The Multidimensional Filter Diagonalization Method: I. Theory and Numerical Implementation

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various “ersatz” or “hybrid” spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N_total = N₁ × N₂ × × N_D, where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357–366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

The multidimensional filter diagonalization method

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various "ersatz" or "hybrid" spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N(total) = N(1) x N(2) x vertical ellipsis x N(D), where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357-366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

Reinvestigation of the (4, 20) band of the O2 second negative system

High resolution absorption spectra of the (4, 20) band in the second negative system (A²Π_u–X²Π_g) of O₂⁺ cation were measured and analyzed in the range of 11 900–12 300 cm^–1 via optical heterodyne velocity modulation spectroscopy. Precise molecular constants of the levels involved were obtained by a nonlinear least-squares fitting procedure combining with our previous spectra of the (4, 19) and (6, 20) bands.     

Sample Optimization and Identification of Signal Patterns of Amino Acid Side Chains in 2D RFDR Spectra of the α-Spectrin SH3 Domain

Future structural investigations of proteins by solid-state CPMAS NMR will rely on uniformly labeled protein samples showing spectra with an excellent resolution. NMR samples of the solid α-spectrin SH3 domain were generated in four different ways, and their ¹³C CPMAS spectra were compared. The spectrum of a [u-¹³C, ¹⁵N]-labeled sample generated by precipitation shows very narrow ¹³C signals and resolved scalar carbon–carbon couplings. Linewidths of 16–19 Hz were found for the three alanine C^β signals of a selectively labeled [70% 3-¹³C]alanine-enriched SH3 sample. The signal pattern of the isoleucine, of all prolines, valines, alanines, and serines, and of three of the four threonines were identified in 2D ¹³C–¹³C RFDR spectra of the [u-¹³C,¹⁵N]-labeled SH3 sample. A comparison of the ¹³C chemical shifts of the found signal patterns with the ¹³C assignment obtained in solution shows an intriguing match.     

Regularization of the two-dimensional filter diagonalization method: FDM2K

We outline an important advance in the problem of obtaining a two-dimensional (2D) line list of the most prominent features in a 2D high-resolution NMR spectrum in the presence of noise, when using the Filter Diagonalization Method (FDM) to sidestep limitations of conventional FFT processing. Although respectable absorption-mode spectra have been obtained previously by the artifice of “averaging” several FDM calculations, no 2D line list could be directly obtained from the averaged spectrum, and each calculation produced numerical artifacts that were demonstrably inconsistent with the measured data, but which could not be removed a posteriori. By regularizing the intrinsically ill-defined generalized eigenvalue problem that FDM poses, in a particular quite plausible way, features that are weak or stem from numerical problems are attenuated, allowing better characterization of the dominant spectral features. We call the new algorithm FDM2K.     

Ultra-high resolution 3D NMR spectra from limited-size data sets

The advantage of the filter diagonalization method (FDM) for analysis of triple-resonance NMR experiments is demonstrated by application to a 3D constant time (CT) HNCO experiment. With a 15N-,13C-labeled human ubiquitin sample (1.0 mM), high spectral resolution was obtained at 500 MHz in 25 min with only 6-8 increments in each of the CT dimensions. This data set size is about a factor of 50-100 smaller than typically required, yet FDM analysis results in a fully resolved spectrum with a sharp peak for each HNCO resonance. Unlike Fourier transform (FT) processing, in which spectral resolution in each dimension is inversely proportional to the acquisition time in this dimension, FDM is a true multi-dimensional method; the resolution in all dimensions is determined by the total information content of the entire signal. As the CT dimensions of the 3D HNCO signal have approximate time-reversal symmetry, they can each be doubled by combining the usual four hyper-complex data sets. This apparent quadrupling of the data is important to the success of the method. Thus, whenever raw sensitivity is not limiting, well-resolved n-dimensional spectra can now be obtained in a small fraction of the usual time. Alternatively, to maximize sensitivity, evolution periods of faster relaxing nuclei may be radically shortened, the total required resolution being obtained through chemical shift encoding of other, more slowly relaxing, spins. Improvements similar to those illustrated with a 3D HNCO spectrum are expected for other triple-resonance spectra, where CT evolution in the indirect dimensions is implemented.     

