Three-dimensional Distribution Function Research Articles

Resolving Photoinduced Femtosecond Three-Dimensional Solute-Solvent Dynamics through Surface Hopping Simulations.

Photoinduced dynamics in solution is governed by mutual solute-solvent interactions, which give rise to phenomena like solvatochromism, the Stokes shift, dual fluorescence, or charge transfer. Understanding these phenomena requires simulating the solute's photoinduced dynamics and simultaneously resolving the three-dimensional solvent distribution dynamics. If using trajectory surface hopping (TSH) to this aim, thousands of trajectories are required to adequately sample the time-dependent three-dimensional solvent distribution functions, and thus resolve the solvent dynamics with sub-Ångstrom and femtosecond accuracy and sufficiently low noise levels. Unfortunately, simulating thousands of trajectories with TSH in the framework of hybrid quantum mechanical/molecular mechanical (QM/MM) can be prohibitively expensive when employing ab initio electronic structure methods. To tackle this challenge, we recently introduced a computationally efficient approach that combines efficient linear vibronic coupling models with molecular mechanics (LVC/MM) via electrostatic embedding [Polonius et al., JCTC 2023, 19, 7171-7186]. This method provides solvent-embedded, nonadiabatically coupled potential energy surfaces while scaling similarly to MM force fields. Here, we employ TSH with LVC/MM to unravel the photoinduced dynamics of two small thiocarbonyl compounds solvated in water. We describe how to estimate the number of trajectories required to produce nearly noise-free three-dimensional solvent distribution functions and present an analysis based on approximately 10,000 trajectories propagated for 3 ps. In the electronic ground state, both molecules exhibit in-plane hydrogen bonds to the sulfur atom. Shortly after excitation, these bonds are broken and reform perpendicular to the molecular plane on timescales that differ by an order of magnitude due to steric effects. We also show that the solvent relaxation dynamics is coupled to the electronic dynamics, including intersystem crossing. These findings are relevant to advance the understanding of the coupled solute-solvent dynamics of solvated photoexcited molecules, e.g., biologically relevant thio-nucleobases.

Quantitative three-dimensional local order analysis of nanomaterials through electron diffraction

Structure-property relationships in ordered materials have long been a core principle in materials design. However, the introduction of disorder into materials provides structural flexibility and thus access to material properties that are not attainable in conventional, ordered materials. To understand disorder-property relationships, the disorder – i.e., the local ordering principles – must be quantified. Local order can be probed experimentally by diffuse scattering. The analysis is notoriously difficult, especially if only powder samples are available. Here, we combine the advantages of three-dimensional electron diffraction – a method that allows single crystal diffraction measurements on sub-micron sized crystals – and three-dimensional difference pair distribution function analysis (3D-ΔPDF) to address this problem. In this work, we compare the 3D-ΔPDF from electron diffraction data with those obtained from neutron and x-ray experiments of yttria-stabilized zirconia (Zr0.82Y0.18O1.91) and demonstrate the reliability of the proposed approach.

A gated-time-of-flight top-hat electrostatic analyzer for low energy ion measurements

When incorporated into a top-hat electrostatic analyzer, a gate electrode enables the separation of ions by their mass-per-charge with modest mass resolution (M/∆M ∼ 10). Gated-time-of-flight (TOF) instruments avoid the energy straggling and angular scattering effects prevalent in foil-based detection systems, providing more pristine measurements of three-dimensional distribution functions of incident ions. Gated-TOF implementations are ideal for measuring the properties of low-energy (i.e., <100 eV) thermal ions in various space environments. We present an instrument prototype capable of separating H+, He+, O+, and O2+ in Earth’s ionosphere and demonstrate that in addition to providing species determination, precise operation of the gate electrode provides an electronically adjustable geometric factor that can extend a single instrument’s dynamic range by several orders of magnitude.

PRESENTATION OF THE GRAVITY FIELD OF CELESTIAL BODIES USING THE POTENTIALS OF FLAT ELLIPSOIDAL DISCS

One of the possible ways for representing the external gravitational field of the planet by the potentials of flat discs, based on the classical potential theory, is proposed. At the same time, the potentials of a single- and double-layer are used for the description with the placement of the integration regions in the equatorial plane. The coefficients of the series expansion of these functions are linear combinations of the Stokes constants of the gravitational field and are uniquely expressed in terms of them. Series terms are single- or double-layer potentials. This makes it possible to calculate these terms using the results of the ellipsoid potential theory. The convergence of such a series, in contrast to the traditional one for spherical functions, is much wider and practically covers the effect of the external potential excluding the region of integration, including in the superficial parts of the surface. Since there is no problem with the convergence of the obtained expansions, we can interpret the obtained results more fully. The construction of flat density distributions for the potentials of a single and double layer is an additional tool in the study of the internal structure of the celestial body, as it is essentially a projection of the volume density of the planet’s interior onto the equatorial plane. Therefore, the extrema of these functions combine the features of the three-dimensional distribution function of the planet’s interior

Direct interpretation of the X-ray and neutron three-dimensional difference pair distribution functions (3D-ΔPDFs) of yttria-stabilized zirconia.

