Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: a new approach

Ultrafast two-dimensional infrared (2D-IR) vibrational echo spectroscopy can probe structural dynamics under thermal equilibrium conditions on time scales ranging from femtoseconds to approximately 100 ps and longer. One of the important uses of 2D-IR spectroscopy is to monitor the dynamical evolution of a molecular system by reporting the time dependent frequency fluctuations of an ensemble of vibrational probes. The vibrational frequency-frequency correlation function (FFCF) is the connection between the experimental observables and the microscopic molecular dynamics and is thus the central object of interest in studying dynamics with 2D-IR vibrational echo spectroscopy. A new observable is presented that greatly simplifies the extraction of the FFCF from experimental data. The observable is the inverse of the center line slope (CLS) of the 2D spectrum. The CLS is the inverse of the slope of the line that connects the maxima of the peaks of a series of cuts through the 2D spectrum that are parallel to the frequency axis associated with the first electric field-matter interaction. The CLS varies from a maximum of 1 to 0 as spectral diffusion proceeds. It is shown analytically to second order in time that the CLS is the T(w) (time between pulses 2 and 3) dependent part of the FFCF. The procedure to extract the FFCF from the CLS is described, and it is shown that the T(w) independent homogeneous contribution to the FFCF can also be recovered to yield the full FFCF. The method is demonstrated by extracting FFCFs from families of calculated 2D-IR spectra and the linear absorption spectra produced from known FFCFs. Sources and magnitudes of errors in the procedure are quantified, and it is shown that in most circumstances, they are negligible. It is also demonstrated that the CLS is essentially unaffected by Fourier filtering methods (apodization), which can significantly increase the efficiency of data acquisition and spectral resolution, when the apodization is applied along the axis used for obtaining the CLS and is symmetrical about tau=0. The CLS is also unchanged by finite pulse durations that broaden 2D spectra.     

Dynamics of a myoglobin mutant enzyme: 2D IR vibrational echo experiments and simulations

Myoglobin (Mb) double mutant T67R/S92D displays peroxidase enzymatic activity in contrast to the wild type protein. The CO adduct of T67R/S92D shows two CO absorption bands corresponding to the A(1) and A(3) substates. The equilibrium protein dynamics for the two distinct substates of the Mb double mutant are investigated by using two-dimensional infrared (2D IR) vibrational echo spectroscopy and molecular dynamics (MD) simulations. The time-dependent changes in the 2D IR vibrational echo line shapes for both of the substates are analyzed using the center line slope (CLS) method to obtain the frequency-frequency correlation function (FFCF). The results for the double mutant are compared to those from the wild type Mb. The experimentally determined FFCF is compared to the FFCF obtained from molecular dynamics simulations, thereby testing the capacity of a force field to determine the amplitudes and time scales of protein structural fluctuations on fast time scales. The results provide insights into the nature of the energy landscape around the free energy minimum of the folded protein structure.     

Extracting 2D IR frequency-frequency correlation functions from two component systems

The center line slope (CLS) method is often used to extract the frequency-frequency correlation function (FFCF) from 2D IR spectra to delineate dynamics and to identify homogeneous and inhomogeneous contributions to the absorption line shape of a system. While the CLS method is extremely efficient, quite accurate, and immune to many experimental artifacts, it has only been developed and properly applied to systems that have a single vibrational band, or to systems of two species that have spectrally resolved absorption bands. In many cases, the constituent spectra of multiple component systems overlap and cannot be distinguished from each other. This situation creates ambiguity when analyzing 2D IR spectra because dynamics for different species cannot be separated. Here a mathematical formulation is presented that extends the CLS method for a system consisting of two components (chemically distinct uncoupled oscillators). In a single component system, the CLS corresponds to the time-dependent portion of the normalized FFCF. This is not the case for a two component system, as a much more complicated expression arises. The CLS method yields a series of peak locations originating from slices taken through the 2D spectra. The slope through these peak locations yields the CLS value for the 2D spectra at a given T(w). We derive analytically that for two component systems, the peak location of the system can be decomposed into a weighted combination of the peak locations of the constituent spectra. The weighting depends upon the fractional contribution of each species at each wavelength and also on the vibrational lifetimes of both components. It is found that an unknown FFCF for one species can be determined as long as the peak locations (referred to as center line data) of one of the components are known, as well as the vibrational lifetimes, absorption spectra, and other spectral information for both components. This situation can arise when a second species is introduced into a well characterized single species system. An example is a system in which water exists in bulk form and also as water interacting with an interface. An algorithm is presented for back-calculating the unknown FFCF of the second component. The accuracy of the algorithm is tested with a variety of model cases in which all components are initially known. The algorithm successfully reproduces the FFCF for the second component within a reasonable degree of error.     