Al satellite transition spectroscopy (SATRAS) of polycrystalline aluminium borate 9Al2O3 · 2B2O3

²⁷Al NMR spectra of polycrystalline aluminium borate 9Al₂O₃ · 2B₂O₃ have been measured at 104, 130 and 156 MHz. The parameters of the quadrupole interaction and the isotropic chemical shifts have been obtained by fitting the CT/MAS pattern and consideration of the inner satellite transitions m = 3/2 ↔ 1/2 and m = −1/2 ↔ −3/2. The gain in spectral resolution concerned with the observation of the MAS lines of the inner satellites leads to complete separation of the signals of AlO₆, AlO₅ and AlO₄ polyhedra. Also signals of structural groups of one and the same coordination number can be distinguished. Experimental and theoretical lineshape calculations are compared.     

Electron Energy Loss Spectra of the Organic Complexes of Europium

In the electron energy loss spectra (EELS) of the organic europium complexes Eu³⁺ (BTFA)₃TPPO and Eu³⁺(BrBTFA)₃TPPO in a gas phase obtained on excitation by monokinetic beams of electrons of different energies in the range 12–50 eV, we have identified the bands associated with the electron transitions S ₀–S ₁, S ₀–S ₂, and S ₀–S ₃. The connection of these transitions with the structural groups of the complexes is established. The addition of the bromine atom to the phenyl ring of diketonate leads to the rise in the relative intensity of the S ₀–S ₂ band. The singlettriplet transitions manifest themselves in the region 2.5–3.2 eV and contribute to the S ₀–S ₂ band of the electron energy loss spectra.     

New High-Resolution Semi-discrete Central Schemes for Hamilton–Jacobi Equations

We introduce a new high-resolution central scheme for multidimensional Hamilton–Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/Δt. By letting Δt↓0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge–Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme.     

The single basis filter diagonalization method: a rapid multidimensional data processing scheme

A new way to apply the filter diagonalization method (FDM) that results in a large increase in the speed of calculation of multidimensional NMR spectra is presented. The speed increase is accompanied by slight differences in spectral lineshapes, although frequency estimates remain essentially identical. For contoured spectra, the method does not result in appreciable differences from the full FDM calculation. Optimal parameter sets for an FDM calculation can be estimated far more rapidly, which makes the FDM more straightforward to employ in practice. The performance of the method versus the full FDM was investigated for both model and experimental signals. The effect of noise on the method was also studied.     

Direct Determination of Motional Correlation Times by 1D MAS and 2D Exchange NMR Techniques

One- and two-dimensional static and magic-angle spinning (MAS) exchange NMR experiments for quantifying slow (τ_c> 1 ms) molecular reorientation dynamics are analyzed, emphasizing the extent to which motional correlation times can be extracteddirectlyfrom the experimental data. The static two-dimensional (2D) exchange NMR experiment provides geometric information, as well as exchange time scales via straightforward and model-free application of Legendre-type orientational autocorrelation functions, particularly for axially symmetric interaction tensors, as often encountered in solid-state²H and¹³C NMR. Under conditions of MAS, increased sensitivity yields higher signal-to-noise spectra, with concomitant improvement in the precision and speed of correlation time measurements, although at the expense of reduced angular (geometric) resolution. For random jump motions, one-dimensional (1D)exchange-inducedsidebands (EIS)¹³C NMR and the recently developed ODESSA and time-reverse ODESSA experiments complement the static and MAS two-dimensional exchange NMR experiments by providing faster means of obtaining motional correlation times. For each of these experiments, the correlation time of a dynamic process may be obtained from a simple exponential fit to the integrated peak intensities measured as a function of mixing time. This is demonstrated on polycrystalline dimethylsulfone, where the reorientation rates from EIS, ODESSA, time-reverse ODESSA, and 2D exchange are shown to be equivalent and consistent with literature values. In the analysis, the advantages and limitations of the different methods are compared and discussed.     

Sum-frequency spectroscopy of physisorbed and chemisorbed molecules at liquid and solid surfaces using band-width-limited picosecond pulses

We studied the vibrational Sum-Frequency (SF) spectra of long chain fatty alcohols and amines physisorbed at liquid/air interfaces and of OctadecylTrichlorSilan (OTS) chemisorbed on glass/air interfaces in situ, i. e., in normal laboratory environment. The intense, band-width-limited IR pulses generated by our laser system are tunable from 2600 to 4000 cm^–1 with a constant pulse duration of 3 ps and a band width of 5 cm^–1 (FWHM) over the entire tuning range, thus covering the CH-, NH-, and OH-stretching regions.Using suitable polarization geometries, information on the molecular orientation is obtained from the amplitudes of the symmetric and degenerate methyl-stretching modes. Opposite phase of adjacent vibrational modes can lead to destructive interference in the SF signal, as analyzed theoretically. This interference effect is observed experimentally for the first time, due to the superior spectral resolution and signal-to-noise ratio of our spectra.     

