Center Line Slope Research Articles

Theory of Center‐Line Slope in 2D Electronic Spectroscopy with Static Disorder

Abstract2D electronic spectroscopy (2DES) is a powerful tool for investigating the dynamics of complex systems. However, analyzing the resulting spectra can be challenging, and thus may require the use of theoretical modeling techniques. The center‐line slope (CLS) method is one of such approaches, which aims to extract the time correlation function (TCF) from 2DES with minimal error. Since static disorder is widely observed in complex systems, it may be interesting to ask whether the CLS approach still work in the presence of the static disorder. In this paper, the effect of the static disorder on the TCF obtained through the CLS method is investigated. It is found that the steady‐state value of the CLS increases monotonically with respect to the static disorder, which suggests that the amplitude of the static disorder can be determined using the CLS in the long‐time limit. Additionally, as the static disorder rises, the decay rate of the CLS first decreases to a certain value and remains at this value until the static disorder is sufficiently large. Afterward, the CLS begins to fluctuate significantly and thus results in obtaining the decay rate through the CLS method unreliable. Based on these discoveries, the authors propose a method to fix the error and obtain the TCF. The findings may pave the way for obtaining reliable system‐bath information by analyzing 2DES in the practical situations.

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.

A new method based on pseudo-Zernike polynomials to analyze and extract dynamical and spectral information from the 2DIR spectra.

Ultrafast two-dimensional infrared (2DIR) spectroscopy is a relatively new methodology, which has now been widely used to study the molecular structure and dynamics of molecular processes occurring in solution. Typically, in 2DIR spectroscopy the dynamics of a system is inferred from the evolution of 2DIR spectral features over waiting times. One of the most important metrics derived from the 2DIR is the frequency-frequency correlation function (FFCF), which can be extracted using different methods, including center and nodal line slope. However, these methods struggle to correctly describe the dynamics in 2DIR spectra with multiple and overlapping transitions. Here, a new approach, utilizing pseudo-Zernike moments, is introduced to retrieve the FFCF dynamics of each spectral component from complex 2DIR spectra. The results show that this new method not only produces equivalent results to more established methodologies in simple spectra but also successfully extracts the FFCF dynamics of individual component from very congested and unresolved 2DIR spectra. In addition, this new methodology can be used to locate the individual frequency components from those complex spectra. Overall, a new methodology for analyzing the 2D spectra is presented here, which allows us to retrieve previously unattainable spectral features from the 2DIR spectra.

An Evaluation of Maximum Determination Methods for Center Line Slope Analysis

Ultrafast molecular dynamics are frequently extracted from two-dimensional (2D) spectra via the center line slope (CLS) method. The CLS method depends on the accurate determination of frequencies where the 2D signal is at a maximum, and multiple approaches exist for the determination of that maximum. Various versions of peak fitting for CLS analyses have been utilized; however, the impact of peak fitting on the accuracy and precision of the CLS method has not been reported in detail. Here, we evaluate several versions of CLS analyses using both simulated and experimental 2D spectra. The CLS method was found to be significantly more robust when fits were used to extract the maxima, particularly fitting methods that utilize pairs of opposite-sign peaks. However, we also observed that pairs of opposite-signed peaks required more assumptions than single peaks, which are important to check when interpreting experimental spectra using peak pairs.

Development of Two-Dimensional Electronic-Vibrational Sum Frequency Generation (2D-EVSFG) for Vibronic and Solvent Couplings of Molecules at Interfaces and Surfaces.