Conformational preferences and vibrational frequency distributions of short peptides in relation to multidimensional infrared spectroscopy.

Molecular dynamics simulations of the structural distributions and the associated amide-I vibrational modes are carried out for dialanine peptide in water and carbon tetrachloride. The various manifestations in nonlinear-infrared spectroscopic experiments of the distributions of conformations of solvated dialanine are examined. The two-dimensional infrared (2D-IR) spectrum of dialanine exhibits the coupling between the amide oscillators and the correlations of the frequency fluctuations. An internally hydrogen-bonded conformation exists in CCl(4) but not in H(2)O where two externally hydrogen-bonded forms are preferred. Simulations of solvated dialanine show how the 2D-IR spectra expose the underlying structural distributions and dynamics that are not deducible from linear-infrared spectra. In H(2)O the 2D-IR shows cross-peaks from large coupling in the alpha-helical conformer and an elongated higher frequency diagonal peak, reflecting the broader distribution of structures for the more flexible acetyl end. In CCl(4), the computed cross-peak portion of the 2D-IR shows evidence of two amide-I transitions in the high-frequency region which are not apparent from the diagonal peak profile. The vibrational frequency inhomogeneity of the amide-I band arises from fluctuations of the instantaneous normal modes of these conformers rather than the shifts induced by hydrogen bonding. The simulation shows that there are correlations between fluctuations of the acetyl and amino end frequencies in H(2)O that arise from mechanical coupling and not from hydrogen bonding at the two ends of the molecule. The angular relationships between the two amide units which also show up in 2D-IR were computed, and spectral manifestations of them are discussed. The simulations also permit a calculation of the rate of energy transfer from one side of the molecule to the other. From these calculations, 2D-IR spectroscopy in conjunction with simulations is seen to be a promising tool for determining dynamics of structure changes in dipeptides.     

Effects of intermolecular vibrational coupling and liquid dynamics on the polarized Raman and two-dimensional infrared spectral profiles of liquid N,N-dimethylformamide analyzed with a time-domain computational method

A time-domain method for calculating polarized Raman and two-dimensional infrared (2D-IR) spectra that includes the effects of both the diagonal frequency modulations (of individual molecules in the system) and the off-diagonal (intermolecular) vibrational coupling is presented and applied to the case of the amide I band of liquid N,N-dimethylformamide. It is shown that the effect of the resonant off-diagonal vibrational coupling and the resulting delocalization of vibrational modes is clearly seen as the noncoincidence effect in the polarized Raman spectrum and some spectral features (especially as asymmetric intensity patterns) in the 2D-IR spectra. The type of 2D-IR spectra (concerning the polarization condition) most appropriate for observing this effect is discussed. On the basis of the agreement between the observed and calculated band profiles of the polarized Raman spectrum, the time dependence of the transient IR absorption anisotropy is also calculated. The method of evaluating the extent of delocalization of vibrational modes that is relevant to the features of these optical signals in the time and frequency domains is discussed. The nature of the molecular motions (concerning the liquid dynamics) that are effective on the diagonal frequency modulations is also examined.     

