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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 433 records · Page 24
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Observation and characterisation of trapped electron modes in Wendelstein 7-X

In the past, quasi coherent (QC) modes were reported for nearly all tokamaks. The general definition describes modes as QC when the magnitude squared coherence is in the range of 0.3 to 0.6. QC modes are observed in the plasma core as well as in the plasma edge and can have quite different physical origins. The one in the core are observed in plasmas with low collisionality, where the electron temperature exceeds the ion temperature in the plasma core. This is the case for electron cyclotron heating in general. The origin of these modes are electrons trapped within a magnetic mirror, as reported in the past from various fusion devices. The so-called trapped-electron modes (TEMs) belong to drift wave instabilities and can be destabilized by electron-temperature gradients in the plasma core. From the diagnostic point of view, QC modes appear as fluctuations in electron density and temperature. Therefore, the microwave reflectometer is very well suited to monitor these modes. This paper describes experiments, conducted at the Wendelstein 7-X stellarator (W7-X), which aim at detecting QC modes at low wave numbers. A poloidal correlation reflectometer installed at W7-X, is able to measure low wave numbers ($k_\perp\unicode{x2A7D}{3.5}\,\textrm{cm}^{-1}$). For medium line-averaged densities ($\int n_\textrm{e}\unicode{x2A7D}{6 \times10^{19}}\,\textrm{m}^{-2}$) the plasma core is accessible for this diagnostic. For different magnetic configurations and plasma parameters, broad QC structures are observed in the coherence spectra. From the analysis of the rotation and the poloidal structure, these QC modes show the properties of electron-temperature-gradient driven TEMs. A linear relation between the mode velocity and the rotation frequency is found. The relation is uniform and confirms the nature of QC-mode observation as TEM in tokamaks, too.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Interplay between Cation “Coloring” and Stereochemically Active Lone Pairs in AgBiS 2 Thin Films

Solution-processed AgBiS 2 thin films were fabricated using novel thiol− amine precursor inks to investigate the stereochemical activity of Bi 3+ 6s 2 lone pairs and their impact on the structure. A dual-space analysis combining Bragg diffraction and hard X-ray photoelectron spectroscopy (HAXPES) revealed a rock salt-like average structure with local distortions linked to cation coloring. Density functional theory (DFT) and crystal orbital Hamilton population (COHP) analyses confirmed that local Bi-rich and Ag-rich nanodomains amplify stereochemical activity, whereas more mixed and cation-order nanodomains are less stereochemically active. This local, nanoscopic mixing of segregated and ordered domains would indeed explain an average Fm3̅m structure that is rock salt-like and that does not manifest the full anharmonicity and noncentrosymmetry evidenced in canonical structures with stereochemical expression. These findings provide insights into the local structural and electronic complexities governing the optoelectronic properties of AgBiS 2 thin films.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Distinguishing isotropic and anisotropic signals for X-ray total scattering using machine learning

Understanding structure–property relationships is essential for advancing technologies based on thin films. X-ray pair distribution function (PDF) analysis can access relevant atomic structure details spanning local-, mid- and long-range structure. While X-ray PDF has been adapted for thin films on amorphous substrates, measurements on single-crystal substrates are necessary to accurately determine structure origins for some thin film materials, especially those for which the substrate changes the accessible structure and properties. However, when measuring films on single-crystal substrates, high-intensity anisotropic Bragg spots saturate 2D detector images, overshadowing the thin films' isotropic scattering signal. This renders previous data processing methods for films on amorphous substrates unsuitable for films on single-crystal substrates. To address this measurement need, we developed IsoDAT2D, an innovative data processing approach using unsupervised machine learning algorithms. The program combines dimensionality reduction and clustering algorithms to separate thin film and single-crystal substrate X-ray scattering signals. We use SimDAT2D , a program we developed to generate simulated thin film data, to validate IsoDAT2D . Here we also use IsoDAT2D to isolate X-ray total scattering signal from a thin film on a single-crystal substrate. The resulting PDF data are compared with similar data processed using previous methods, especially substrate subtraction for single-crystal and amorphous substrates. PDF data from IsoDAT2D -identified X-ray total scattering data are significantly better than from single-crystal substrate subtraction, but not as reliable as PDF data from amorphous substrate subtraction. With IsoDAT2D , there are new opportunities to expand PDF to a wider variety of thin films, including those on single-crystal substrates, with which new structure–property relationships can be elucidated to enable fundamental understanding and technological advances.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Metasurface‐Based Mueller Matrix Microscope

