Rotational Averaging Research Articles

Manipulating one- and two-dimensional stimulated-x-ray resonant-Raman signals in molecules by pulse polarizations

The simulation of spontaneous (RIXS) and stimulated x-ray Raman scattering (SXRS) signals in isotropic samples requires rotational averaging of a fourth-rank tensor product of two polarizabilities. Attosecond stimulated x-ray Raman spectroscopy excites multiple valence transitions covered by the pulse bandwidths. These excitations depend on the orientation of the molecule with respect to the pulse polarizations in the laboratory frame, making the response a high-rank tensor operator. Many contributions to the response coming from different tensor components complicate the analysis and interpretation of these measurements. By using the magic angle between the excitation and detection fields these signals may be expressed as correlation functions of the scalar isotropic polarizabilities, which greatly simplifies their interpretation. We show that a similar simplification of three-pulse two-dimensional stimulated x-ray Raman scattering (2D-SXRS), which depends on a rotationally averaged sixth-rank tensor, is possible by a super magic angle (SMA) combination of two measurements with specific pulse polarization configurations. Calculated SMA 2D-SXRS signals for trans-$N$-methylacetamide (NMA) at the nitrogen $K$ edge reveal different features compared with the all-parallel polarization configuration.

Computation of fluctuation scattering profilesviathree-dimensional Zernike polynomials

Ultrashort X-ray pulses from free-electron laser X-ray sources make it feasible to conduct small- and wide-angle scattering experiments on biomolecular samples in solution at sub-picosecond timescales. During these so-called fluctuation scattering experiments, the absence of rotational averaging, typically induced by Brownian motion in classic solution-scattering experiments, increases the information content of the data. In order to perform shape reconstruction or structure refinement from such data, it is essential to compute the theoretical profiles from three-dimensional models. Based on the three-dimensional Zernike polynomial expansion models, a fast method to compute the theoretical fluctuation scattering profiles has been derived. The theoretical profiles have been validated against simulated results obtained from 300 000 scattering patterns for several representative biomolecular species.

Theoretical Modeling of the Surface-Enhanced Raman Optical Activity.

Surface-enhanced Raman optical activity (SEROA) is a new technique combining the sensitivity of the surface-enhanced Raman scattering (SERS) with the detailed information about molecular structure provided by the chiral spectroscopies. So far, experimental SEROA spectra have been reported in several studies, but the interpretation and theoretical background are rather limited. In this work, general expressions for the electromagnetic contribution to SEROA are derived using the matrix polarization theory and used to investigate the enhancement in model systems. The results not only reveal a strong dependence of the enhancement on the distance between the molecule and a metal part but also the dependence of the ratio of ROA and Raman intensities (circular intensity difference, CID) on the distance and rotational averaging. For a ribose model, an optimal molecule-colloid distance was predicted which provided the highest CIDs. However, the CID maximum disappeared after a rotational averaging. For cysteine zwitterion, the simulated SEROA and SERS spectra provided a qualitative agreement with previous experiments.

A qualitative confocal microscopy study on a range of colloidal processes by simulating microgravity conditions through slow rotations

We report qualitatively on the differences between colloidal systems left to evolve in the Earth's gravitational field and the same systems for which a slow continuous rotation averaged out the effects of particle sedimentation on a distance scale small compared to the particle size. Several systems of micron-sized colloidal particles were studied: a hard sphere fluid, colloids interacting via long-range electrostatic repulsion above the freezing volume fraction, an oppositely charged colloidal system close to either gelation and/or crystallization, colloids with a competing short-range depletion attraction and a long-range electrostatic repulsion, colloidal dipolar chains, and colloidal gold platelets under conditions where they formed stacks. Important differences in structure formation were observed between the experiments where the particles were allowed to sediment and those where sedimentation was averaged out. For instance, in the case of colloids interacting via long-range electrostatic repulsion, an unusual sequence of dilute-fluid–dilute-crystal–dense-fluid–dense-crystal phases was observed throughout the suspension under the effect of gravity. This was related to the volume fraction dependence of the colloidal interactions, whereas the system stayed homogeneously crystallized with rotation. For the oppositely charged colloids, a gel-like structure was found to collapse under the influence of gravity with a few crystalline layers grown on top of the sediment, whereas when the colloidal sedimentation was averaged out, the gel completely transformed into crystallites that were oriented randomly throughout the sample. Rotational averaging out of gravitational sedimentation is an effective and cheap way to estimate the importance of gravity for colloidal self-assembly processes.

