Rotational Averaging Research Articles

Irreducible Cartesian tensor decomposition: A computational approach

Cartesian tensors are widely used in physics and chemistry, especially for the formulation of linear and nonlinear spectroscopies as well as for molecular response properties. In this work, we review the problem of irreducible Cartesian tensor (ICT) decomposition of a generic Cartesian tensor of rank n into its irreducible parts, each characterized by a specific symmetry. The matrix formulation of the ICT decomposition is structurally similar to the problem of rotational averaging using isotropic Cartesian tensors. Analogously to the latter, the ICT decomposition can be considered as a problem of selecting a set of permutations of n indices that provides a linearly independent set of mappings between Cartesian tensor subspaces. This selection can be performed using a simple computational approach based on the reduced row echelon form (rref) algorithm. This protocol has been implemented in a computer code used to re-derive the already known ICT decomposition for 2 ≤ n ≤ 4. Finally, for the first time, we performed the explicit ICT decomposition of a Cartesian tensor of rank n = 5.

Three-Dimensional Rotational Averaging Using Irreducible Sets of Linearly Independent Fundamental Isotropic Cartesian Tensors: A Computational Approach.

The theoretical formulation of linear and nonlinear molecular spectroscopies applied to isotropic samples (e.g., liquid or gas solutions) goes through a fundamental step known as the rotational averaging of Cartesian tensors. Rotational averaging of Cartesian tensors is a mathematical procedure from which the expressions for the rotationally invariant observables (e.g., rates or intensities), associated with a given spectroscopic process, can be found. In this work, the mathematical/computational procedure for finding the rotational averages of Cartesian tensors of any rank n, which is based on the use of the fundamental isotropic Cartesian tensors (FICTs), is discussed. Moreover, for the first time, a heuristic computational method for finding a set of linearly independent FICTs is proposed. The procedure has been tested for 2 ≤ n ≤ 12, where most of the linear and nonlinear molecular spectroscopies apply (e.g., one-photon and multiphoton absorption, emission, electronic circular dichroism, Raman optical activity, coherent and incoherent mth-harmonic generation, etc.). Finally, it is shown how this computational procedure can be extended for n > 12.

Radicals from tributyl phosphate decomposition: a combined electron paramagnetic resonance spectroscopic and computational chemistry investigation.

The radiation- and chemically-induced radicals from tributyl phosphate (TBP) have been characterized by EPR spectroscopy and theoretical calculations. The yield of X-ray-generated TBP radicals measured by a PBN spin trap is 0.22 μmol J-1 (2.1 radicals/100 eV) at room temperature (298 K). The EPR spectra obtained by irradiating TBP with an electron beam at 77 K are in close agreement with literature data for samples irradiated with gamma- and X-rays [https://doi.org/10.1007/BF02165504, https://doi.org/10.1016/1359-0197(89)90319-6]. Possible conformers of alkyl-type, TBP-derived radicals were analyzed by Density Functional Theory calculations. The main contribution to the experimental spectrum at 77 K is shown to be made by a conformer of the CH3˙CHCH2-radical, which contains all carbon atoms of the butyl group in the same plane. The EPR spectra of TBP radicals induced by the OH radical in aqueous solution were measured for the first time using a continuous flow system. The formation of the alkyl-type TBP radicals CH3˙CHCH2-, ˙CH2CH2-, and -CH2˙CHO- in the ratio of 5/4/1 was detected; their spectral assignment was based on quantum chemical calculations with rotational averaging of HFC constants for the corresponding beta- and alpha-protons.