In conventional optical microscopes, image contrast of objects mainly results from the differences in light intensity and/or color. Muller matrix optical microscopes (MMMs), on the other hand, can provide significantly enhanced image contrast and rich information about objects by analyzing their interactions with polarized light. However, state-of-the-art MMMs are fundamentally limited by bulky and slow polarization state generators and analyzers. Here, the study demonstrates a metasurface-based MMM, i.e., Meta-MMM, which is equipped with a chip-integrated, single-shot metasurface polarization state analyzer (Meta-PSA). The Meta-MMM is featured with high-speed measurement (≈2s per Muller matrix (MM) image), superior operation stability, dual-color operation, and high measurement accuracy (measurement error 1–2%) for MM imaging. The Meta-MMM is applied to nanostructure characterization, surface morphology analysis, and discovering birefringent structures in honeybee wings. As a result, the Meta-MMMs hold the promise to revolutionize various applications from biological imaging, medical diagnosis, and material characterization to industry inspection and space exploration.

36 MATERIALS SCIENCE↗
DOE OSTI2024

3D Lead-Organoselenide-Halide Perovskites and their Mixed-Chalcogenide and Mixed-Halide Alloys

Here, we incorporate Se into the 3D halide perovskite framework using the zwitterionic ligand: SeCYS ( + NH 3 (CH 2 ) 2 Se - ), which occupies both the X - and A + sites in the prototypical ABX 3 perovskite. The new organoselenide-halide perovskites: (SeCYS)PbX 2 (X=Cl, Br) expand upon the recently discovered organosulfide-halide perovskites. Single-crystal X-ray diffraction and pair distribution function analysis reveal the average structures of the organoselenide-halide perovskites, whereas the local lead coordination environments and their distributions were probed through solid-state 77 Se and 207 Pb NMR, complemented by theoretical simulations. Density functional theory calculations illustrate that the band structures of (SeCYS)PbX 2 largely resemble those of their S analogs, with similar band dispersion patterns, yet with a considerable band gap decrease. Optical absorbance measurements indeed show band gaps of 2.07 and 1.86 eV for (SeCYS)PbX 2 with X=Cl and Br, respectively. We further demonstrate routes to alloying the halides (Cl, Br) and chalcogenides (S, Se) continuously tuning the band gap from 1.86 to 2.31 eV–straddling the ideal range for tandem solar cells or visible-light photocatalysis. The comprehensive description of the average and local structures, and how they can fine-tune the band gap and potential trap states, respectively, establishes the foundation for understanding this new perovskite family, which combines solid-state and organo-main-group chemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Entropy-defect synergy for dual luminescence mechanism in spinel: Time-resolved anti-counterfeiting and fingerprint visualization

Multimodal luminescent materials, while promising for anti-counterfeiting, often lack dynamic time-dependent responses and controllable spatial distribution, limiting their encryption capabilities in the spatiotemporal dimension. Here, this work presents a coordinated control strategy based on entropy and defect engineering, and uses a backpropagation (BP) neural network for material screening to successfully prepare spinel Mg 0.8 (Fe 0.04 Co 0.04 Ni 0.04 Cu 0.04 Zn 0.04 )Cr 2 O 4 (MgA 5 CO) phosphors with time-dependent dynamic luminescence behavior. This phosphor simultaneously activated the d-d transition luminescence (∼618 nm) derived from Co 2+ /Cr 3+ and the defect luminescence (∼398 nm) related to zinc vacancies (V Zn ) in a single-phase solid solution. The phosphor exhibits a time-dependent color evolution from pink to purple under fixed-wavelength excitation, due to the different excited-state dynamics and decay lifetimes associated with the d-d transition and defect luminescence. Structural characterization and spectral analysis confirmed the existence of V Zn and its significant role in defect luminescence process. The fluorescent and dynamic luminescent properties of entropy-based spinel oxide enable its use in advanced anti-counterfeiting applications like fingerprint recognition and color-changing dedicated anti-counterfeiting mark, showing promise in high-end and time-dynamic anti-counterfeiting fields. This research not only developed a new type of fluorescent dynamic anti-counterfeiting material, but also provided a new idea for constructing advanced optical functional materials with multiple luminescence mechanisms.