Native and unfolded cytochrome c--comparison of dynamics using 2D-IR vibrational echo spectroscopy

Unfolded vs native CO-coordinated horse heart cytochrome c (h-cyt c) and a heme axial methionine mutant cyt c552 from Hydrogenobacter thermophilus ( Ht-M61A) are studied by IR absorption spectroscopy and ultrafast 2D-IR vibrational echo spectroscopy of the CO stretching mode. The unfolding is induced by guanidinium hydrochloride (GuHCl). The CO IR absorption spectra for both h-cyt c and Ht-M61A shift to the red as the GuHCl concentration is increased through the concentration region over which unfolding occurs. The spectra for the unfolded state are substantially broader than the spectra for the native proteins. A plot of the CO peak position vs GuHCl concentration produces a sigmoidal curve that overlays the concentration-dependent circular dichroism (CD) data of the CO-coordinated forms of both Ht-M61A and h-cyt c within experimental error. The coincidence of the CO peak shift curve with the CD curves demonstrates that the CO vibrational frequency is sensitive to the structural changes induced by the denaturant. 2D-IR vibrational echo experiments are performed on native Ht-M61A and on the protein in low- and high-concentration GuHCl solutions. The 2D-IR vibrational echo is sensitive to the global protein structural dynamics on time scales from subpicosecond to greater than 100 ps through the change in the shape of the 2D spectrum with time (spectral diffusion). At the high GuHCl concentration (5.1 M), at which Ht-M61A is essentially fully denatured as judged by CD, a very large reduction in dynamics is observed compared to the native protein within the approximately 100 ps time window of the experiment. The results suggest the denatured protein may be in a glassy-like state involving hydrophobic collapse around the heme.     

Nonperturbative vibrational energy relaxation effects on vibrational line shapes

A general formulation of nonperturbative quantum dynamics of solutes in a condensed phase is proposed to calculate linear and nonlinear vibrational line shapes. In the weak solute-solvent interaction limit, the temporal absorption profile can be approximately factorized into the population relaxation profile from the off-diagonal coupling and the pure-dephasing profile from the diagonal coupling. The strength of dissipation and the anharmonicity-induced dephasing rate are derived in Appendix A. The vibrational energy relaxation (VER) rate is negligible for slow solvent fluctuations, yet it does not justify the Markovian treatment of off-diagonal contributions to vibrational line shapes. Non-Markovian VER effects are manifested as asymmetric envelops in the temporal absorption profile, or equivalently as side bands in the frequency domain absorption spectrum. The side bands are solvent-induced multiple-photon effects which are absent in the Markovian VER treatment. Exact path integral calculations yield non-Lorentzian central peaks in absorption spectrum resulting from couplings between population relaxations of different vibrational states. These predictions cannot be reproduced by the perturbative or the Markovian approximations. For anharmonic potentials, the absorption spectrum shows asymmetric central peaks and the asymmetry increases with anharmonicity. At large anharmonicities, all the approximation schemes break down and a full nonperturbative path integral calculation that explicitly accounts for the exact VER effects is needed. A numerical analysis of the O-H stretch of HOD in D(2)O solvent reveals that the non-Markovian VER effects generate a small recurrence of the echo peak shift around 200 fs, which cannot be reproduced with a Markovian VER rate. In general, the nonperturbative and non-Markovian VER contributions have a stronger effect on nonlinear vibrational line shapes than on linear absorption.     