29Si NMR in solid state with CPMG acquisition under MAS

A remarkable enhancement of sensitivity can be often achieved in ²⁹Si solid-state NMR by applying the well-known Carr–Purcell–Meiboom–Gill (CPMG) train of rotor-synchronized π pulses during the detection of silicon magnetization. Here, several one- and two-dimensional (1D and 2D) techniques are used to demonstrate the capabilities of this approach. Examples include 1D ²⁹Si{X} CPMAS spectra and 2D ²⁹Si{X} HETCOR spectra of mesoporous silicas, zeolites and minerals, where X = ¹H or ²⁷Al. Data processing methods, experimental strategies and sensitivity limits are discussed and illustrated by experiments. The mechanisms of transverse dephasing of ²⁹Si nuclei in solids are analyzed. Fast magic angle spinning, at rates between 25 and 40 kHz, is instrumental in achieving the highest sensitivity gain in some of these experiments. In the case of ²⁹Si–²⁹Si double-quantum techniques, CPMG detection can be exploited to measure homonuclear J-couplings.     

1H NMR spectroscopy of rat brain in vivo at 14.1Tesla: improvements in quantification of the neurochemical profile

Ultra-short echo-time proton single voxel spectra of rat brain were obtained on a 14.1 T 26 cm horizontal bore system. At this field, the fitted linewidth in the brain tissue of adult rats was about 11 Hz. New, separated resonances ascribed to phosphocholine, glycerophosphocholine and N-acetylaspartate were detected for the first time in vivo in the spectral range of 4.2–4.4 ppm. Moreover, improved separation of the resonances of lactate, alanine, γ-aminobutyrate, glutamate and glutathione was observed. Metabolite concentrations were estimated by fitting in vivo spectra to a linear combination of simulated spectra of individual metabolites and a measured spectrum of macromolecules (LCModel). The calculated concentrations of metabolites were generally in excellent agreement with those obtained at 9.4 T. These initial results further indicated that increasing magnetic field strength to 14.1 T enhanced spectral resolution in ¹H NMR spectroscopy. This implies that the quantification of the neurochemical profile in rodent brain can be achieved with improved accuracy and precision.     

Mathematical modeling of tunable TEA CO2 lasers

A mathematical model, based on the Landau–Teller equations of six-temperature model for the CO₂–N₂–He–CO system, to describe the process of dynamic emission in tunable TEA CO₂ lasers is introduced. In this model, the Landau–Teller equations are rewritten with regard to fine longitudinal mode frequencies in the laser resonator. These revised equations can be utilized to estimate the laser output spectra as well as other laser output pulse parameters. Examples are given to show the modeling results of non-tunable, grating tuned or injection-locking TEA CO₂ lasers.     

Spectral‐Luminescent Properties of Oxazoles and Oxadiazoles in a Gas Phase on Excitation by Electrons

The energyloss spectra of electrons, fluorescence excitation functions, and the fluorescence spectra on excitation of the vapors of a number of oxazoles and oxadiazoles by monokinetic beams of electrons of various energies are determined. In contrast to optical absorption spectra, in the energyloss spectra of a number of studied substances a band associated with the S ₀–T ₁ singlettriplet transition is observed. The –^*type transitions are fixed up to S ₀–S ₅ on excitation of molecules by highenergy electrons, including the region of vacuum ultraviolet. The cross sections of elastic and inelastic collisions of electrons of different energies with POPOP molecules have been measured. The dependences obtained differ substantially from those calculated in the Born approximation. The cross section of elastic scattering is in a rather good correspondence with the geometric section of the molecule.     

