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DOE OSTI2024

Addressing Experimental Challenges in Probing Dark Energy with Accuracy and Precision with the Rubin Observatory Legacy Survey of Space & Time (LSST)

With support from this award, the Stanford group under the leadership of PI Patricia Burchat focused on pixel-level algorithms that are key to the investigation of dark energy with the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST). The work was carried out through strong engagement in the DOE-supported LSST Dark Energy Science Collaboration (DESC), for which the PI served as Deputy Spokesperson (07/2019 - 07/2021) and Technical Coordinator (07/2021 - 07/2023). Members of the Burchat group are Full DESC members and are among scientists who have been accepted to join the Rubin Observatory System Integration, Test, and Commissioning (SIT-Com) effort. The specific intellectual focus of the group was the optimization of gravitational lensing as a probe of the evolution of the distribution of dark matter in space and time to learn about the nature of dark energy. The Stanford group pursued an integrated research program to understand in detail, and then remediate or calibrate, systematic effects that could limit the sensitivity of the LSST to cosmic shear as a dark energy probe -- particularly those related to our understanding of the atmospheric point spread function (PSF), challenges associated with blended galaxy images, and impacts of residual camera artifacts. Key to this work was developing, enhancing, and validating the fidelity of simulations that play a key role in the development of our understanding of subtle systematic biases and then using these simulations to develop techniques to characterize and mitigate biases, and associated algorithms for implementation in the LSST or DESC science pipelines. The group’s work included contributions to the open-source GalSim simulation package, which is widely used in the community; simulations and studies of spatial correlations of atmospheric PSFs; studies of temporal correlations of the atmospheric PSF; impacts of chromatic PSFs on cosmic shear calibration; continued studies of anisotropic Gaussian process interpolation of PSF models across the focal plane; and measuring and mitigating impacts of residual camera artifacts on cosmic shear. A primary impact of this work is that it increases the scientific discovery potential of the DOE and NSF investments in the Rubin Observatory Legacy Survey of Space \& Time, by addressing challenges in interpreting astronomical images through more accurate descriptions of the blurring of images due to the atmosphere. Specifically, this work impacts our ability to understand the fundamental nature of dark energy.

Burchat, Patricia Rose↗
DOE OSTI2026

Robustness of the Ensemble Score Filter to the Type of Assimilated Observation Networks

Recent advances in data assimilation (DA) have focused on developing more flexible approaches that can better accommodate nonlinearities in models and observations. However, it remains unclear how the performance of these advanced methods depends on the observation network characteristics. In this study, we present initial experiments with the surface quasi‐geostrophic model, in which we compare a recently developed ensemble filter using score‐based diffusion models with the standard Local Ensemble Transform Kalman Filter (LETKF). Our results show that the analysis solutions respond differently to the number, spatial distribution, and nonlinear fraction of assimilated observations. We also find notable changes in the multiscale characteristics of the analysis errors. Given that standard DA techniques will eventually be replaced by more advanced methods, we hope this study sets the ground for future efforts to reassess the value of Earth observing systems in the context of newly emerging algorithms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Poisson-response Tensor-on-Tensor Regression and Applications

We introduce Poisson-response tensor-on-tensor regression (PToTR), a novel regression framework designed to handle tensor responses composed element-wise of random Poisson-distributed counts. Tensors, or multi-dimensional arrays, composed of counts are common data in fields such as inter national relations, social networks, epidemiology, and medical imaging, where events occur across multiple dimensions like time, location, and dyads. PToTR accommodates such tensor responses alongside tensor covariates, providing a versatile tool for multi dimensional data analysis. We propose algorithms for maximum likelihood estimation under a canonical polyadic (CP) structure on the regression coefficient tensor that satisfy the positivity of Poisson parameters and then provide an initial theoretical error analysis for PToTR estimators. We also demonstrate the utility of PToTR through three concrete applications: longitudinal data analysis of the Integrated Crisis Early Warning System database, positron emission tomography (PET) image reconstruction, and change-point detection of communication patterns in longitudinal dyadic data. These applications highlight the versatility of PToTR in addressing complex, structured count data across various domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Planar Collisionless Shock Simulations with the Semi-implicit Particle-in-cell Model FLEKS