Closed form of the rotational Crofton formula

AbstractThe closed form of a rotational version of the famous Crofton formula is derived. In the case where the sectioned object is a compact d‐dimensional C2 manifold with boundary, the rotational average of intrinsic volumes (total mean curvatures) measured on sections passing through a fixed point can be expressed as an integral over the boundary involving hypergeometric functions. In the more general case of a compact subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R}^d$\end{document} with positive reach, the rotational average also involves hypergeometric functions. For convex bodies, we show that the rotational average can be expressed as an integral with respect to a natural measure on supporting flats. It is an open question whether the rotational average of intrinsic volumes studied in the present paper can be expressed as a limit of polynomial rotation invariant valuations.

Conformational analysis of the chemical shifts for molecules containing diastereotopic methylene protons

Quantum chemistry SCF/GIAO calculations were carried out on a set of compounds containing diastereotopic protons. Five molecules, including recently synthesized 1,3-di(2,3-epoxypropoxy)benzene, containing the chiral or pro-chiral center and the neighboring methylene group, were chosen. The rotational averages (i.e. normalized averages with respect to the rotation about the torsional angle τ with the exponential energy weight at temperature T) calculated individually for each of the methylene protons in 1,3-di(2,3-epoxypropoxy)benzene differ by ca. 0.6 ppm, which is significantly less than the value calculated for the lowest energy conformer. This value turned out to be low enough to guarantee the proper ordering of theoretical chemical shifts, supporting the interpretation of the 1H NMR spectrum of this important compound. The rotational averages of chemical shifts for methylene protons for a given type of conformer are shown to be essentially equal to the Boltzmann averages (here, the population-weighted averages for the individual conformers representing minima on the E( τ) cross-section). The calculated Boltzmann averages in the representative conformational space may exhibit completely different ordering as compared to the chemical shifts calculated for the lowest-energy conformer. This is especially true in the case of molecules, for which no significant steric effects are present. In this case, only Boltzmann averages account for the experimental pattern of proton signals. In addition, better overall agreement with experiment (lower value of the root-mean-square deviation between calculated and measured chemical shifts) is typically obtained when Boltzmann averages are used.

Comparison of CC Triple and Double Bonds as Spacers in Push–Pull Chromophores

AbstractWe report the synthesis and properties of two series of homologous donor–acceptor (D–A) chromophores in which N,N‐dimethylanilino (DMA) or N,N‐dihexylanilino (DHA) donors and dicyanovinyl acceptors are separated by up to four C≡C triple‐bond spacers or up to three C=C double‐bond spacers. The intramolecular charge‐transfer (CT) interactions of the new D–A oligoynes and the known all‐trans D–A oligoenes were investigated by X‐ray crystallography, electrochemistry, UV/Vis spectroscopy, and theoretical calculations. In both series, the optical and electrochemical HOMO–LUMO gaps decrease with increasing spacer length. The HOMO–LUMO gaps for the D–A oligoynes and oligoenes with a given spacer length are nearly identical. The effect of the spacer length was found to level‐off for spacers with more than six carbon atoms. The third‐order optical nonlinearity of both series of molecules was determined by measuring the rotational averages of the third‐order polarizabilities γrot by degenerate four‐wave mixing.