Application of Dey–Mittra conformal boundary algorithm to 3D electromagnetic modeling

The Dey–Mittra conformal boundary conditions have been implemented for the finite-difference time-domain (FDTD) electromagnetic solver of the VORPAL plasma simulation framework and studied in the context of three-dimensional, large-scale computations. The maximum stable time step when using these boundary conditions can be arbitrarily small, due to the presence of small fractional cells inside the vacuum region. Use of the Gershgorin Circle theorem allows the determination of a rigorous criterion for exclusion of small cells in order to have numerical stability for particular values of the ratio f_DM≡Δt/Δt_CFL of the time step to the Courant–Friedrichs–Lewy value for the infinite system. Application to a spherical cavity shows that these boundary conditions allow computation of frequencies with second-order error for sufficiently small f_DM. However, for sufficiently fine resolution, dependent on f_DM, the error becomes first order, just like the error for stair-step boundary conditions. Nevertheless, provided one does use a sufficiently small value of f_DM, one can obtain third-order accuracy through Richardson extrapolation. Computations for the TESLA superconducting RF cavity design compare favorably with experimental measurements.     

Signal Enhancement Using 45° Water Flipback for 3D N-Edited ROESY and NOESY HMQC and HSQC

A novel implementation of the water flipback technique employing a 45° flip-angle water-selective pulse is presented. The use of this water flipback technique is shown to significantly enhance signal in 3D ¹⁵N-edited ROESY in a 20 kDa complex of the vnd/NK-2 homeodomain bound to DNA. The enhancement is seen relative to the same experiment using weak water presaturation during the recovery delay. This enhancement is observed for the signals from both labile and nonlabile protons. ROESY and NOESY pulse sequences with 45° water flipback are presented using both HMQC and HSQC for the ¹⁵N dimension. The 45° flipback pulse is followed by a gradient, a water selective 180° pulse, and another gradient to remove quadrature images and crosspeak phase distortion near the water frequency. Radiation damping of the water magnetization during the t₁ and t₂ evolution periods is suppressed using gradients. Water resonance planes from NOESY–HMQC and NOESY–HSQC spectra show that the HMQC version of the pulse sequences can provide stronger signal for very fast exchanging protons. The HSQC versions of the ROESY and NOESY pulse sequences are designed for the quantitative determination of protein–water crossrelaxation rates, with no water-selective pulses during the mixing time and with phase cycling and other measures for reducing axial artifacts in the water signal.     

Progress on the two-dimensional filter diagonalization method. An efficient doubling scheme for two-dimensional constant-time NMR

An efficient way to treat two-dimensional (2D) constant-time (CT) NMR data using the filter diagonalization method (FDM) is presented. In this scheme a pair of N- and P-type data sets from a 2D CT NMR experiment are processed jointly by FDM as a single data set, twice as large, in which the signal effectively evolves in time for twice as long. This scheme is related to "mirror-image" linear prediction, but with the distinction that the data are directly used, without any preprocessing such as Fourier transformation along one dimension, or point-by-point reflection. As the signal has nearly perfect Lorentzian line shape in the CT dimension, it can be efficiently handled by the FDM approach. Applied to model and experimental signals, the scheme shows significant resolution improvement, and appears to tolerate noise reasonably well. Other complex aspects of multidimensional FDM are discussed and illustrated.     

Phase-sensitive spectral estimation by the hybrid filter diagonalization method

A more robust way to obtain a high-resolution multidimensional NMR spectrum from limited data sets is described. The Filter Diagonalization Method (FDM) is used to analyze phase-modulated data and cast the spectrum in terms of phase-sensitive Lorentzian "phase-twist" peaks. These spectra are then used to obtain absorption-mode phase-sensitive spectra. In contrast to earlier implementations of multidimensional FDM, the absolute phase of the data need not be known beforehand, and linear phase corrections in each frequency dimension are possible, if they are required. Regularization is employed to improve the conditioning of the linear algebra problems that must be solved to obtain the spectral estimate. While regularization smoothes away noise and small peaks, a hybrid method allows the true noise floor to be correctly represented in the final result. Line shape transformation to a Gaussian-like shape improves the clarity of the spectra, and is achieved by a conventional Lorentzian-to-Gaussian transformation in the time-domain, after inverse Fourier transformation of the FDM spectra. The results obtained highlight the danger of not using proper phase-sensitive line shapes in the spectral estimate. The advantages of the new method for the spectral estimate are the following: (i) the spectrum can be phased by conventional means after it is obtained; (ii) there is a true and accurate noise floor; and (iii) there is some indication of the quality of fit in each local region of the spectrum. The method is illustrated with 2D NMR data for the first time, but is applicable to n-dimensional data without any restriction on the number of time/frequency dimensions.