This study investigates the applicability of the semi-implicit particle-in-cell code FLexible Exascale Kinetic Simulator (FLEKS) to heliospheric shock simulations. We examine one- and two-dimensional local planar shock simulations, initialized using MHD states with upstream conditions representative of plasmas in the hypersonic, β ∼ 1 regime, for both quasi-perpendicular and quasi-parallel configurations. The refined algorithm in FLEKS proves robust, enabling accurate shock simulations with a grid resolution on the order of the electron inertial length d e . Our simulations successfully capture key shock features, including shock structures (foot, ramp, overshoot, and undershoot), upstream and downstream waves (fast magnetosonic, whistler, Alfvén ion-cyclotron, and mirror modes), and non-Maxwellian particle distributions. Crucially, we find that at least two spatial dimensions are critical for accurately reproducing downstream-wave physics in quasi-perpendicular shocks and capturing the complex dynamics of quasi-parallel shocks, including surface rippling, shocklets, short, large-amplitude magnetic structures, magnetic reconnection, and jets. Furthermore, our parameter studies demonstrate the impact of mass ratio and grid resolution on shock physics. This work provides valuable guidance for selecting appropriate physical and numerical parameters for shock simulations using a semi-implicit PIC method, paving the way for incorporating kinetic shock processes into large-scale collisionless plasma simulations with the MHD-AEPIC model.

plasma astrophysics↗
DOE OSTI2024

Deep-Learning-Based Multi-Timescale Load Forecasting in Buildings: Opportunities and Challenges from Research to Deployment

Electricity load forecasting for buildings and campuses is becoming increasingly important as the penetration of distributed energy resources (DERs) grows. Efficient operation and dispatch of DERs require reasonably accurate predictions of future energy consumption in order to conduct near-real-time optimized dispatch of on-site generation and storage assets. Electric utilities have traditionally performed load forecasting for load pockets spanning large geographic areas, and therefore, forecasting has not been a common practice by buildings and campus operators. Given the growing trends of research and prototyping in the grid-interactive efficient buildings domain, characteristics beyond simple algorithm forecast accuracy are important in determining the algorithm's true utility for smart buildings. Other characteristics include the overall design of the deployed architecture and the operational efficiency of the forecasting system. In this work, we present a deep-learning-based load forecasting system that predicts the building load at 1-hour intervals for 18 hours in the future. We also discuss challenges associated with the real-time deployment of such systems as well as the research opportunities presented by a fully functional forecasting system that has been developed within the National Renewable Energy Laboratory's Intelligent Campus program.

building load forecasting↗
DOE OSTI2025

Efficient analysis of small-angle scattering curves for large biomolecular assemblies using Monte Carlo methods

Structure elucidation from small-angle scattering curves of large biomolecular assemblies is notoriously challenging. This is because the simulation of high-resolution features in the structure of large macromolecular assemblies, such as de novo protein assemblies, is computationally demanding when it needs to cover a broad range of length scales. Conventional methods, such as the numerical approximation to the Debye equation or the use of spherical harmonics, do not scale well as the size of the assembly increases, which limits their application to small structures (e.g. individual proteins). This work explores the effectiveness of a Monte Carlo method to simulate and fit scattering curves for large biomolecular assemblies spanning over ranges covering atomic and molecular detail (e.g. spacing and orientation of proteins in an assembly) as well as large-scale (hundreds of nanometres) features. Owing to its speed and scalability, it can be combined with a fitting algorithm to extract structural features from experimental small-angle scattering curves in biomolecular assemblies that are otherwise intractable for interpretation. This work first demonstrates the effectiveness of the tool using experimental small-angle X-ray scattering (SAXS) data from tile-like proteins that assemble into 1D tube-like macromolecular structures. Here, the diameter distribution of tubes is extracted from SAXS fits, and this is quantitatively compared with distributions from electron microscopy. SAXS data are also obtained from 2D sheet-like protein assemblies, and the proposed method is used to quantify structural features such as the separation distance between protein building blocks and the flexing of the sheet. An open-source implementation of the methodology is provided for use in a broad range of biological systems involving multi-scale scattering analysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Variance-Reduced Accelerated First-Order Methods: Central Limit Theorems and Confidence Statements

In this paper, we consider a strongly convex stochastic optimization problem and propose three classes of variable sample-size stochastic first-order methods: (i) the standard stochastic gradient descent method, (ii) its accelerated variant, and (iii) the stochastic heavy-ball method. In each scheme, the exact gradients are approximated by averaging across an increasing batch size of sampled gradients. We prove that when the sample size increases at a geometric rate, the generated estimates converge in mean to the optimal solution at an analogous geometric rate for schemes (i)–(iii). Based on this result, we provide central limit statements, whereby it is shown that the rescaled estimation errors converge in distribution to a normal distribution with the associated covariance matrix dependent on the Hessian matrix, the covariance of the gradient noise, and the step length. If the sample size increases at a polynomial rate, we show that the estimation errors decay at a corresponding polynomial rate and establish the associated central limit theorems (CLTs). Under certain conditions, we discuss how both the algorithms and the associated limit theorems may be extended to constrained and nonsmooth regimes. As a result, we provide an avenue to construct confidence regions for the optimal solution based on the established CLTs and test the theoretical findings on a stochastic parameter estimation problem.