Characterization by ENDOR Spectroscopy of the Iron–Alkyl Bond in a Synthetic Counterpart of Organometallic Intermediates in Radical SAM Enzymes

Members of the radical S-adenosyl-l-methionine (SAM) enzyme superfamily initiate a broad spectrum of radical transformations through reductive cleavage of SAM by a [4Fe-4S]1+ cluster it coordinates to generate the reactive 5'-deoxyadenosyl radical (5'-dAdo•). However, 5'-dAdo• is not directly liberated for reaction and instead binds to the unique Fe of the cluster to create the catalytically competent S = 1/2 organometallic intermediate Ω. An alternative mode of reductive SAM cleavage, especially seen photochemically, instead liberates CH3•, which forms the analogous S = 1/2 organometallic intermediate with an Fe-CH3 bond, ΩM. The presence of a covalent Fe-C bond in both structures was established by the ENDOR observation of 13C and 1H hyperfine couplings to the alkyl groups that show isotropic components indicative of Fe-C bond covalency. The synthetic [Fe4S4]3+-CH3 cluster, M-CH3, is a crystallographically characterized analogue to ΩM that exhibits the same [Fe4S4]3+ cluster state as Ω and ΩM, and thus an analysis of its spectroscopic properties─and comparison with those of Ω and ΩM─can be grounded in its crystal structure. We report cryogenic (2 K) EPR and 13C/1/2H ENDOR measurements on isotopically labeled M-CH3. At low temperatures, the complex exhibits EPR spectra from two distinct conformers/subpopulations. ENDOR shows that at 2 K, one contains a static methyl, but in the other, the methyl undergoes rapid tunneling/hopping rotation about the Fe-CH3 bond. This generates an averaged hyperfine coupling tensor whose analysis requires an extended treatment of rotational averaging. The methyl group 13C/1/2H hyperfine couplings are compared with the corresponding values for Ω and ΩM.

Investigating temporal changes in Jupiter’s aerosol structure with rotationally-averaged 2015–2020 HST WFC3 images

Annual imaging observations of Jupiter from 2015 onward by the Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) as part of the Outer Planet Atmospheres Legacy (OPAL) program (Simon et al., 2015) have provided a valuable data set from which Jupiter’s temporal variability can be characterized and modeled. Spectral samples at discrete filter bands from the UV to short near-IR wavelengths also provide an opportunity to connect temporal changes in reflectivity with temporal changes in cloud structure. To improve constraints on that cloud structure we used center-to-limb variations that were noise-reduced by rotational averaging during each annual OPAL observation sequence. Fitting observations at multiple and distinctly different observer zenith angles helps to make up for the lack of multiple methane-band filters in discerning the vertical distribution of aerosols. The short-wavelength limit of the OPAL observations also provides better constraints than ground-based observation on the properties of Jupiter’s chromophore, i.e., the compound that creates Jupiter’s variety of red colorations to clouds and hazes composed mainly of colorless condensates. We chose four latitudes to investigate with radiative transfer models based on a simple 3-layer aerosol structure consisting of a deep putative NH3 cloud layer, with a physically thin chromophore layer that has a fitted location at or near the top of the deep cloud, and an overlying stratospheric haze layer. Our chosen planetocentric latitudes were 5.5°S (Equatorial Zone South,EZS), 0° (Equatorial Zone, EZ), 10°N (North Equatorial Belt, NEB), and 21.75°N (North Temperate Belt, NTB). Two major color transformations occurred during the 2015–2020 period we analyzed. One is a dramatic dimming of the center of the Equatorial Zone at short wavelengths, especially after 2017, and a significant reddening of that region. The second is a significant reddening of part of the North Temperate Belt (near 22°N) between 2016 and 2017. Both reddening events can be modeled by doubling of the optical depth of the chromophore layer during the transitions. Comparison with prior 2017 model results for the NEB, EZ, and NTB have revealed agreement with Braude et al. (2020) on the vertical location of the chromophore layer but disagreements on cloud-top pressures and the vertical distribution of stratospheric haze particles. We identified what seems to be common tendency of models based on HST/WFC3 observations to either infer very little stratospheric haze opacity or to put that opacity at too high a pressure.

Measuring Exciton Fine-Structure in Randomly Oriented Perovskite Nanocrystal Ensembles Using Nonlinear Optical Spectroscopy: Theory.