Many photoinduced excited states' relaxation processes and chemical reactions occur at interfaces and surfaces, including charge transfer, energy transfer, proton transfer, proton-coupled electron transfer, configurational dynamics, conical intersections, etc. Of them, interactions of electronic and vibrational motions, namely, vibronic couplings, are the main determining factors for the relaxation processes or reaction pathways. However, time-resolved electronic-vibrational spectroscopy for interfaces and surfaces is lacking. Here we develop interface/surface-specific two-dimensional electronic-vibrational sum frequency generation spectroscopy (2D-EVSFG) for time-dependent vibronic coupling of excited states at interfaces and surfaces. We further demonstrate the fourth-order technique by investigating vibronic coupling, solvent correlation, and time evolution of the coupling for photoexcited interface-active molecules, crystal violet (CV), at the air/water interface as an example. The two vibronic absorption peaks for CV molecules at the interface from the 2D-EVSFG experiments were found to be more prominent than their counterparts in bulk from 2D-EV. Quantitative analysis of the vibronic peaks in 2D-EVSFG suggested that a non-Condon process participates in the photoexcitation of CV at the interface. We further reveal vibrational solvent coupling for the zeroth level on the electronic state with respect to that on the ground state, which is directly related to the magnitude of its change in solvent reorganization energy. The change in the solvent reorganization energy at the interface is much smaller than that in bulk methanol. Time-dependent center line slopes (CLSs) of 2D-EVSFG also showed that kinetic behaviors of CV at the air/water interface are significantly different from those in bulk methanol. Our ultrafast 2D-EVSFG experiments not only offer vibrational information on both excited states and the ground state as compared with the traditional doubly resonant sum frequency generation and electronic-vibrational coupling but also provide vibronic coupling, dynamical solvent effects, and time evolution of vibronic coupling at interfaces.

Measuring the Ultrafast Spectral Diffusion and Vibronic Coupling Dynamics in CdSe Colloidal Quantum Wells using Two-Dimensional Electronic Spectroscopy.

We measure the ultrafast spectral diffusion, vibronic dynamics, and energy relaxation of a CdSe colloidal quantum wells (CQWs) system at room temperature using two-dimensional electronic spectroscopy (2DES). The energy relaxation of light-hole (LH) excitons and hot carriers to heavy-hole (HH) excitons is resolved with a time scale of ∼210 fs. We observe the equilibration dynamics between the spectroscopically accessible HH excitonic state and a dark state with a time scale of ∼160 fs. We use the center line slope analysis to quantify the spectral diffusion dynamics in HH excitons, which contains an apparent sub-200 fs decay together with oscillatory features resolved at 4 and 25 meV. These observations can be explained by the coupling to various lattice phonon modes. We further perform quantum calculations that can replicate and explain the observed dynamics. The 4 meV mode is observed to be in the near-critically damped regime and may be mediating the transition between the bright and dark HH excitons. These findings show that 2DES can provide a comprehensive and detailed characterization of the ultrafast spectral properties in CQWs and similar nanomaterials.

SH-It happens: S-H bonds as intrinsic 2D-IR labels in proteins.

Cysteine S-H bonds have a spectroscopically convenient stretching frequency of ∼2550cm-1. However, their cross section is low, and the band can be strongly broadened in heterogeneous environments, making detection very challenging. With two-dimensional infrared (2D-IR) setups achieving ever higher sensitivities in recent years, systematic use of the weak cysteine sulfhydryls (Cys-SHs) absorption band is now within reach, even at low millimolar protein concentrations. Here, we demonstrate the capabilities of Cys-SH as an intrinsic 2D-IR label in pyruvate oxidase from E. coli, an enzyme with ten cysteines in its native sequence. 1D-IR measurements on the wild-type and individual cysteine knock-out variants show that two such residues have especially narrow SH signatures, caused by their intrahelical hydrogen bonding. 2D-IR analysis of these bands reveals an extraordinarily high anharmonicity (∼110cm-1) and a long vibrational lifetime (∼4ps). This allows monitoring spectral diffusion via center line slope analysis for up to 10 ps-separately for both the ground and excited states. The unique spectroscopic features and its ease of introduction make Cys-SH a useful IR spectroscopic label.

Concerted Electron-Nuclear Motion in Proton-Coupled Electron Transfer-Driven Grotthuss-Type Proton Translocation.