Defect project↗
DOE OSTI2025

Backup power or bill savings? How electricity tariffs impact residential solar-plus-storage usage in the United States

Adoption of paired solar-plus-storage systems has accelerated in recent years, driven by both the demand for backup power and a desire to manage utility bills. Tradeoffs between those two uses can arise through the reserve setting on the battery storage system, which serves to maintain a minimum state of charge in case of a power interruption. Our paper applies an economic framework to evaluate this tradeoff in terms of changes in bill savings and customer reliability value across reserve levels, considering how those tradeoffs depend on the underlying electricity rate structure and levels. The analysis is based on a representative set of load profiles, solar profiles, tariff designs, and stochastic power interruption events across ten different regions in the United States. We find that the opportunity cost of holding storage capacity in reserve, in terms of foregone bill reductions, outweighs any gains in reliability value from mitigated power interruptions in the majority of customer situations. Higher storage reserve levels increase total customer value only in specific circumstances, such as for customers with inferior reliability (10x average interruptions), with a very high value of lost load ($50/kWh), and with tariff or interconnection rules that disallow grid charging. However, even this result is dampened when considering tariff designs with higher price differentials that increase the opportunity cost of holding storage in reserve (e.g. import/export or time-of-use rates). Allowing grid charging in tariffs essentially eliminates the necessity to hold any storage in reserve in all sensitivity cases explored.

Electric resilience↗
DOE OSTI2024

Failure investigation and mitigation after experimental research reactor fuel plate deformation in an irradiation device

Experimental research reactor fuel testing is conducted in the Belgian Reactor 2 (BR2) of the Belgian Nuclear Research Centre (SCK CEN) in dedicated irradiation vehicles or rigs. One such vehicle allows flat full-size fuel plates to be irradiated by inserting them into slotted baskets that captures a narrow portion of the longitudinal edges of the plates. The motion of the fuel plates within the baskets is possible within the narrow slots and thus, the plate is considered to be unattached. The design intentionally omits fixing mechanisms of the fuel plates to the baskets to facilitate the inspection and repositioning of the plates between the irradiation cycles and the accommodation of thermal expansion of the plates in the lateral direction. However, loosely inserted fuel plates have weak structural boundary conditions allowing for larger out-of-plane deflections caused by hydrodynamic loads exerted by the flowing coolant, as compared to those of fixed plates. Unexpected large deformations of plates occurred in several irradiation cycles that further resulted in a loss of cladding integrity. These deformations could not be attributed to a single source. This triggered a series of thermal hydraulic, structural, and fluid-structure interaction analyses aiming at understanding the observed phenomenon. The analyses revealed that, for a certain combination of unfavorable manufacturing and assembly tolerances, fuel plate edges could escape out of the slots in the irradiation basket due to the hydrodynamic load. Subsequently, the plate could become wedged inside the basket coolant channel opening. This resulted in reduced coolant flow and accelerated temperature increase and thermal expansion of the plate while under irradiation. This unfavorable feedback loop could then lead to excessive plate surface temperatures, deformed plates and cladding failure, as was observed in the experiments. These analyses not only provided a probable cause of the fuel plate failures, but also resulted in a new and improved design of the irradiation basket to avoid these issues in the future. In conclusion, a series of recent successful irradiations confirm that the sources of failures were identified correctly, and the implemented mitigations were adequate.