Three-dimensional difference pair distribution functions (3D-ΔPDFs) from X-ray and neutron diffraction experiments are reported for yttria-stabilized zirconia (Zr0.82Y0.18O1.91). A quantitative analysis of the signatures in the three-dimensional difference pair distribution functions is used to establish that oxygen ions neighbouring a vacancy shift by 0.525 (5) Å along ⟨1, 0, 0⟩ towards the vacancy while metal ions neighbouring a vacancy shift by 0.465 (2) Å along ⟨1, 1, 1⟩ away from the vacancy. The neutron 3D-ΔPDF shows a tendency for vacancies to cluster along ⟨½, ½, ½⟩, which results in sixfold coordinated metal ions.

Dynamic correlations and possible diffusion pathway in the superionic conductor Cu2-xSe.

The superionic conductor Cu2-xSe has regained interest as a thermoelectric material owing to its low thermal conductivity, suggested to arise from a liquid-like Cu substructure, and the material has been coined a phonon-liquid electron-crystal. Using high-quality three-dimensional X-ray scattering data measured up to large scattering vectors, accurate analysis of both the average crystal structure as well as the local correlations is carried out to shed light on the Cu movements. The Cu ions show large vibrations with extreme anharmonicity and mainly move within a tetrahedron-shaped volume in the structure. From the analysis of weak features in the observed electron density, the possible diffusion pathway of Cu is identified, and it is clear from its low density that jumps between sites are infrequent compared with the time the Cu ions spend vibrating around each site. These findings support the conclusions drawn from recent quasi-elastic neutron scattering data, casting doubt on the phonon-liquid picture. Although there is diffusion of Cu ions in the structure, making it a superionic conductor, the jumps are infrequent and probably not the origin of the low thermal conductivity. From three-dimensional difference pair distribution function analysis of the diffuse scattering data, strongly correlated movements are identified, showing atomic motions which conserve interatomic distances at the cost of large changes in angles.

The Solar Probe ANalyzer—Ions on the Parker Solar Probe

The Solar Probe ANalyzer for Ions (SPAN-I) onboard NASA’s Parker Solar Probe spacecraft is an electrostatic analyzer with time-of-flight capabilities that measures the ion composition and three-dimensional distribution function of the thermal corona and solar-wind plasma. SPAN-I measures the energy per charge of ions in the solar wind from 2 eV to 30 keV with a field of view of 247.°5 × 120° while simultaneously separating H+ from He++ to develop 3D velocity distribution functions of individual ion species. These observations, combined with reduced distribution functions measured by the Sun-pointed Solar Probe Cup, will help us further our understanding of the solar-wind acceleration and formation, the heating of the corona, and the acceleration of particles in the inner heliosphere. This paper describes the instrument hardware, including several innovative improvements over previous time-of-flight sensors, the data products generated by the experiment, and the ground calibrations of the sensor.

Gr Predictor: A Deep Learning Model for Predicting the Hydration Structures around Proteins.

Among the factors affecting biological processes such as protein folding and ligand binding, hydration, which is represented by a three-dimensional water site distribution function around the protein, is crucial. The typical methods for computing the distribution functions, including molecular dynamics simulations and the three-dimensional reference interaction site model (3D-RISM) theory, require a long computation time ranging from hours to tens of hours. Here, we propose a deep learning (DL) model that rapidly estimates the distribution functions around proteins obtained using the 3D-RISM theory from the protein 3D structure. The distribution functions predicted using our DL model are in good agreement with those obtained using the 3D-RISM theory. Particularly, the coefficient of determination between the distribution function obtained by the DL model and that obtained using the 3D-RISM theory is approximately 0.98. Furthermore, using a graphics processing unit, the prediction by the DL model is completed in less than 1 min, more than 2 orders of magnitude faster than the calculation time of the 3D-RISM theory. The position of water molecules around the protein was estimated based on the distribution function obtained by our DL model, and the position of waters estimated by our DL model was in good agreement with that of water molecules estimated using the 3D-RISM theory and of crystallographic waters. Therefore, our DL model provides a practical and efficient way to calculate the three-dimensional water site distribution functions and to estimate the position of water molecules around the protein. The program called "gr Predictor" is available under the GNU General Public License from https://github.com/YoshidomeGroup-Hydration/gr-predictor.