Lead halide perovskite nanocrystals (PNCs) exhibit unique optoelectronic properties, many of which originate from a purported bright-triplet exciton fine-structure. A major impediment to measuring this fine-structure is inhomogeneous spectral broadening, which has limited most experimental studies to single-nanocrystal spectroscopies. It is shown here that the linearly polarized single-particle selection rules in PNCs are preserved in nonlinear spectroscopies of randomly oriented ensembles. Simulations incorporating rotational averaging demonstrate that techniques such as transient absorption and two-dimensional coherent spectroscopy are capable of resolving exciton fine-structure in PNCs, even in the presence of inhomogeneous broadening and orientation disorder.

A general method for rotational averages

The theory of nonlinear spectroscopy on randomly oriented molecules leads to the problem of averaging molecular quantities over random rotation. We solve this problem for arbitrary tensor rank by deriving a closed-form expression for the rotationally invariant tensor of averaged direction cosine products. From it, we obtain some useful new facts about this tensor. Our results serve to speed the inherently lengthy calculations of nonlinear optics.

Rovibrationally state-specific collision model for the O2(Σg-3) + O(P3) system in DSMC.

A rovibrationally state-specific collision model for the O2(Σg-3)+O(P3) system is presented for direct simulation Monte Carlo, including rotation-vibration-translation energy transfer, exchange, dissociation, and recombination processes. The two-step binary collision approach is employed to model recombination reactions. Two available cross section databases by Andrienko/Boyd and Esposito/Capitelli are employed for the rovibrationally resolved model (rv-STS) and vibrationally resolved model (v-STS), respectively. The difference between rv-STS and v-STS comes from two contributions: the multisurface factor of dissociation (fMS) and the rotational averaging process. The dissociation cross section with the constant fMS is typically larger than with the variable fMS, especially for the low vibrational energy states. On the other hand, the cross sections resulting from the rotationally averaged database are found to underpredict the dissociation rate coefficient at low temperatures. In the rovibrational heating case, the rv-STS predicts faster relaxation than the v-STS, which also shows a lower quasi-steady-state temperature than v-STS. In the rovibrational cooling case, the rv-STS shows a faster relaxation than v-STS, which also presents a thermal non-equilibrium between rovibrational and translational mode during the cooling process.

Physical Principles of Circular Dichroism

Circular dichroism (CD) is the differential absorption of left and right circularly polarized light. It arises from molecular electron oscillations that are driven by both the light’s electric and magnetic fields, where the effects are in phase for one circular polarization and out of phase for the other. We describe these interactions, which have not been presented at an intuitive level before, with classical and quantum treatments. A rotational average of the latter solution leads to the Rosenfeld equation, which underlies most CD research. We illustrate its use with a chiral version of a harmonic oscillator, which we then apply to the far-ultraviolet CD of protein α-helices. This presentation is aimed at the level of upper-division undergraduates and could be usefully incorporated into a physical chemistry course.

The Spatial Frequency Content of Urban and Indoor Environments as a Potential Risk Factor for Myopia Development.

PurposeTo examine the hypothesis that the spatial frequency spectra of urban and indoor environments differ from the natural environment in ways that may promote the development of myopia.MethodsA total of 814 images were analyzed from three datasets; University of California Berkeley (UCB), University of Texas (UT), and Botswana (UPenn). Images were processed in Matlab (Mathworks Inc) to map the camera color characteristics to human cone sensitivities. From the photopic luminance images generated, two-dimensional spatial frequency (SF) spectra were calculated and converted to one-dimensional spectra by rotational averaging. The spatial filtering profile of a 0.4 Bangerter foil, which has been shown to induce myopia experimentally, was also determined.ResultsThe SF slope for natural scenes followed the recognized 1/fα relationship with mean slopes of −1.08, −0.90, and −1.04 for the UCB, UT and UPenn image sets, respectively. Indoor scenes had a significantly steeper slope (−1.48, UCB; −1.52, UT; P < 0.0001). Urban environments showed an intermediate slope (−1.29, UCB; −1.22, UT) that was significantly different from the slopes derived from the natural scenes (P < 0.0001). The change in SF content between natural outdoor scenes and indoors was comparable to that induced by a 0.4 Bangerter foil, which reduced the SF slope of a natural scene from −0.88 to −1.47.ConclusionsCompared to natural outdoor images, man-made outdoor and indoor environments have spatial frequency characteristics similar to those known to induce form-deprivation myopia in animal models. The spatial properties of the man-made environment may be one of the missing drivers of the human myopia epidemic.

Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory

We present three schemes to go beyond the electric-dipole approximation in x-ray absorption spectroscopy calculations within a four-component relativistic framework. The first is based on the full semi-classical light-matter interaction operator and the two others on a truncated interaction within the Coulomb gauge (velocity representation) and multipolar gauge (length representation). We generalize the derivation of the multipolar gauge to an arbitrary expansion point and show that the potentials corresponding to different expansion points are related by a gauge transformation, provided that the expansion is not truncated. This suggests that the observed gauge-origin dependence in the multipolar gauge is more than just a finite-basis set effect. The simplicity of the relativistic formalism enables arbitrary-order implementations of the truncated interactions, with and without rotational averaging, allowing us to test their convergence behavior numerically by comparison to the full formulation. We confirm the observation that the oscillator strength of the electric-dipole allowed ligand K-edge transition of TiCl4, when calculated to the second order in the wave vector, becomes negative but also show that inclusion of higher-order contributions allows convergence to the result obtained using the full light-matter interaction. However, at higher energies, the slow convergence of such expansions becomes dramatic and renders such approaches at best impractical. When going beyond the electric-dipole approximation, we therefore recommend the use of the full light-matter interaction.

Noise Reduction for Digital Communications—The Masterpiece, a Modified Costas Loop

An efficient way of noise reduction has been presented: A modified Costas loop called as Masterpiece. The basic version of the Costas loop has been developed for SSB SC demodulation, but the same circuit can be applied for QAM (quadrature amplitude modulation) demodulation as well. Noise sensitivity of the basic version has been decreased. One trick is the transformation of the real channel input into complex signal, the other one is the application of our folding algorithm. The result is that the Masterpiece provides a 4QAM symbol error rate (SER) of 6 × 10−4 for input signal to noise ratio (SNR) of −1 dB. In this paper, an improved version of the original Masterpiece is introduced. The complex channel input signal is normalized, and rotational average is applied. The 4QAM result is SER of 3 × 10−4 for SNR of −1 dB. At SNR of 0 dB, the improved version produces 100 times better SER than that the original Costas loop does. In our times, this topic has a special importance because by application of our Masterpiece, all dangerous field strengths from 5G and WiFi, could be decreased by orders of magnitude. The Masterpiece can break the Shannon formula.

Noise reduction for digital communications – A modified Costas loop

An efficient way of noise reduction has been presented: A modified Costas loop called as Masterpiece. The basic version of the Costas loop has been developed for SSB SC demodulation, but the same circuit can be applied for QAM demodulation as well. Noise sensitivity of the basic version has been decreased. One trick is the transformation of the real channel input into complex signal, the other one is the application of our folding algorithm. The result is that the Masterpiece provides a 4QAM symbol error rate (SER) of 6*10-4 for input signal to noise ratio (SNR) of -1 dB. In this paper, an improved version of the original Masterpiece is introduced. The complex channel input signal is normalized, and rotational average is applied. The 4QAM result is SER of 3*10-4 for SNR of -1 dB. At SNR of 0 dB, the improved version produces 100 times better SER than that the original Costas loop does.