Controlling low-level image properties: The SHINE toolbox

Visual perception can be influenced by top-down processes related to the observer's goals and expectations, as well as by bottom-up processes related to low-level stimulus attributes, such as luminance, contrast, and spatial frequency. When using different physical stimuli across psychological conditions, one faces the problem of disentangling the contributions of low- and high-level factors. Here, we make available the SHINE (spectrum, histogram, and intensity normalization and equalization) toolbox for MATLAB, which we have found useful for controlling a number of image properties separately or simultaneously. The toolbox features functions for specifying the (rotational average of the) Fourier amplitude spectra, for normalizing and scaling mean luminance and contrast, and for exact histogram specification optimized for perceptual visual quality. SHINE can thus be employed for parametrically modifying a number of image properties or for equating them across stimuli to minimize potential low-level confounds in studies on higher level processes.

Acoustic phonon strain induced mixing of the fine structure levels in colloidal CdSe quantum dots observed by a polarization grating technique

Acoustic phonon modes in colloidal semiconductor nanocrystals are of significant interest due to their role in dephasing and as the main component of homogeneous line broadening. Despite their importance, these modes have proven elusive and have only recently been experimentally observed. This paper expands on results presented in our earlier paper [V. M. Huxter, A. Lee, S. S. Lo, et al., Nano Lett. 9, 405 (2008)], where a cross polarized heterodyne detected ultrafast transient grating (CPH-3TG) technique was used to observe the acoustic phonon mode. In the present work, we explain the origin of the observed quantum beat in the CPH-3TG signal. Further experiments are presented that show that the observed quantum beat, which arises from a coherent acoustic phonon mode in the nanocrystals, appears in anisotropy-type signals. The action of this mode induces a periodic strain in the nanocrystal that lowers the symmetry of the unit cell, mixing the fine structure states and their transition dipole moments. This mixing is manifested in anisotropy signals as a depolarization, which periodically modifies the rotational averaging factors. Through observation of the acoustic phonon mode using the CPH-3TG optical technique, it is possible to access its microscopic (atomic-level) basis and to use it as a probe to quantify changing macroscopic (whole particle) material parameters.

A new rotational integral formula for intrinsic volumes in space forms

A new rotational version of Crofton's formula is derived for the intrinsic volumes of a domain Y in a space form. More precisely, a functional is defined on the intersection between Y and a totally geodesic submanifold (plane) through a fixed point, such that the rotational average of this functional is equal to the intrinsic volumes of Y. Particular cases of interest in stereology are considered for the Euclidean case.

Effects of Restricted Rotations and Dynamic Averaging on the Calculated Isotropic Hyperfine Coupling Constants of the bis-Dimethyl and bis-Di(trifluoromethyl) Nitroxide Radicals

The rotational effects of the CH(3) and CF(3) groups on the electronic structure and nuclear hyperfine coupling constants (HFCCs) in dimethylnitroxide (DMNO*) and ditrifluoro-methynitroxide (TFMNO*) are investigated using the UB1LYP hybrid density functional method. The CH(3) and CF(3) HFCCs of both radicals are found to obey the McConnell relation during rotation. The two CH(3) groups of the DMNO* do not gear with each other, but the rotation of the first CH(3) group induces only a small rocking effect ( approximately 7 degrees ) in the second group. However, in TFMNO*, the fluorine atoms from different CF(3) groups are close enough so that the steric repulsion between them causes them to act as two interlocked gears, where one drives the other. Therefore, both CF(3) groups undergo full rotation. To the best of our knowledge, this is only the second example of "gearing" to be studied. Stabilization due to hyperconjugation is also a major factor that affects the magnitudes of the HFCCs of the CF(3) groups during rotational averaging. Stable configurations at specific CF(3) group orientations have a large overlap with the NO pi-electron cloud because the lobes of the hybridized p(sigma)(F(2)), p(sigma)(F(3)), p(sigma)(F(5)), and p(sigma)(F(6)) orbitals along the F-C bonds have cylindrical symmetry and are of the correct phases for hyperconjugation to occur. The calculated TFMNO* C(1)-N and C(2)-N bond orders range from 0.91 to 0.95 as the CF(3) groups are rotated. Therefore, the C-N bonds are essentially single bonds. This, in conjunction with the low rotational energy barrier of approximately 50 cm(-1), explains why the EPR intensities of the (19)F hyperfine splittings, in the range of 163-297 K, are characteristic of six equivalent rapidly rotating fluorine atoms. The TFMNO* out-of-plane NO vibrations induce additional s character at the (14)N nucleus. This increases the magnitude of the (14)N HFCC and decreases the (19)F HFCCs. As the temperature increases and because of mixing of the first excited out-of-plane vibrational state, the NO vibrational amplitudes also increase. This leads to an increased (14)N HFCC and decreased (19)F HFCCs, which is in agreement with experiment.