BR2↗
DOE OSTI2026

Comparison of URANS and LES predictions for the open phase of the OECD NEA CSNI fluid structure interaction CFD benchmark

The OECD NEA CSNI WGAMA CFD Task Group ran a benchmark in 2020 and 2021 to assess the predictive capabilities of coupled fluid structure interaction (FSI) CFD analysis methods. This paper presents the predictions made for the open phase of the benchmark using URANS and LES turbulence modelling approaches, and a comparison of the results to the experimental data. The benchmark comprised a channel containing two inline cylinders in cross-flow. The cylinders were fixed at one end, free at the other, and had measured resonant frequencies and damping properties. The URANS modelling used ANSYS Fluent 2-way coupled to ANSYS Mechanical. The LES modelling used Nek5000, 1-way coupled to Diablo. Comparisons with cross-channel velocity profiles are presented, both for the mean flow and its RMS. Comparisons are also made to the frequency spectra for point measurements of fluid velocity and pressure, and for the accelerations of the free end of each cylinder. URANS predicts the average velocity profiles relatively well, and is able to predict the velocity and acceleration spectra at the shedding frequency. However, the frequency content at the 4th harmonic of the shedding frequency is low in the URANS flow fields, and so does not excite accelerations at the resonant frequency of the cylinders. LES makes better predictions of the average profiles, and the velocity spectra agree well at both the shedding frequency and at higher frequencies. In conclusion, the 1-way coupled LES results show good agreement for acceleration spectra.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

Enhancement of Uranium Ionization Efficiencies Using Zn-MOF-74 Derived Nanoporous Ion Emitters for Thermal Ionization Mass Spectrometry

The recent introduction of nano-porous ion emitters (nano-PIEs) formed from metal organic frameworks (MOFs) has demonstrated the potential to enhance sensitivity for thermal ionization mass spectrometry (TIMS). Nano-PIEs take advantage of the parent MOF’s chemical and structural tunability to form scaffolds for ion emitters. A study by Barpaga et al. 2023 on MOF-74 as the parent material found that with high volatility metals in their framework uranium sample utilization efficiency (SUE) increases by up to nine-fold (e.g., Zn-MOF-74) compared to that of the analyte on a bare filament (~0.05%). Here, in this study, we investigate the performance of Zn-MOF-74 to maximize uranium efficiencies at the trace level (= 10 -12 g) by altering the parent MOF morphology and chemistry (i.e., MOF crystal sizes and thermal degradation) and optimizing its integration with TIMS (i.e., MOF mass on a filament and ramp conditions). We observed improvement in SUE up to 20 times (≤1.0%) that of a bare filament load when nano-PIEs derived from nanocrystals of Zn-MOF-74 were heated under a specific current ramp condition. This demonstrates that rates of nano-PIE structural collapse during TIMS analysis and the subsequently formed nanomaterials (and their features) can be tuned to control analyte ionization.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

First-Principles Simulations Correlating X-ray Absorption Spectroscopy Features to Point Defects in h -BN

Hexagonal boron nitride (h-BN) is a promising material for a range of emerging applications in electronics, quantum information technology, and energy storage. Soft X-ray absorption spectroscopy (XAS) is powerful to reveal atomic details of BN, especially in the presence of defects. However, correlating XAS spectral features with specific defect types remains elusive. In this Letter, we report B K-edge XAS measurements of sputter-deposited turbostratic h-BN films and use a combination of first-principles spectroscopic simulations and analysis of detailed electronic structure and local charge transfer characteristics to elucidate their unique spectroscopic features. Our results show that the two main defect-related peaks, between the main π* resonances of h-BN and B2O3, as typically observed in BN films deposited by energetic condensation or bombarded with energetic ions, are associated with electronic states of H-passivated B atoms bonded to one and two oxygen impurity atoms, respectively. These conclusions hold significant implications for applications relying on defect-mediated properties of h-BN.

chemical structure↗
DOE OSTI2025

Advancing AI-Driven Analysis in X-ray Absorption Spectroscopy: Spectral Domain Mapping and Universal Models