Transfer and Amplification of Chirality Within the “Ring of Fire” Observed in Resonance Raman Optical Activity Experiments

We report extremely strong chirality transfer from a chiral nickel complex to solvent molecules detected as Raman optical activity (ROA). Electronic energies of the complex were in resonance with the excitation-laser light. The phenomenon was observed for a wide range of achiral and chiral solvents. For chiral 2-butanol, the induced ROA was even stronger than the natural one. The observations were related to so-called quantum (molecular) plasmons that enable a strong chiral Rayleigh scattering of the resonating complex. According to a model presented here, the maximal induced ROA intensity occurs at a certain distance from the solute, in a three-dimensional "ring of fire", even after rotational averaging. Most experimental ROA signs and relative intensities could be reproduced. The effect might significantly increase the potential of ROA spectroscopy in bioimaging and sensitive detection of chiral molecules.

Transfer and Amplification of Chirality Within the “Ring of Fire” Observed in Resonance Raman Optical Activity Experiments

AbstractWe report extremely strong chirality transfer from a chiral nickel complex to solvent molecules detected as Raman optical activity (ROA). Electronic energies of the complex were in resonance with the excitation‐laser light. The phenomenon was observed for a wide range of achiral and chiral solvents. For chiral 2‐butanol, the induced ROA was even stronger than the natural one. The observations were related to so‐called quantum (molecular) plasmons that enable a strong chiral Rayleigh scattering of the resonating complex. According to a model presented here, the maximal induced ROA intensity occurs at a certain distance from the solute, in a three‐dimensional “ring of fire”, even after rotational averaging. Most experimental ROA signs and relative intensities could be reproduced. The effect might significantly increase the potential of ROA spectroscopy in bioimaging and sensitive detection of chiral molecules.

Photoelectron Spectroscopy of Molecules Beyond the Electric Dipole Approximation.

A methodology implemented to compute photoionization cross sections beyond the electric dipole approximation using Gaussian type orbitals for the initial state and plane waves for the final state is applied to molecules of various sizes. The molecular photoionization cross sections computed for valence molecular orbitals as a function of photon energy present oscillations due to the wave-like nature of both the outgoing photoelectron and of the incoming photon. These oscillations are damped by rotational and vibrational averaging or by performing a k-point summation for the solid state case. For core orbitals, the corrections introduced by going beyond the electric dipole approximation are comparable to the atomic case. For valence orbitals, nondipole corrections to the total photoinization cross sections can reach up to 20% at photon energies above 1 keV. The corrections to the differential cross sections calculated at the magic angle are larger, reaching values between 30% and 50% for all molecules included. Our findings demonstrate that photoelectron spectroscopy, especially angle-resolved, on, e.g., molecules and clusters on surfaces, using high photon energies, must be accompanied by theories that go beyond the electric dipole approximation.

On three-dimensional rotational averages of odd-rank tensors**We dedicate this paper to Professor Wolfgang Schleich, the impact of whose abundant research accomplishments on people and their careers, and on science itself, we salute.

The recent growing interest in nonlinear optical spectroscopy in optically active medium demands the three-dimensional rotational average of high-rank tensors. For example, calculations of the coherent anti-Stokes Raman scattering in optically active medium requires the averaging of the ninth rank tensor. Following the classic work of Andrews et al (1977 J. Chem. Phys. 67 5026) we present a method for finding the rotational average of odd-rank tensors in an overcomplete basis of isotropic tensors. The method is successfully applied to the rotational averages of tensors of rank n = 5, 7, 9, 11. Explicit new results for ninth- and eleventh-rank tensors are given.

The Perpendicular Orientation of Amphotericin B Methyl Ester in Hydrated Lipid Bilayers Supports the Barrel-Stave Model.