CTF determination and correction for low dose tomographic tilt series

The resolution of cryo-electron tomography can be limited by the first zero of the microscope's contrast transfer function (CTF). To achieve higher resolution, it is critical to determine the CTF and correct its phase inversions. However, the extremely low signal-to-noise ratio (SNR) and the defocus gradient in the projections of tilted specimens make this process challenging. Two programs, CTFPLOTTER and CTFPHASEFLIP, have been developed to address these issues. CTFPLOTTER obtains a 1D power spectrum by periodogram averaging and rotational averaging and it estimates the noise background with a novel approach, which uses images taken with no specimen. The background-subtracted 1D power spectra from image regions at different defocus values are then shifted to align their first zeros and averaged together. This averaging improves the SNR sufficiently that it becomes possible to determine the defocus for subsets of the tilt series rather than just the entire series. CTFPHASEFLIP corrects images line-by-line by inverting phases appropriately in thin strips of the image at nearly constant defocus. CTF correction by these methods is shown to improve the resolution of aligned, averaged particles extracted from tomograms. However, some restoration of Fourier amplitudes at high frequencies is important for seeing the benefits from CTF correction.

Axial Asymmetry of the Charge- and Spin-Density Distributions in Π States. Molecular Quadrupole Moments and Hyperfine Coupling Constants of CH, NH, OH, CF, LiO, NO, and FO

The axial asymmetry of the charge- and spin-density distributions in Pi states is studied via second-rank traceless tensors P(ii) (ii = xx, yy, zz), namely, quadrupole moments (Theta(ii)), electric field gradients (q(ii)), and magnetic dipolar (T(ii)) hyperfine coupling constants (hfcc's). In linear molecules, it holds that P(xx) does not = P(yy) does not = P(zz) for Pi, but P(xx) = P(yy) does not = P(zz) for Sigma, Delta, Phi,..., states. Thus, traceless P(ii) in Pi states have two independent parameters, P(parallel) = P(zz) is proportional to [r(m)(3 cos2 theta - 1] and deltaP(perpendicular) = |P(xx) - P(yy)| is proportional to [r(m) sin2 theta], with m = 2(Theta(ii)) or -3(q(ii), T(ii)). All linear states have P(parallel) does not = 0, but only Pi states exhibit deltaP(perpendicular) does not = 0, as shown by hfcc's like c = (3/2)T(zz), and d = |T(xx) - T(yy)|, as well as q0 = (-q(zz)) and |q(2)/2| = |q(xx) - q(yy)|. Little is known about Theta(zz) and deltaTheta(perpendicular) = |Theta(xx) - Theta(yy)| in Pi states since most experimental values (gas-phase) are rotational averages, and several theoretical studies have reported Theta(zz) but assumed deltaTheta(perpendicular) = 0. The diatomics studied here have X2Pi(1/2)(pi1) ground states, like CH and NO, or are of type X2Pi(3/2)(pi3), like OH, CF, LiO, and FO. The A3Pi(sigma pi3) state of NH is also included. Our P(parallel) and deltaP(perpendicular) values--calculated at the experimental R(e)'s with the B3LYP/aug-cc-pVQZ method--reproduce well the available literature data. The properties of the CF and FO radicals are not well-known so that our {c, d} and {q0, q2} values should help future experimental studies of their hyperfine spectra. Excluding OH, the complete quadrupole sets {Theta(zz), deltaTheta(perpendicular)} are new for all species discussed here. For comparison purposes, Theta(zz) of a low-lying Sigma state is also calculated for each X2Pi radical.

