. It is found that for the monomer the frequency fluctuation correlation function (FFCF) and the one-dimensional infrared spectra are position sensitive. The spectra of the -CO probes at the dimerization interface (residues Phe24, Phe25, and Tyr26) split and indicate an asymmetry despite the overall (formal) point symmetry of the dimer structure. Also, the decay times of the FFCF for the same -CO probe in the monomer and the dimer can differ by up to 1 order of magnitude, for example, for residue PheB24, which is solvent exposed for the monomer but at the interface for the dimer. The spectroscopic shifts correlate approximately with the average number of hydration waters and the magnitude of the FFCF at time zero. However, this correlation is only qualitative due to the heterogeneous and highly dynamical environment. Based on density functional theory calculations, the dominant contribution for solvent-exposed -CO is found to arise from the surrounding water (∼75%), whereas the protein environment contributes considerably less. The results suggest that infrared spectroscopy is a positionally sensitive probe of insulin dimerization, in particular in conjunction with isotopic labeling of the probe.

Vibrational Spectroscopy of N3- in the Gas and Condensed Phase.

Azido-derivatized amino acids are potentially useful, positionally resolved spectroscopic probes for studying the structural dynamics of proteins and macromolecules in solution. To this end, a computational model for the vibrational modes of N3- based on accurate electronic structure calculations and a reproducing kernel Hilbert space representation of the potential energy surface for the internal degrees of freedom is developed. Fully dimensional quantum bound state calculations yield the antisymmetric stretch vibration at 1974 cm-1 compared with 1986 cm-1 from experiment. This mode shifts by 64 cm-1 (from the frequency distribution) and 74 cm-1 (from the IR line shape) to the blue, respectively, compared with 61 cm-1 from experiment for N3- in water. The decay time of the frequency fluctuation correlation function is 1.1 ps, which is in good agreement with experiment (1.2-1.3 ps) and the full width at half maximum of the asymmetric stretch in solution is 18.5 cm-1 compared with 25.2 cm-1 from experiment. A computationally more efficient analysis based on instantaneous normal modes is shown to provide comparable, albeit somewhat less quantitative results compared to solving the three-dimensional Schrödinger equation for the fundamental vibrations.

Measuring the Ultrafast Correlation Dynamics between the Q x and Q y Bands in Chlorophyll Molecules.

We use two-dimensional electronic spectroscopy to measure the ultrafast correlation dynamics between the Q x and Q y transitions in chlorophyll molecules. We derive a variation to the center line slope method to quantify the frequency fluctuation cross-correlation function, C xy( Tw). Compared with the frequency fluctuation correlation function of the Q y transition, we observe that there is only a minimal correlation between the Q x and Q y transition, even at the ultrashort timescale of ∼100 fs, which then decays to zero in a time scale of ∼2 ps.

Measuring the Ultrafast Spectral Diffusion Dynamics of Colloidal CdSe Nanomaterials

AbstractWe use ultrafast coherent two-dimensional electronic spectroscopy (2DES) to study the ultrafast spectral diffusion dynamics of colloidal CdSe quantum dots (QDs) and CdSe nanoplatelets (NPLs). The Center Line Slope (CLS) and Nodal Line Slope (NLS) techniques were employed to analyse the 2DES spectra. We show that no spectral diffusion dynamics occurs for the CdSe QDs. On the other hand, spectral diffusion was observed in the CdSe 5 mono-layers NPLs heavy-hole transition. The normalized Frequency Fluctuation Correlation Function (FFCF) of the CdSe NPLs heavy-hole transition was measured to have a major fast decay component at 140 fs.

The effect of laser pulse bandwidth on the measurement of the frequency fluctuation correlation functions in 2D electronic spectroscopy

Several methods have been proposed to obtain the frequency fluctuation correlation function (FFCF) of a transition from the 2DES spectra without complicated fitting procedures. Although all were shown to give correct values in idealized conditions, this might not be applicable in real experimental circumstances. Here, we observe the effects of using different bandwidth excitation pulses in three commonly used methods: ellipticity, center line slope of slices along excitation (CLSω1) and detection (CLSω3) frequencies. The studies were performed on the Qy transition of chlorophyll a. We show that the CLSω3 method gives consistent results for all bandwidths used. For the other two methods, the recovered FFCF values decrease in magnitude with decreasing excitation pulse bandwidths. We also show that by applying the simplified expressions derived by Do et al., incorporating finite pulses in 2DES spectra simulation, we can recover accurate FFCFs using any of these methods.

Reorientation-induced spectral diffusion of non-isotropic orientation distributions.

When reorientation of a vibrational chromophore is faster than the relaxation of its local environment, the frequency fluctuation correlation function (FFCF) measured by two-dimensional infrared spectroscopy (2D-IR) spectroscopy is an interplay of scalar structural spectral diffusion and vectorial reorientation-induced spectral diffusion (RISD). Theory has been established to calculate the RISD component of different polarization configurations with the assumption that the molecule orients randomly in a local electric field. We show here that in the [BMIM][SCN]/AOT/chlorobenzene system, where the local electric field is strong, this assumption is incapable of reproducing the experimental results. We modify the current theory by assuming a Boltzmann distribution of transition dipoles of the chromophores in a local electric field and use a Markov chain model to calculate the RISD component of 2D-IR with different polarization configurations. The result reproduces key features of the experiment and suggests a potential ion-exchange in the [BMIM][SCN]/AOT/chlorobenzene system.