Temperature dependent equilibrium native to unfolded protein dynamics and properties observed with IR absorption and 2D IR vibrational echo experiments

Dynamic and structural properties of carbonmonoxy (CO)-coordinated cytochrome c(552) from Hydrogenobacter thermophilus (Ht-M61A) at different temperatures under thermal equilibrium conditions were studied with infrared absorption spectroscopy and ultrafast two-dimensional infrared (2D IR) vibrational echo experiments using the heme-bound CO as the vibrational probe. Depending on the temperature, the stretching mode of CO shows two distinct bands corresponding to the native and unfolded proteins. As the temperature is increased from low temperature, a new absorption band for the unfolded protein grows in and the native band decreases in amplitude. Both the temperature-dependent circular dichroism and the IR absorption area ratio R(A)(T), defined as the ratio of the area under the unfolded band to the sum of the areas of the native and unfolded bands, suggest a two-state transition from the native to the unfolded protein. However, it is found that the absorption spectrum of the unfolded protein increases its inhomogeneous line width and the center frequency shifts as the temperature is increased. The changes in line width and center frequency demonstrate that the unfolding does not follow simple two-state behavior. The temperature-dependent 2D IR vibrational echo experiments show that the fast dynamics of the native protein are virtually temperature independent. In contrast, the fast dynamics of the unfolded protein are slower than those of the native protein, and the unfolded protein fast dynamics and at least a portion of the slower dynamics of the unfolded protein change significantly, becoming faster as the temperature is raised. The temperature dependence of the absorption spectrum and the changes in dynamics measured with the 2D IR experiments confirm that the unfolded ensemble of conformers continuously changes its nature as unfolding proceeds, in contrast to the native state, which displays a temperature-independent distribution of structures.     

Collective solvent coordinates for the infrared spectrum of HOD in D2O based on an ab initio electrostatic map

An ab initio MP2 vibrational Hamiltonian of HOD in an external electrostatic potential parametrized by the electric field and its gradient-tensor is constructed. By combining it with the fluctuating electric field induced by the D(2)O solvent obtained from molecular dynamics simulations, we calculate the infrared absorption of the O-H stretch. The resulting solvent shift and infrared line shape for three force fields (TIP4P, SPC/E, and SW) are in good agreement with the experiment. A collective coordinate response for the solvent effect is constructed by identifying the main electrostatic field and gradient components contributing to the line shape. This allows a realistic stochastic Liouville equation simulation of the line shapes which is not restricted to Gaussian frequency fluctuations.     

N_2O分子C~1Ⅱ态的吸收光谱及解离动力学

通过四波混频差频的方法产生高分辨的真空紫外激光,用以测量143.6至146.9 nm范围内的射流冷却N_2O分子吸收光谱,对应于C~1Ⅱ←Ⅹ~1∑~+的吸收跃迁.谱图显示出三个分立的振动谱峰叠加在宽吸收背景上,谱峰间隔分别是521和482 cm~(-1).前人的高精度量子化学计算表明C~1Ⅱ态在N-O键长方向表现为无势垒的排斥态.而在N-N键伸缩及N_2O弯曲振动方向则表现为束缚态,因此观测到的振动谱峰被归属为激发态的Feshbach共振.通过反Fourier变换可以得到Feshbach共振对应的非稳定周期轨道的特征周期为61 fs,相应的振动频率为546 cm~(-1).鉴于这一频率与弯曲振动频率非常接近,非稳定周期轨道被认为是由C~1Ⅱ态的弯曲振动与解离运动相互作用而形成的,N-N伸缩振动没有参与形成非稳定周期轨道.由此,N_2O分子C~1Ⅱ态光激发-解离过程得以清晰地阐述.     

Vibrational spectral diffusion of azide in water

Vibrational spectral diffusion denotes the time-dependent fluctuations of a solute's vibrational frequencies due to local environmental dynamics. Vibrational line shapes are weakly sensitive to spectral diffusion, whereas three-pulse vibrational echoes are much more sensitive. We report here on theoretical studies of spectral diffusion of the asymmetric stretch of the azide anion in heavy water. We run a classical molecular dynamics simulation of rigid azide in rigid water, and at every time step we calculate the azide's anharmonic asymmetric stretch frequency using an optimized quantum mechanics/molecular mechanics method developed earlier. This generates a frequency trajectory, which we use to calculate the absorption line shape and integrated three-pulse echo intensity. Our results for both the line width and the integrated echo intensity are in excellent agreement with experiment. Our calculated frequency time-correlation function is in excellent agreement with experiment for long times (greater than 250 fs) but differs considerably from experiment at short times; our theoretical correlation function has a very pronounced oscillation, presumably due to intermolecular azide-water hydrogen-bond stretching dynamics.     