Role of the Geometry, Restricted Rotations and Solvents on the Computed 2,2′-Diphenyl-1-picrylhydrazyl Hyperfine Tensors

The nuclear hyperfine tensor (A) components of the 2,2'-diphenyl-1-picrylhydrazyl neutral radical are computed using the UB1LYP hybrid density functional method. Solvent interactions via hydrogen bonding are found to play a crucial role in the position of the two phenyl rings relative to the picryl moiety. Under these conditions, the calculated isotropic hyperfine tensor components of the N 1 and N 2 hydrazyl backbone are within approximately 1.3 Gauss (G) of the experimental values determined by EPR and ENDOR spectroscopy. Just as important are the effects of restricted rotations of the phenyl rings on these tensors. Rotational averaging using a Maxwell-Boltzmann type distribution improves the agreement between theory and experiment to less than 1.0 G. In addition, rotational averaging of the twelve isotropic proton coupling constants has also been performed. They come within 0.3 G of the experimental values. Thus, for the first time, all the nuclear hyperfine tensor components of this large class of molecules are accurately calculated without resorting to post Hartree-Fock techniques.

Asymptotics of variance of the lattice point count

The variance of the number of lattice points inside the dilated bounded set rD with random position in ℝ d has asymptotics ∼ r d−1 if the rotational average of the squared modulus of the Fourier transform of the set is O(ϰ −d−1). The asymptotics follow from Wiener’s Tauberian theorem.

A rotational integral formula for intrinsic volumes

A rotational version of the famous Crofton formula is derived. The motivation for deriving the formula comes from local stereology, a new branch of stereology based on sections through fixed reference points. The formula shows how rotational averages of intrinsic volumes measured on sections passing through fixed points are related to the geometry of the sectioned object. In particular it is shown how certain weighting factors, appearing in the rotational integral formula, can be expressed in terms of hypergeometric functions. Close connections to geometric tomography will be pointed out. Applications to stereological particle analysis are discussed.

Interatomic contributions to high-energy electron-molecule scattering

Within the independent atom scattering model, we derive an approximate formula for the rotationally and vibrationally averaged three-atom terms in the series expansion of the electron scattering cross section. This formula uses the atomic scattering factors as well as the atomic scattered wavefunctions as input, and rotational averaging is performed numerically. We compare our results to previous theoretical multiple scattering approaches for the molecules F(3), F(4), and SF(6) and to experimental data for TeF(6). Our results are consistent with those of previous calculations and inclusion of the three-atom term produces a dramatically better least squares fit of the TeF(6) data. The algorithm presented here is sufficiently fast and simple to be incorporated easily into existing electron diffraction codes.

Draft Crystal Structure of the Vault Shell at 9-Å Resolution

Vaults are the largest known cytoplasmic ribonucleoprotein structures and may function in innate immunity. The vault shell self-assembles from 96 copies of major vault protein and encapsulates two other proteins and a small RNA. We crystallized rat liver vaults and several recombinant vaults, all among the largest non-icosahedral particles to have been crystallized. The best crystals thus far were formed from empty vaults built from a cysteine-tag construct of major vault protein (termed cpMVP vaults), diffracting to about 9-Å resolution. The asymmetric unit contains a half vault of molecular mass 4.65 MDa. X-ray phasing was initiated by molecular replacement, using density from cryo-electron microscopy (cryo-EM). Phases were improved by density modification, including concentric 24- and 48-fold rotational symmetry averaging. From this, the continuous cryo-EM electron density separated into domain-like blocks. A draft atomic model of cpMVP was fit to this improved density from 15 domain models. Three domains were adapted from a nuclear magnetic resonance substructure. Nine domain models originated in ab initio tertiary structure prediction. Three C-terminal domains were built by fitting poly-alanine to the electron density. Locations of loops in this model provide sites to test vault functions and to exploit vaults as nanocapsules.