Photoinduced proton-coupled electron transfer and long-range two-proton transport via a Grotthuss-type mechanism are investigated in a biomimetic construct. The ultrafast, nonequilibrium dynamics are assessed via two-dimensional electronic vibrational spectroscopy, in concert with electrochemical and computational techniques. A low-frequency mode is identified experimentally and found to promote double proton and electron transfer, supported by recent theoretical simulations of a similar but abbreviated (non-photoactive) system. Excitation frequency peak evolution and center line slope dynamics show direct evidence of strongly coupled nuclear and electronic degrees of freedom, from which we can conclude that the double proton and electron transfer processes are concerted (up to an uncertainty of 24 fs). The nonequilibrium pathway from the photoexcited Franck–Condon region to the E2PT state is characterized by an ∼110 fs time scale. This study and the tools presented herein constitute a new window into hot charge transfer processes involving an electron and multiple protons.

Least-Squares Fitting of Multidimensional Spectra to Kubo Line-Shape Models.

We report a comprehensive study of the efficacy of least-squares fitting of multidimensional spectra to generalized Kubo line-shape models and introduce a novel least-squares fitting metric, termed the scale invariant gradient norm (SIGN), that enables a highly reliable and versatile algorithm. The precision of dephasing parameters is between 8× and 50× better for nonlinear model fitting compared to that for the centerline-slope (CLS) method, which effectively increases data acquisition efficiency by 1–2 orders of magnitude. Whereas the CLS method requires sequential fitting of both the nonlinear and linear spectra, our model fitting algorithm only requires nonlinear spectra but accurately predicts the linear spectrum. We show an experimental example in which the CLS time constants differ by 60% for independent measurements of the same system, while the Kubo time constants differ by only 10% for model fitting. This suggests that model fitting is a far more robust method of measuring spectral diffusion than the CLS method, which is more susceptible to structured residual signals that are not removable by pure solvent subtraction. Statistical analysis of the CLS method reveals a fundamental oversight in accounting for the propagation of uncertainty by Kubo time constants in the process of fitting to the linear absorption spectrum. A standalone desktop app and source code for the least-squares fitting algorithm are freely available, with example line-shape models and data. We have written the MATLAB source code in a generic framework where users may supply custom line-shape models. Using this application, a standard desktop fits a 12-parameter generalized Kubo model to a 106 data-point spectrum in a few minutes.

Spectral Diffusion of Excitons in 3,4,9,10-Perylenetetracarboxylic-diimide (PTCDI) Thin Films.

In this work, we study spectral diffusion of molecular excitons in thin films of 3,4,9,10-perylenetetracarboxylic-diimide by using two-dimensional electronic spectroscopy (2DES). Temperature dependence of the spectral diffusion is studied from 105 to 471 K by analyzing the center line slope (CLS) of the ground-state bleach in the 2DES signal. A significant acceleration of the decay of the CLS with increasing the temperature is observed, which cannot be explained by a linear system-bath coupling model with a harmonic bath. We propose an anharmonic coupling model as the underlying mechanism, in which the exciton energy gap fluctuations by a high-frequency intramolecular vibration are enhanced by coupling with a low-frequency phonon mode.

Two-dimensional electronic-vibrational spectroscopy: Exploring the interplay of electrons and nuclei in excited state molecular dynamics.

Two-dimensional electronic-vibrational spectroscopy (2DEVS) is an emerging spectroscopic technique which exploits two different frequency ranges for the excitation (visible) and detection (infrared) axes of a 2D spectrum. In contrast to degenerate 2D techniques, such as 2D electronic or 2D infrared spectroscopy, the spectral features of a 2DEV spectrum report cross correlations between fluctuating electronic and vibrational energy gaps rather than autocorrelations as in the degenerate spectroscopies. The center line slope of the spectral features reports on this cross correlation function directly and can reveal specific electronic-vibrational couplings and rapid changes in the electronic structure, for example. The involvement of the two types of transition moments, visible and infrared, makes 2DEVS very sensitive to electronic and vibronic mixing. 2DEV spectra also feature improved spectral resolution, making the method valuable for unraveling the highly congested spectra of molecular complexes. The unique features of 2DEVS are illustrated in this paper with specific examples and their origin described at an intuitive level with references to formal derivations provided. Although early in its development and far from fully explored, 2DEVS has already proven to be a valuable addition to the tool box of ultrafast nonlinear optical spectroscopy and is of promising potential in future efforts to explore the intricate connection between electronic and vibrational nuclear degrees of freedom in energy and charge transport applications.