The clinically important antibiotic amphotericin B (AmB) is a membrane-active natural product that targets membrane sterol. The antimicrobial activity of AmB is generally attributed to its membrane permeabilization, which occurs when a pore is formed across a lipid bilayer. In this study, the molecular orientation of AmB was investigated using solid-state nuclear magnetic resonance (NMR) to better understand the mechanism of antifungal activity. The methyl ester of AmB (AME) labeled with NMR isotopes, d3-AME, and its fluorinated and/or 13C-labeled derivatives were prepared. All of the AmB derivatives showed similar membrane-disrupting activities and ultraviolet spectra in phospholipid liposomes, suggesting that their molecular assemblies in membranes closely mimic those of AmB. Solid-state 2H NMR measurements of d3-AME in a hydrated membrane showed that the mobility of AME molecules depends on concentration and temperature. At a 1:5:45 AME:Erg:dimyristoylphosphatidylcholine ratio, AME became sufficiently mobilized to observe the motional averaging of quadrupole coupling. On the basis of the rotational averaging effect of 19F chemical shift anisotropy, 2H quadrupolar splitting, and 13C-19F dipolar coupling of 14β-F-AMEs, we deduced that the molecular axis of AME is predominantly parallel to the normal of a lipid bilayer. This result supports the barrel-stave model as a molecular assembly of AmB in membranes.

Reliable Biomolecular Structural Modelling with Small-Angle Scattering

The small-angle scattering (SAS) of X-rays or neutrons from biomolecules in solution can provide (1) unique structural insights, (2) information on the range of potential conformations present, (3) and powerful restraints for integrative/hybrid structural modelling (IHM). The rotational and ensemble averaging of the signal presents multiple challenges to the reliable utilisation of SAS data: (1) demonstrating SAS profile represents the species of interest, (2) reliably evaluating model fits, and (3) over-interpretation and over-fitting of data. A community driven project has been underway for more than a decade now to define the information that should be made available regarding sample quality, data acquisition and reduction, presentation of scattering data and validation, and modelling for biomolecular SAS experiments. These guidelines (Trewhella et al. (2017) Acta D73, 710) provide a reporting framework so that “readers can independently assess the quality of the data and the basis for any interpretations presented” concerning the results of bimolecular SAS experiments, including the generation or testing of 3D models. The validation task forces for SAS (Trewhella et al. (2013) Structure 21, 875) and for IHM (Sali et al. (2015) Structure 23, 1156) of the world-wide Protein Data Bank are additionally addressing the complex issues concerning the archiving and validation of IHM models that depend upon computational methods and disparate data from multiple techniques (Burley et al. (2017) Structure 25, 1317). Finally, the question of how many parameters are justified by the model evidence inherent in the SAS profile is an important question to answer when modelling conformational ensembles (Potrzebowski, Trewhella and Andr⌈ (2018) in review PLOS Computational Biology). This presentation will address the recommended reporting framework and its rationale, the reliable incorporation of SAS data in IHM, and the strengths and limits of SAS in ensemble modelling.

Configuration determination by residual dipolar couplings: accessing the full conformational space by molecular dynamics with tensorial constraints.

Residual dipolar couplings (RDCs) and other residual anisotropic NMR parameters provide valuable structural information of high quality and quantity, bringing detailed structural models of flexible molecules in solution in reach. The corresponding data interpretation so far is directly or indirectly based on the concept of a molecular alignment tensor, which, however, is ill-defined for flexible molecules. The concept is typically applied to a single or a small set of lowest energy structures, ignoring the effect of vibrational averaging. Here, we introduce an entirely different approach based on time averaged molecular dynamics with dipolar couplings as tensorial orientational restraints that can be used to solve structural problems in molecules of any size without the need of introducing an explicit molecular alignment tensor into the computation. RDC restraints are represented by their full 3D interaction tensor in the laboratory frame, for which pseudo forces are calculated using a secular dipolar Hamiltonian as the target. The resulting rotational averaging of each individual tensorial restraint leads to structural ensembles that best fulfil the experimental data. Using one-bond RDCs, the approach has been implemented in the force field procedures of the program COSMOS and extensively tested. A concise theoretical introduction, including the special treatment of force fields for stable and fast MD simulations, as well as applications regarding configurational analyses of small to medium-sized organic molecules with different degrees of flexibility, is given. The observed results are discussed in detail.