Lei, Jinlong↗
DOE OSTI2025

Efficient quantum simulation of QCD jets on the light front

Quark and gluon jets provide one of the best ways to probe the matter produced in ultrarelativistic high-energy collisions, from cold nuclear matter to hot quark-gluon plasma. In this work, we propose a unified framework for efficient quantum simulation of many-body dynamics using the ( 3 + 1 )-dimensional QCD Hamiltonian on the light front, particularly suited for studying the scattering of quark and gluon jets on nuclear matter in heavy-ion collisions. We describe scalable methods for mapping physical degrees of freedom onto qubits and for simulating in-medium jet evolution. We then validate our framework by implementing an algorithm that directly maps second-quantized Fock states onto qubits and uses Trotterized simulation for simulating time dynamics. Using a classical emulator, we investigate the evolution of quark and gluon jets with up to three particles in Fock states, extending prior studies. These calculations enable the study of key observables, including jet momentum broadening, particle production, and parton distribution functions. Published by the American Physical Society 2025

Qian, Wenyang (ORCID:0000000155250996)↗
DOE OSTI2025

CAFE AU LAIT: Compute-Aware Federated Augmented Low-Rank AI Training

Federated finetuning is crucial for unlocking the knowledge embedded in pretrained Large Language Models (LLMs) when data are geographically distributed across clients. Unlike finetuning with data from a single institution, federated finetuning allows collaboration across multiple institutions, enabling the utilization of diverse and decentralized datasets while preserving data privacy. Given the high computing costs of LLM training and the emphasis on energy efficiency in Federated Learning (FL), Low-Rank Adaptation (LoRA) has emerged as a widely adopted algorithm due to its significantly reduced number of trainable parameters. However, this assumes that all data silos have the necessary computing resources to compute local updates of LLMs. Nevertheless, in practice, the computing resources across clients are highly heterogeneous: while some may have access to hundreds of GPUs, others might have limited or no GPU access. Recently, federated finetuning using synthetic data has been proposed, allowing clients to participate in a collaborative training run without training LLMs locally. However, our experimental results reveal a performance gap between models trained using synthetic data and those trained using local updates. Motivated by the observed heterogeneity in computing resources and the performance gap, we propose a novel two-stage algorithm that leverages the storage and computing capabilities of a strong server. In the first stage, under the coordination of the strong server, clients with limited computing resources collaborate to generate synthetic data, which is transferred to and stored on the strong server. In the second stage, the strong server uses this synthetic data on behalf of the resource-constrained clients to perform federated LoRA finetuning alongside clients with sufficient computing resources. This approach ensures that all clients can participate in the finetuning process. Experimental results demonstrate that incorporating local updates from even a small fraction of clients improves performance compared to using synthetic data for all clients. Furthermore, we incorporate the Gaussian mechanism in both stages to guarantee client-level differential privacy.

Wang, Jiayi [ORNL]↗
DOE OSTI2025

eMosaic: Electrification Mosaic Platform for Grid Informed Smart Charging Management (Final Scientific/Technical Report)

ABB (Prime Contractor), in collaboration with its partners at the Utah State University (USU), Idaho National Laboratory, Rocky Mountain Power (RMP), and Electric Power Engineers (EPE), have performed research, development, and wide scale demonstration of a scalable and resilient Electrification Mosaic (eMosaic) platform for Smart Charge Management (SCM) for Electric Vehicle Infrastructure. Work was completed under DE EE0009194, titled “eMosaic Electrification Mosaic Platform for Grid Informed Smart Charging Management”, funded by the US Department of Energy. The project members developed algorithms that provide localized and bulk grid services and that reduce and stabilize costs all the way down the supply chain to the PEV owner through SCM. This platform aggregates telemetry from multiple data sources as pieces of the larger picture including personal, private fleet or transportation EVs, fast chargers and other supply equipment, weather service information, and geographically distributed charging sites such as public lots, garage and retail, and private or shared usage depots. ABB and the project team designed, tested, and improved a charging management system at local/edge and cloud levels. The ultimate objective of the project was to convincingly demonstrate that the developed secure eMosaic plat-form can be readily and favorably adopted by diverse utilities and site owners at scale. This was achieved through a demonstration plan with field deployment at several physical sites across 4 states and additional scalable simulation from high fidelity charging models.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