Characterization of the ultrafast spectral diffusion and vibronic coherence of TIPS-pentacene using 2D electronic spectroscopy.

TIPS-pentacene is a small-molecule organic semiconductor that is widely used in optoelectronic devices. It has been studied intensely owing to its ability to undergo singlet fission. In this study, we aim to develop further understanding of the coupling between the electronic and nuclear degrees of freedom of TIPS-pentacene (TIPS-Pn). We measured and analyzed the 2D electronic spectra of TIPS-Pn in solutions. Using center line slope (CLS) analysis, we characterized the frequency-fluctuation correlation function of the 0-0 vibronic transition. Strong oscillations in the CLS values were observed for up to 5ps with a frequency of 264cm-1, which are attributable to a large vibronic coupling with the TIPS-Pn ring-breathing vibrational mode. In addition, detailed analysis of the CLS values allowed us to retrieve two spectral diffusion lifetimes, which are attributed to the inertial and diffusive dynamics of solvent molecules. Amplitude beating analysis also uncovered couplings with another vibrational mode at 1173cm-1. The experimental results can be described using the displaced harmonic oscillator model. By comparing the CLS values of the simulated data with the experimental CLS values, we estimated a Huang-Rhys factor of 0.1 for the ring-breathing vibrational mode. The results demonstrated how CLS analysis can be a useful method for characterizing the strength of vibronic coupling.

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.

Intramolecular O-H⋯S hydrogen bonding in threefold symmetry: Line broadening dynamics from ultrafast 2DIR-spectroscopy and ab initio calculations.

The dynamics of intramolecular hydrogen-bonding involving sulfur atoms as acceptors is studied using two-dimensional infrared (2DIR) spectroscopy. The molecular system is a tertiary alcohol whose donating hydroxy group is embedded in a hydrogen-bond potential with torsional C3-symmetry about the carbon-oxygen bond. The linear and 2DIR-spectra recorded in the OH-stretching region of the alcohol can be simulated very well using Kubo's line shape theory based on the cumulant expansion for evaluating the linear and nonlinear optical response functions. The correlation function for OH-stretching frequency fluctuations reveals an ultrafast component decaying with a time constant of 700 fs, which is in line with the apparent decay of the center line slopes averaged over absorption and bleach/emission signals. In addition, a quasi-static inhomogeneity is detected, which prevents the 2DIR line shape to fully homogenize within the observation window of 4 ps. The experimental data were then analyzed in more detail using a full ab initio approach that merges time-dependent structural information from classical molecular dynamics (MD) simulations with an OH-stretching frequency map derived from density functional theory (DFT). The latter method was also used to obtain a complementary transition dipole map to account for non-Condon effects. The 2DIR-spectra obtained from the MD/DFT method are in good agreement with the experimental data at early waiting delays, thereby corroborating an assignment of the fast decay of the correlation function to the dynamics of hydrogen-bond breakage and formation.

Two-Dimensional Electronic-Vibrational Spectroscopy Reveals Cross-Correlation between Solvation Dynamics and Vibrational Spectral Diffusion.

Two-dimensional electronic-vibrational spectroscopy (2DEVS) is a useful technique for studying the structure and dynamics of photoexcited molecules via monitoring of the vibrational spectrum in real-time. However, quantitative modeling or prediction of experimental spectra has been hampered by the lack of a firm theoretical basis for this quantity. Here, we develop a useful theory of 2DEVS and show that the time-dependent line shape of the 2DEVS spectrum provides invaluable information on the cross-correlation function of the solvation dynamics and vibrational spectral diffusion. The center and nodal line slopes of each 2DEVS peak are determined by the associated cross-electronic-vibrational frequency-frequency correlation function, which is shown to be related to the intermolecular interactions and vibrational anharmonicities. The present theory of 2DEVS would thus be of use for a refined understanding of the 2DEVS spectra of reactive chemical and biological systems. We anticipate that further development of the expressions developed here will illuminate the application of 2DEVS studies of vibronically induced energy and electron transfer in functional materials.