In recent years, rapid progress has been made in developing artificial intelligence (AI) and machine learning (ML) methods for X-ray absorption spectroscopy (XAS) analysis. Compared to traditional XAS analysis methods, AI/ML approaches offer dramatic improvements in efficiency and help eliminate human bias. To advance this field, we advocate an AI-driven XAS analysis pipeline that features several interconnected key building blocks: benchmarks, workflows, databases, and AI/ML models. Specifically, we present two case studies for XAS ML. In the first study, we demonstrate the importance of reconciling the discrepancies between simulation and experiment using spectral domain mapping (SDM). Our ML model, which is trained solely on simulated spectra, predicts an incorrect oxidation state trend for Ti atoms in a combinatorial zinc titanate film. After transforming the experimental spectra into a simulation-like representation using SDM, the same model successfully recovers the correct oxidation state trend. In the second study, we explore the development of universal XAS ML models that are trained on the entire periodic table, which enables them to leverage common trends across elements. Looking ahead, we envision that an AI-driven pipeline can unlock the potential of real-time XAS analysis to accelerate scientific discovery.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Autonomous phase mapping of gold nanoparticles synthesis with differentiable models of spectral shape

Autonomous experimentation–or self-driving labs–offers a systematic approach to accelerate materials discovery by integrating automated synthesis, characterization, and data-driven decision-making. We present a closed-loop workflow for the on-demand synthesis and structural characterization of colloidal gold nanoparticles, enabling direct mapping from composition to nanoscale structure. Our framework leverages differentiable models of spectral shape to address two central tasks in self-driving labs: (a) phase mapping, or identifying compositional regions with distinct structural behavior; and (b) material retrosynthesis, or optimizing compositions for target structure. Using functional data analysis, we develop a data-driven model with generative pre-training, active learning, and high-throughput experiments to predict spectral responses across composition space. We demonstrate the approach on seed-mediated growth of gold nanoparticles, showcasing its ability to extract design rules, reveal secondary interactions, and efficiently navigate morphology space. Gradient-based optimization of the models enables inverse design, making this a unified platform.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Unconventional nonlinear Hall effects in twisted multilayer 2D materials

We present the first investigation of unusual nonlinear Hall effects in twisted multilayer 2D materials. Contrary to expectations, our study shows that these nonlinear effects are not merely extensions of their monolayer counterparts. Instead, we find that stacking order and pairwise interactions between neighboring layers, mediated by Berry curvatures, play a pivotal role in shaping their collective nonlinear optical response. By combining large-scale Real-Time Time-Dependent Density Functional Theory (RT-TDDFT) simulations with model Hamiltonian analyses, we demonstrate a remarkable second-harmonic transverse response in hexagonal boron nitride four-layers, even in cases where the total Berry curvature cancels out. Furthermore, our symmetry analysis of the layered structures provides a simplified framework for predicting nonlinear responses in multilayer materials in general. Our investigation challenges the prevailing understanding of nonlinear optical responses in layered materials and opens new avenues for the design and development of advanced materials with tailored optical properties.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Advanced spectroscopic studies of (PPh 4 ) 2 [Co(N 3 ) 4 ], a field-induced single-ion magnet

The high-spin Co II complex (PPh 4 ) 2 [Co(N 3 ) 4 ] (Co-N 3 )has been investigated using advanced spectroscopic techniques [far-IR magneto-spectroscopy (FIRMS), high-frequency and high-field EPR (HFEPR), and inelastic neutron scattering (INS)] to study its zero-field-splitting (ZFS), giving spin-Hamiltonian (SH) parameters. The analysis of multi-frequency HFEPR reveals the easy-axis anisotropy with a D value of −10.39(5) cm −1 and a rhombic ratio (E/D) of 0.21(1). The magnetic properties have also been probed by direct-current (DC) magnetometry, suggesting minor differences in anisotropy from the previously reported polymorph (Co-N 3 ′). Ligand-field theory (LFT) analysis indicates that the structures of Co-N 3 and Co-N 3 ′ are closer to D 2d symmetry than other symmetries considered. Alternate-current (AC) susceptibility reveals slow magnetic relaxation under an applied field, indicating that Co-N 3 is a field-induced single-ion magnet (SIM). Here, while both Co-N 3 and Co-N 3 ′ were studied by DC magnetometry, one unusual aspect of the current work on Co-N 3 is that advanced spectroscopies HFEPR, FIRMS, and INS were used to directly observe transitions between ZFS split states, giving accurate SH parameters.