Short-range order in the quasiliquid phases of alkane substructures within aperiodic urea inclusion crystals

$n$-alkane/urea inclusion compounds are prototypical examples of aperiodic crystals with uniformly aligned alkane guest molecules contained within linear channels formed by the supramolecular urea host. Here, we investigate single-crystal diffuse scattering, which is present in the form of sharp layers, from short-chain guest molecules (from octane to dodecane). The strong modulation of diffuse scattering within these layers shows that the positions of alkane molecules in neighboring channels are correlated. Using the three-dimensional difference pair distribution function method, we have extracted the effective interaction potentials and have shown that the interaction is mediated by the relaxation of urea molecules.

Order-Disorder Transitions in (Ca_{x}Sr_{1-x})_{3}Rh_{4}Sn_{13}.

The classification of structural phase transitions as displacive or order-disorder in character is usually based on spectroscopic data above the transition. We use single crystal x-ray diffraction to investigate structural correlations in the quasiskutterudites, (Ca_{x}Sr_{1-x})_{3}Rh_{4}Sn_{13}, which have a quantum phase transition at x∼0.9. Three-dimensional pair distribution functions show that the amplitudes of local atomic displacements are temperature independent below the transition and persist to well above the transition, a signature of order-disorder behavior. The implications for the associated electronic transitions are discussed.

Longitudinal and transverse polarizations of Λ hyperon in unpolarized SIDIS and e+e− annihilation

We make a systematic study of $\mathrm{\ensuremath{\Lambda}}$ hyperon polarizations in unpolarized lepton induced semi-inclusive reactions such as ${e}^{\ensuremath{-}}N\ensuremath{\rightarrow}{e}^{\ensuremath{-}}\mathrm{\ensuremath{\Lambda}}X$ and ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}hX$. We present the general form of cross sections in terms of structure functions obtained from a general kinematic analysis. This already shows that the produced hyperons can be polarized in three orthogonal directions, i.e., the longitudinal direction along the hyperon momentum, the normal direction of the production plane, and the transverse direction in the production plane. We present the parton model results at the leading twist and leading order in perturbative QCD using transverse momentum dependent factorization and provide the expressions for these structure functions and polarizations in terms of three-dimensional parton distribution functions and fragmentation functions. We emphasize in particular that by studying the longitudinal polarization and the transverse polarization in the production plane, we can extract the corresponding chiral-odd fragmentation functions ${H}_{1Lq}^{\ensuremath{\perp}\mathrm{\ensuremath{\Lambda}}}$, ${H}_{1Tq}^{\mathrm{\ensuremath{\Lambda}}}$, and ${H}_{1Tq}^{\ensuremath{\perp}\mathrm{\ensuremath{\Lambda}}}$. We also present numerical results of rough estimates utilizing available parametrizations of fragmentation functions and approximations. We discuss how to measure these polarizations and point out in particular that they can be carried out in future EIC and/or ${e}^{+}{e}^{\ensuremath{-}}$ annihilation experiments such as Belle.

Development of a deep-learning model for predicting the hydration structures around proteins

Hydration is an important factor in understanding a variety of biological processes, such as protein folding and ligand binding. With the use of all-atom molecular dynamics simulation, the three-dimensional distribution function of water molecules around a protein can be obtained as a hydration structure. However, its high computational cost hinders its application to a large number of proteins. In the present paper, we propose that a hybrid of the 3D-RISM theory and a deep-learning technique can resolve this issue.