CLS Next Gen: Accurate Frequency-Frequency Correlation Functions from Center Line Slope Analysis of 2D Correlation Spectra Using Artificial Neural Networks.

The center line slope (CLS) observable has become a popular method for characterizing spectral diffusion dynamics in two-dimensional (2D) correlation spectroscopy because of its ease of implementation, robustness, and clear theoretical relationship to the frequency-frequency correlation function (FFCF). The FFCF relates the frequency fluctuations of an ensemble of chromophores to coupled bath modes of the chemical system and is used for comparison to molecular dynamics simulations and for calculating 2D spectra. While in the appropriate limits, the CLS can be shown to be the normalized FFCF, from which the full FFCF can be obtained, in practice the assumptions that relate the CLS to the normalized FFCF are frequently violated. These violations are due to the presence of homogeneous broadening and motional narrowing. The generalized problem of relating the CLS to the FFCF is reanalyzed by introducing a new set of dimensionless parameters for both the CLS and FFCF. A large data set was generated of CLS parameters derived from numerically modeled 2D line shapes with known FFCF parameters. This data set was used to train feedforward artificial neural networks that act as functions, which take the CLS parameters as inputs and return FFCF parameters. These neural networks were deployed in an algorithm that is able to quickly and accurately determine FFCF parameters from experimental CLS parameters and the fwhm of the absorption line shape. The method and necessary inputs to accurately obtain the FFCF from the CLS are presented.

Dynamics of Water Molecules and Ions in Concentrated Lithium Chloride Solutions Probed with Ultrafast 2D IR Spectroscopy.

Water and ion dynamics in concentrated LiCl solutions were studied using ultrafast 2D IR spectroscopy with the methyl thiocyanate (MeSCN) CN stretch as the vibrational probe. The IR absorption spectrum of MeSCN has two peaks, one peak for water associated with the nitrogen lone pair of MeSCN (W) and the other peak corresponding to Li+ associated with the lone pair (L). To extract the spectral diffusion (structural dynamics) of W and L species, we developed a method that isolates the peak of interest by subtracting the 2D Gaussian proxies of multiple interfering peaks. Center line slope data (normalized frequency-frequency correlation function) for 2D bands from the W and L are fit with triexponential functions. The fastest component (1.1-1.6 ps) is assigned to local hydrogen bond length fluctuations. The intermediate timescale (∼4.0 ps) corresponds to the hydrogen bond network rearrangement. The slowest component decays in ∼40 ps and corresponds to ion pair and ion cluster dynamics. The very similar W and L spectral diffusion indicates that the motions of the water and ions are strongly coupled. Orientational relaxations of the W and L species were extracted using a new method to eliminate the effects of overlapping peaks. The results show that MeSCN bound to water undergoes orientational relaxation significantly faster than MeSCN bound to Li+. The orientational and spectral diffusion results are compared. A Stark coupling model is used to extract the root mean square average electric field caused by the ion clouds along the CN moiety as a function of concentration.

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.

Two-dimensional electronic vibrational spectroscopy and ultrafast excitonic and vibronic photosynthetic energy transfer.

Two-dimensional electronic-vibrational (2DEV) spectroscopy is a new coherent spectroscopic technique, which shows considerable promise for unravelling complex molecular dynamics. In this Discussion we describe an application to the energy transfer pathway in the major light harvesting protein, LHCII, providing new data on the center line slopes (CLS) of the spectral peaks. The CLS provides information that appears unique to the 2DEV method. We then outline a general approach to calculating 2DEV spectra which is valid for strongly and weakly coupled molecular systems. We conclude with some prospects for the future development of 2DEV spectroscopy and its theoretical analysis.

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.