Heterodyned fifth-order 2D-IR spectroscopy of the azide ion in an ionic glass

A heterodyned fifth-order infrared pulse sequence has been used to measure a two-dimensional infrared (2D-IR) spectrum of azide in the ionic glass 3KNO3:2Ca(NO3)2. By rephasing a two-quantum coherence, a process not possible with third-order spectroscopy, the 2D-IR spectra are line narrowed, allowing the frequencies, anharmonicities, and their correlations to be measured for the first four (nu=0-3) antisymmetric stretch vibrational levels. In this glass, the vibrational levels are extremely inhomogeneously broadened. Furthermore, the glass shifts the energy of the nu=3 state more than the others, causing an inhomogeneous distribution in the anharmonic constants that are partially correlated to the inhomogeneous distribution of the fundamental frequency. These effects are discussed in light of the strong interactions that exist between the charged solute and solvent. Since this is the first example of a heterodyned fifth-order infrared pulse sequence, possible cascaded contributions to the signal are investigated. No evidence of cascaded signals is found. Compared to third-order spectroscopies, fifth-order pulse sequences provide advanced control over vibrational coherence and population times that promise to extend the capabilities of ultrafast infrared spectroscopy.     

N_2O分子C~1Π态的吸收光谱及解离动力学

通过四波混频差频的方法产生高分辨的真空紫外激光,用以测量143.6至146.9 nm范围内的射流冷却N2O分子吸收光谱,对应于C1Π←X1Σ+的吸收跃迁.谱图显示出三个分立的振动谱峰叠加在宽吸收背景上,谱峰间隔分别是521和482 cm-1.前人的高精度量子化学计算表明C1Π态在N—O键长方向表现为无势垒的排斥态,而在N—N键伸缩及N2O弯曲振动方向则表现为束缚态,因此观测到的振动谱峰被归属为激发态的Feshbach共振.通过反Fourier变换可以得到Feshbach共振对应的非稳定周期轨道的特征周期为61 fs,相应的振动频率为546 cm-1.鉴于这一频率与弯曲振动频率非常接近,非稳定周期轨道被认为是由C1Π态的弯曲振动与解离运动相互作用而形成的,N—N伸缩振动没有参与形成非稳定周期轨道.由此,N2O分子C1Π态光激发-解离过程得以清晰地阐述.     

Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes

In this and the following paper, we describe the ultrafast structural fluctuations and rearrangements of the hydrogen bonding network of water using two-dimensional (2D) infrared spectroscopy. 2D IR spectra covering all the relevant time scales of molecular dynamics of the hydrogen bonding network of water were studied for the OH stretching absorption of HOD in D2O. Time-dependent evolution of the 2D IR line shape serves as a spectroscopic observable that tracks how different hydrogen bonding environments interconvert while changes in spectral intensity result from vibrational relaxation and molecular reorientation of the OH dipole. For waiting times up to the vibrational lifetime of 700 fs, changes in the 2D line shape reflect the spectral evolution of OH oscillators induced by hydrogen bond dynamics. These dynamics, characterized through a set of 2D line shape analysis metrics, show a rapid 60 fs decay, an underdamped oscillation on a 130 fs time scale induced by hydrogen bond stretching, and a long time decay constant of 1.4 ps. 2D surfaces for waiting times larger than 700 fs are dominated by the effects of vibrational relaxation and the thermalization of this excess energy by the solvent bath. Our modeling based on fluctuations with Gaussian statistics is able to reproduce the changes in dispersed pump-probe and 2D IR spectra induced by these relaxation processes, but misses the asymmetry resulting from frequency-dependent spectral diffusion. The dynamical origin of this asymmetry is discussed in the companion paper.     