Hand, Adam T. [Univ. of Tennessee, Knoxville, TN (↗
DOE OSTI2025

Topological perturbation to a standard dehydrogenation catalyst, Pt 3 Sn

Topological materials, which exhibit protected topological surface states (TSS) near the Fermi level, have been proposed to be good catalysts. Topological catalysis may be more prevalent than we suspect, and not limited to exotic new materials. Here we study a known dehydrogenation catalyst, Pt 3 Sn alloy, which happens to be a topological semimetal, and probe the participation of TSSs in catalytic dehydrogenation of methane catalyzed by this material. Through first principle modeling and detailed analysis of the electronic structure for topological and non-topological surfaces of Pt 3 Sn, we find that TSS get significantly altered by the binding of reaction intermediates, particularly H. However, this effect of TSS on the binding of the reagents is merely perturbative, as the majority of the adsorbate binding is achieved by not-surface-focused electronic states, located much deeper below the Fermi level. Therefore, the reaction energetics and selectivity are predominantly determined by electronic states other than TSS. The fact that TSS are available for the reagent binding does not alone guarantee that the catalysis is strongly driven by TSS. However, TSS are not to be ignored, as small changes in the energetics along the reaction profile can translate into substantial differences in the reaction rate. Hence, in our view, Pt 3 Sn – a topological material – is first and foremost a standard catalyst, with added topological features, and not purely a topological catalyst. Our results point at the need to carefully consider all the bonding effects at the topological material interface.

Chemistry↗
DOE OSTI2024

High repetition-rate 0.5 Hz broadband neutron source driven by the Advanced Laser Light Source

Neutron beams are an essential tool to investigate material structure and perform nondestructive analysis, as they give unique access to element composition, thus ideally complementing density analysis allowed by standard x-rays investigation. Laser-driven neutron sources, though compact and cost-effective, currently have lower average flux than conventional neutron sources, due to the limited repetition rate of the lasers used so far. However, advancements in laser technology allow nowadays to address this challenge. Here, we report results obtained at the Advanced Laser Light Source characterizing stable production of broadband (0.1–2 MeV) neutrons produced at a high repetition rate (0.5 Hz). The interaction of laser pulses of 22 fs duration and 3.2 J on-target energy with 2-μm-thick tantalum targets produced protons in the Target Normal Sheath Acceleration (TNSA) regime up to 7.3 MeV. These protons were subsequently converted into neutrons by (p,n) reactions in lithium fluoride (LiF). Activation measurements and bubble detectors were used to characterize neutron emissions, with a neutron fluence of up to ∼1.4×105 neutrons/shot/sr and energies mainly between a few hundred of kilo-electron volt and 2 MeV. The total neutron yield was ∼5×105 neutrons/shot. This paves the way for numerous applications, e.g., in homeland security, materials science, or cultural heritage.

Physics↗
DOE OSTI2025

On the discretization error of the discrete generalized quantum master equation

The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 112 , 110401 (2014)] can be considered a discrete-time formulation of the Nakajima–Zwanzig quantum master equation (NZ-QME) for modeling non-Markovian quantum dynamics. A recent paper [Makri, J. Chem. Theory Comput. 21 , 5037 (2025)] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time t = 0. Here, this work presents a detailed analysis of the discretization structure of the TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel K N and the continuous-time NZ-QME kernel $\mathscr{K}$( N Δ t ). This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as Δ t → 0. While the TTM provides a consistent discretization, we note that alternative schemes are also viable, such as the midpoint derivative/midpoint integral scheme proposed in Makri’s work. The relative performance of various schemes for either computing accurate $\mathscr{K}$( N Δ t ) from exact dynamics or obtaining accurate dynamics from exact $\mathscr{K}$( N Δ t ) warrants further investigation.

Density-matrix↗