Ultrafast vibrational dynamics of the DNA backbone at different hydration levels mapped by two-dimensional infrared spectroscopy.

DNA oligomers are studied at 0% and 92% relative humidity, corresponding to N < 2 and N > 20 water molecules per base pair. Two-dimensional (2D) infrared spectroscopy of DNA backbone modes between 920 and 1120 cm−1 maps fluctuating interactions at the DNA surface. At both hydration levels, a frequency fluctuation correlation function with a 300 fs decay and a slow decay beyond 10 ps is derived from the 2D lineshapes. The fast component reflects motions of DNA helix, counterions, and water shell. Its higher amplitude at high hydration level reveals a significant contribution of water to the fluctuating forces. The slow component reflects disorder-induced inhomogeneous broadening.

Layerwise decomposition of water dynamics in reverse micelles: A simulation study of two-dimensional infrared spectrum

We present computer simulation study of two-dimensional infrared spectroscopy (2D-IR) of water confined in reverse micelles (RMs) of various sizes. The present study is motivated by the need to understand the altered dynamics of confined water by performing layerwise decomposition of water, with an aim to quantify the relative contributions of different layers water molecules to the calculated 2D-IR spectrum. The 0-1 transition spectra clearly show substantial elongation, due to inhomogeneous broadening and incomplete spectral diffusion, along the diagonal in the surface water layer of different sized RMs. Fitting of the frequency fluctuation correlation functions reveal that the motion of the surface water molecules is sub-diffusive and indicate the constrained nature of their dynamics. This is further supported by two peak nature of the angular analogue of van Hove correlation function. With increasing system size, the water molecules become more diffusive in nature and spectral diffusion almost completes in the central layer of the larger size RMs. Comparisons between experiments and simulations establish the correspondence between the spectral decomposition available in experiments with the spatial decomposition available in simulations. Simulations also allow a quantitative exploration of the relative role of water, sodium ions, and sulfonate head groups in vibrational dephasing. Interestingly, the negative cross correlation between force on oxygen and hydrogen of O-H bond in bulk water significantly decreases in the surface layer of each RM. This negative cross correlation gradually increases in the central water pool with increasing RMs size and this is found to be partly responsible for the faster relaxation rate of water in the central pool.

2D IR spectra of cyanide in water investigated by molecular dynamics simulations

Using classical molecular dynamics simulations, the 2D infrared (IR) spectroscopy of CN(-) solvated in D2O is investigated. Depending on the force field parametrizations, most of which are based on multipolar interactions for the CN(-) molecule, the frequency-frequency correlation function and observables computed from it differ. Most notably, models based on multipoles for CN(-) and TIP3P for water yield quantitatively correct results when compared with experiments. Furthermore, the recent finding that T1 times are sensitive to the van der Waals ranges on the CN(-) is confirmed in the present study. For the linear IR spectrum, the best model reproduces the full widths at half maximum almost quantitatively (13.0 cm(-1) vs. 14.9 cm(-1)) if the rotational contribution to the linewidth is included. Without the rotational contribution, the lines are too narrow by about a factor of two, which agrees with Raman and IR experiments. The computed and experimental tilt angles (or nodal slopes) α as a function of the 2D IR waiting time compare favorably with the measured ones and the frequency fluctuation correlation function is invariably found to contain three time scales: a sub-ps, 1 ps, and one on the 10-ps time scale. These time scales are discussed in terms of the structural dynamics of the surrounding solvent and it is found that the longest time scale (≈10 ps) most likely corresponds to solvent exchange between the first and second solvation shell, in agreement with interpretations from nuclear magnetic resonance measurements.

Measuring the Spectral Diffusion of Chlorophyll a Using Two-Dimensional Electronic Spectroscopy

Ultrafast two-dimensional electronic spectroscopy has been used to study the spectral diffusion of the Qy transition of chlorophyll a in methanol. The two time frequency-fluctuation correlation function (FFCF) of the transition has been measured using the center line slope method, together with optimized fitting of the linear spectrum. The FFCF was measured to decay over four time scales. The three fastest time constants of which were measured to be ∼65 fs, ∼0.5 ps, and ∼7 ps. These are assigned as the inertial component of solvation and spectrally diffusive solvation processes respectively. The fourth time constant (>1 ns) may be attributed to the chromophore structural inhomogeneity.

Three-Dimensional Infrared Spectroscopy of Isotope-Substituted Liquid Water Reveals Heterogeneous Dynamics

The dynamics of the hydrogen bond network of isotopically substituted liquid water are investigated with a new ultrafast nonlinear vibrational spectroscopy, three-dimensional infrared spectroscopy (3D-IR). The 3D-IR spectroscopy is sensitive to three-point frequency fluctuation correlation functions, and the measurements reveal heterogeneous structural relaxation dynamics. We interpret these results as subensembles of water which do not interconvert on a half picosecond time scale. We connect the experimental results to molecular dynamics (MD) simulations, performing a line shape analysis as well as complex network analysis.

Non-Markovian dynamics in a dense atomic vapor

Two- and three-pulse photon echo spectroscopy in a dense potassium vapor reveals a non-Markovian correlation function of frequency fluctuations. Through comparison with calculations using an exciton picture a slowly-decaying component of the correlation function is attributed to long-range resonant interactions. A time-resolved photon echo experiment shows the photon-echo-like behavior at short timescales.