A non-Darcy gas flow model for coalbed methane in mine gobs

The expected extraction efficiency of coalbed methane (CBM) depends significantly on the laws considered to govern its gas flow. This study applies a non-Darcy gas flow model to describe the CBM migration in mine gobs; by mine gob, we mean that it is a fractured zone along with massive cracks and the primary place where gas flows after mining activities. A permeation experiment involving crushed sandstone is first conducted to prove the CBM undergoes Forchheimer-type non-Darcy flow. Subsequently, the three-dimensional continuous distribution functions of the permeability parameters are determined. The non-Darcy flow model includes the influence of inertial force on the gas flow, which is neglected in the Darcy model. A coupling model is established based on the experimental results and the gas flow characteristics in different regions. Thereafter, the model and distribution functions are applied to a series of numerical simulations of CBM extraction at the Sihe coal mine in China, to ascertain the most appropriate location for a ground borehole. These simulations involve boreholes placed in three different zones: the natural accumulation zone (NAZ), the load affected zone (LAZ), and the compaction stable zone (CSZ). The simulation results show that the total extraction quantity expected from the borehole in the NAZ is 2.4 and 13.5 times that from the boreholes in the LAZ and CSZ, respectively. This confirms that the NAZ is the most suitable zone for a borehole. This research ultimately provides a realistic gas flow model for CBM extraction from mine gobs.

On single-crystal total scattering data reduction and correction protocols for analysis in direct space

Data reduction and correction steps and processed data reproducibility in the emerging single-crystal total-scattering-based technique of three-dimensional differential atomic pair distribution function (3D-ΔPDF) analysis are explored. All steps from sample measurement to data processing are outlined using a crystal of CuIr2S4 as an example, studied in a setup equipped with a high-energy X-ray beam and a flat-panel area detector. Computational overhead as pertains to data sampling and the associated data-processing steps is also discussed. Various aspects of the final 3D-ΔPDF reproducibility are explicitly tested by varying the data-processing order and included steps, and by carrying out a crystal-to-crystal data comparison. Situations in which the 3D-ΔPDF is robust are identified, and caution against a few particular cases which can lead to inconsistent 3D-ΔPDFs is noted. Although not all the approaches applied herein will be valid across all systems, and a more in-depth analysis of some of the effects of the data-processing steps may still needed, the methods collected herein represent the start of a more systematic discussion about data processing and corrections in this field.

A method for calculating the nonequilibrium entropy of a flowing polymer melt via atomistic simulation.

Nonequilibrium thermodynamics as applied to polymeric liquids is limited by the inability to quantify the configurational entropy. There is no known experimental method to determine it rigorously. Theoretically, entropy is based entirely on the configurational microstate of the material, but for polymer liquids, the number of available configurations is immense and covers long length scales associated with the chain-like nature of the constituent molecules. In principle, however, it should be possible to calculate the entropy from a realistic molecular dynamics simulation that contains positional data for each atomic unit making up the polymer macromolecules. However, there are two challenges in calculating the entropy from an atomistic simulation: it is necessary to relate atomic positions to configurational mesostates, depending on the degree of coarse-graining assumed (if any), and then to entropy, and considerable computational resources are required to determine the three-dimensional probability distribution functions of the configurational mesostates. In this study, a method was developed to calculate nonequilibrium entropy using 3d probability distributions for a linear, entangled polyethylene melt undergoing steady-state shear and elongational flow. An approximate equation expressed in terms of second moments of the 3d distributions was also examined, which turned out to provide almost identical values of entropy as the fully 3d distributions at the mesoscopic level associated with the end-to-end vector of the polymer chains.

Comprehensive 3D-RISM Analysis of the Hydration of Small Molecule Binding Sites in Ligand-Free Protein Structures

Ligand binding to proteins involves displacing water molecules at the binding site and rearranging water molecules close to the ligand. The hydration state of ligand binding sites is important for the stability of the ligand-protein complex. Using the three-dimensional reference interaction site model (3D-RISM) theory, here we calculated the three-dimensional distribution function of the water oxygen site, gO(r), for 3,706 ligand-free protein structures of small molecule-protein complexes [1]. The 3D-RISM theory allowed the first comprehensive analysis of the hydration state for a large number of protein structures, a difficult subject for existing methods based on molecular dynamics simulations. The probability distribution of gO(r) at the positions of approximately 620,000 crystallographic waters (CWs) was calculated and analyzed. The gO(r) for CWs close to the ligand revealed that several CWs were stabilized by interaction networks formed between the ligand, CW, and protein. Analysis of gO(r) at the position of ligand heavy atoms indicated that, in the crystallographic binding poses (correct poses) of the ligand, hydrogen bond interactions are dominant in the highly hydrated regions whereas weak interactions such as CH-O are dominant in the moderately hydrated regions. Further, polar heteroatoms of the ligand tend to occupy highly hydrated regions in the correct poses and less hydrated regions in the wrongly docked poses, suggesting a way to distinguish these two poses. [1] T. Yoshidome, M. Ikeguchi, and M. Ohta, J. Comp. Chem., 41, 2406 (2020).