Quadrupole contribution to the third-order optical activity spectroscopy

Time-resolved nonlinear optical activity measurement spectroscopy can be a useful tool for studying biomolecular and chemical reaction dynamics of chiral molecules. Only recently, the two-dimensional (2D) circularly polarized photon echo (CP-PE) spectroscopy of polypeptides and a photosynthetic light-harvesting complex were discussed, where the beam configuration was specifically controlled in such a way to eliminate the quadrupole contribution to the CP-PE signal. In this paper, we generalize the CP-PE spectroscopy by including the transition quadrupole contributions from peptide amide I vibrational transition and chlorophyll electronic transition. By using a density functional theory calculation method, the corresponding amide I vibrational and chlorophyll Q(y) electronic transition quadrupole tensor elements are determined. Amplitude of nonlinear optical transition pathway involving a quadrupole transition is found to be comparable to those of magnetic dipole terms for two different cases considered, i.e., dipeptides and photosynthetic antenna complex. However, due to the rotational averaging factors, the overall quadrupole contribution is an order of magnitude smaller than the magnetic dipole contribution. This suggests that the conventional 2D photon echo method and experimental scheme can be directly used to measure the 2D CP-PE signal from proteins and molecular complexes and that the 2D CP-PE signal is mainly dictated by the magnetic dipole contribution.

DFT and NMR Studies of 2JCOH, 3JHCOH, and 3JCCOH Spin-Couplings in Saccharides: C−O Torsional Bias and H-Bonding in Aqueous Solution

Density functional theory (DFT) has been used to investigate the structural dependencies of NMR spin-coupling constants (J-couplings) involving the exchangeable hydroxyl protons of saccharides. 3JHCOH, 3JCCOH, and 2JCOH values were calculated at different positions in model aldopyranosyl rings as a function of one or more torsion angles, and results support the use of a generalized Karplus equation to treat 3JHCOH involving the non-anomeric OH groups. The presence of O5 appended to the H1-C1-O1-H coupling pathway introduces asymmetry in 3JH1,O1H Karplus curves due to internal electronegative substituent effects on the gauche couplings, thus requiring separate equations to treat this coupling. 3JCCOH values depend not only on the C-C-O-H torsion angle but also on the orientation of terminal substituents on the coupled carbon, similar to 3JCOCC studied previously (Bose et al., J. Am. Chem. Soc. 1998, 120, 11158-11173). "In-plane" oxygen increased 3JCCOH by approximately 3-4 Hz, whereas "in-plane" carbon gave more modest enhancements ( approximately 1 Hz). Three Karplus equations were derived for non-anomeric 3JCCOH based on the nature and orientation of substituents on the coupled carbon. Like 3JH1,O1H, 3JC2,O1H is also subject to internal electronegative substituent effects on the gauche couplings, thus necessitating separate equations to treat this coupling. 2JCOH values were found not to be useful probes of C-O torsions as a result of their nonsystematic dependence on these torsions. Experimental measurements of 3JHCOH and 3JCCOH in doubly 13C-labeled methyl beta-lactoside 20 and its constituent 13C-labeled methyl aldopyranosides in H2O/acetone-d6 at -20 degrees C showed that some C-O torsion angles are influenced by molecular context and do not experience complete rotational averaging in solution. A strong bias in the H3-C3-O3-H torsion angle in the Glc residue of 20 favoring a gauche conformation suggests the presence of inter-residue H-bonding between O3HGlc and O5Gal. Quantitative analysis of 3JHCOH and 3JCCOH values in 20 indicates that approximately 85% of the forms in solution have geometries consistent with H-bonding. These results suggest that H-bonding between adjacent and/or remote residues may play a role in dictating preferred glycosidic bond conformation in simple and complex oligosaccharides in aqueous solution.