2D-IR spectroscopy of the sulfhydryl band of cysteines in the hydrophobic core of proteins

We investigate the sulfhydryl band of cysteines as a new chromophore for two-dimensional IR (2D-IR) studies of the structure and dynamics of proteins. Cysteines can be put at almost any position in a protein by standard methods of site-directed mutagenesis and, hence, have the potential to be an extremely versatile local probe. Although being a very weak absorber in aqueous environment, the sulfhydryl group gets strongly polarized when situated in an alpha-helix inside the hydrophobic core of a protein because of a strong hydrogen bond to the backbone carbonyl group. The extinction coefficient (epsilon=150 M(-1) cm(-1)) then is sufficiently high to perform detailed 2D-IR studies even at low millimolar concentrations. Using porcine (carbonmonoxy)hemoglobin as an example, which contains two such cysteines in its wild-type form, we demonstrate that spectral diffusion deduced from the 2D-IR line shapes reports on the overall-breathing of the corresponding alpha-helix. The vibrational lifetime of the sulfhydryl group (T1 approximately 6 ps) is considerably longer than that of the much more commonly used amide I mode (approximately 1.0 ps), thereby significantly extending the time window in which spectral diffusion processes can be observed. The experiments are accompanied by molecular dynamics simulations revealing a good overall agreement.     

Direct identification and decongestion of Fermi resonances by control of pulse time ordering in two-dimensional IR spectroscopy

We show that it is possible to both directly measure and directly calculate Fermi resonance couplings in benzene. The measurement method used was a particular form of two-dimensional infrared spectroscopy (2D-IR) known as doubly vibrationally enhanced four wave mixing. By using different pulse orderings, vibrational cross peaks could be measured either purely at the frequencies of the base vibrational states or split by the coupling energy. This capability is a feature currently unique to this particular form of 2D-IR and can be helpful in the decongestion of complex spectra. Five cross peaks of the ring breathing mode nu13 with a range of combination bands were observed spanning a region of 1500-4550 cm(-1). The coupling energy was measured for two dominant states of the nu13+nu16 Fermi resonance tetrad. Dephasing rates were measured in the time domain for nu13 and the two (nu13+nu16) Fermi resonance states. The electronic and mechanical vibrational anharmonic coefficients were calculated to second and third orders, respectively, giving information on relative intensities of the cross peaks and enabling the Fermi resonance states of the combination band nu13+nu16 at 3050-3100 cm(-1) to be calculated. The excellent agreement between calculated and measured spectral intensities and line shapes suggests that assignment of spectral features from ab initio calculations is both viable and practicable for this form of spectroscopy.     

Transient absorption of vibrationally excited ice Ih

The ultrafast dynamics of HDO:D2O ice Ih at 180 K is studied by midinfrared ultrafast pump-probe spectroscopy. The vibrational relaxation of HDO:D2O ice is observed to proceed via an intermediate state, which has a blueshifted absorption spectrum. Polarization resolved measurements reveal that the intermediate state is part of the intramolecular relaxation pathway of the HDO molecule. In addition, slow dynamics on a time scale of the order of 10-100 ps is observed, related to thermally induced collective reorganizations of the ice lattice. The transient absorption line shape is analyzed within a Lippincott-Schroeder model for the OH-stretch potential. This analysis identifies the main mechanism behind the strong spectral broadening of the v(OH)=1-->2 transition.     

Transient 2D-IR spectroscopy of thiopeptide isomerization

Transient 2D-IR spectra have been recorded during the photoreaction of the thioxopeptide Boc-Ala-Gly(S)-Ala-Aib-OMe. We demonstrate the potential of transient 2D-IR spectroscopy to resolve vibrational bands hidden in conventional pump-probe spectra. Different types of transient cross-peak signals are observed, and their information content is discussed by comparison with model spectra.