ГЕОДИНАМІКА

Purpose. To create an algorithm for constructing a three-dimensional masses distribution function of the planet and its derivatives taking into account the Stokes constants of arbitrary orders. Being based on this method, the task is to perform the research on the internal structure of the Earth. Methodology. The derivatives of the inhomogeneous mass distribution are presented by linear combinations of biorthogonal polynomials which coefficients are obtained from the system of equations. These equations follow from integral transformations of the Stokes constants, the calculation process is carried out by a sequential approximation and for the initial approximation we take a one-dimensional density model that is consistent with Stokes constants up to the second inclusive order. Next, the coefficients of expansion of the potential of higher orders are determined up to a predetermined order. In this case, the information on the power moments of the density of surface integrals makes it possible to analyze and control the iterative process. Results. The results of calculations using the software according to the described algorithm are obtained. A fairly high degree of approximation (sixth order) of three-dimensional mass distributions function is achieved. Carto diagrams were created being based on the values of deviations of the three-dimensional average distributions (“isodens”), which give a fairly detailed picture of the Earth’s internal structure. The presented maps of “inhomogeneity’s” at characteristic depths (2891 km core – mantle, 5150 km internal – external core) allow us to draw preliminary conclusions about global mass movements. At the same time, the information on derivatives is significant for interpretation. First of all, it should be noted that the gradient of “inhomogeneity’s” is directed toward the center of mass. The presented projections of this gradient on a plane perpendicular to the rotation axis (horizontal plane) show the tendency of spatial displacements. Scientific novelty. Vector diagrams of the gradient, in combination with carto diagrams, give a broad picture of the dynamics and possible mechanisms of mass movement within the planet. To a certain extent, these studies confirm the phenomenon of gravitational convection of masses. Practical significance. The proposed algorithm can be used in order to build regional models of the planet, and numerical results can be used to interpret global and local geodynamic processes inside and on the Earth’s surface.

Elastothermodynamic damping modeling of three-layer Kirchhoff–Love microplate considering three-dimensional heat conduction

Elastothermodynamic damping is one of the most important energy loss mechanisms in micro/nano mechanical resonators, which is related to the high sensitivity and high resonance frequency characteristics of resonators. With the wide application of resonators, multilayer microplate resonators have been applied. In this paper, the elastothermodynamic damping models of three-layer of Kirchhoff–Love microplate under three typical boundary support conditions of Clamped-Clamped-Clamped-Clamped (C-C-C-C), Clamped-Free-Clamped-Free (C-F-C-F) and Clamped-Free-Free-Free (C-F-F-F) are established. Firstly, the thermoelastic mechanical equations of the microplate are given, and then the three-dimensional temperature distribution functions of the microplate are obtained based on the generalized orthogonal function method. Finally, the elastothermodynamic damping models are established based on the composite multilayer plate theory. The model in this paper is compared with the previous model and the finite element results.

Comprehensive 3D-RISM analysis of the hydration of small molecule binding sites in ligand-free protein structures.

Hydration is a critical factor in the ligand binding process. Herein, to examine the hydration states of ligand binding sites, the three‐dimensional distribution function for the water oxygen site, , is computed for 3,706 ligand‐free protein structures based on the corresponding small molecule–protein complexes using the 3D‐RISM theory. For crystallographic waters (CWs) close to the ligand, reveals that several CWs are stabilized by interaction networks formed between the ligand, CW, and protein. Based on the for the crystallographic binding pose of the ligand, hydrogen bond interactions are dominant in the highly hydrated regions while weak interactions such as CH‐O are dominant in the moderately hydrated regions. The polar heteroatoms of the ligand occupy the highly hydrated and moderately hydrated regions in the crystallographic (correct) and wrongly docked (incorrect) poses, respectively. Thus, the of polar heteroatoms may be used to distinguish the correct binding poses.

Effect of gas temperature on CO2+ ion transport in CO2

The effect of gas temperature on the transport of the CO2+ ion in its parent gas CO2 is studied. For this purpose, a Monte Carlo (MC) solution technique of the ion transport equation developed in the past by the authors, which was formally proven to provide the exact solution of the transport equation, is used. The study is performed first by benchmarking against experimental data and another MC code, using cross sections from literature and then changing the gas temperature. The solution is characterized through its moments such as reduced mobility, coefficients of its expansion in Legendre polynomials and three-dimensional ion velocity distribution function. It is shown that the gas temperature has a significant effect on all these quantities. This finding has important consequences for plasma modeling which are discussed.