GrainGNN: A dynamic graph neural network for predicting 3D grain microstructure

We propose GrainGNN, a surrogate model for the evolution of polycrystalline grain structure under rapid solidification conditions in metal additive manufacturing. High fidelity simulations of solidification microstructures are typically performed using multicomponent partial differential equations (PDEs) with moving interfaces. The inherent randomness of the PDE initial conditions (grain seeds) necessitates ensemble simulations to predict microstructure statistics, e.g., grain size, aspect ratio, and crystallographic orientation. Here, currently such ensemble simulations are prohibitively expensive and surrogates are necessary.In GrainGNN, we use a dynamic graph to represent interface motion and topological changes due to grain coarsening. We use a reduced representation of the microstructure using hand-crafted features; we combine pattern finding and altering graph algorithms with two neural networks, a classifier (for topological changes) and a regressor (for interface motion). Both networks have an encoder-decoder architecture; the encoder has a multi-layer transformer long-short-term-memory architecture; the decoder is a single layer perceptron.We evaluate GrainGNN by comparing it to high-fidelity phase field simulations for in-distribution and out-of-distribution grain configurations for solidification under laser power bed fusion conditions. GrainGNN results in 80%–90% pointwise accuracy; and nearly identical distributions of scalar quantities of interest (QoI) between phase field and GrainGNN simulations compared using Kolmogorov-Smirnov test. GrainGNN's inference speedup (PyTorch on single x86 CPU) over a high-fidelity phase field simulation (CUDA on a single NVIDIA A100 GPU) is 150×–2000× for 100-initial grain problem. Further, using GrainGNN, we model the formation of 11,600 grains in 220 seconds on a single CPU core.

36 MATERIALS SCIENCE↗
DOE OSTI2024

The gravitational lensing imprints of DES Y3 superstructures on the CMB: a matched filtering approach

Low-density cosmic voids gravitationally lens the cosmic microwave background (CMB), leaving a negative imprint on the CMB convergence |$\kappa$|⁠. This effect provides insight into the distribution of matter within voids, and can also be used to study the growth of structure. We measure this lensing imprint by cross-correlating the Planck CMB lensing convergence map with voids identified in the Dark Energy Survey Year 3 (DES Y3) data set, covering approximately 4200 deg|$^2$| of the sky. We use two distinct void-finding algorithms: a 2D void-finder that operates on the projected galaxy density field in thin redshift shells, and a new code, Voxel, which operates on the full 3D map of galaxy positions. We employ an optimal matched filtering method for cross-correlation, using the Marenostrum Institut de Ciències de l’Espai N-body simulation both to establish the template for the matched filter and to calibrate detection significances. Using the DES Y3 photometric luminous red galaxy sample, we measure |$A_\kappa$|⁠, the amplitude of the observed lensing signal relative to the simulation template, obtaining |$A_\kappa = 1.03 \pm 0.22$| (⁠|$4.6\sigma$| significance) for Voxel and |$A_\kappa = 1.02 \pm 0.17$| (⁠|$5.9\sigma$| significance) for 2D voids, both consistent with Lambda cold dark matter expectations. We additionally invert the 2D void-finding process to identify superclusters in the projected density field, for which we measure |$A_\kappa = 0.87 \pm 0.15$| (⁠|$5.9\sigma$| significance). The leading source of noise in our measurements is Planck noise, implying that data from the Atacama Cosmology Telescope, South Pole Telescope and CMB-S4 will increase sensitivity and allow for more precise measurements.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Cadmus v1.0

This software package is for distributed computation of persistent (co)homology. It is primarily intended for Topological Data Analysis (TDA) audience and scientists who use TDA methods and apply them to datasets that are too big to handle on a single machine. There are very few codes available for this; Cadmus outperforms DIPHA, if the user is interested in cohomology. The accompanying paper explaining the algorithm was accepted to ALENEX 26.

Nigmetov, Arnur [Lawrence Berkeley National Labora↗
DOE OSTI2024

Continuous automatic polarization channel stabilization from heterodyne detection of coexisting dim reference signals

Quantum networking continues to encode information in polarization states due to ease and precision. The variable environmental polarization transformations induced by deployed fiber need correction for deployed quantum networking. Here, we present a method for automatic polarization compensation (APC) and demonstrate its performance on a metropolitan quantum network. Designing an APC involves many design decisions as indicated by the diversity of previous solutions in the literature. Our design leverages heterodyne detection of wavelength-multiplexed dim classical references for continuous high-bandwidth polarization measurements used by newly developed multi-axis (non-)linear control algorithm(s) for complete polarization channel stabilization with no downtime. This enables continuous relatively high-bandwidth correction without significant added noise from classical reference signals. We demonstrate the performance of our APC using a variety of classical and quantum characterizations. Finally, we use C-band and L-band APC versions to demonstrate continuous high-fidelity entanglement distribution on a metropolitan quantum network with an average relative fidelity of 0.94 ± 0.03 for over 30 hrs.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

On the transferability of residence time distributions in two 10-km long river sections with similar hydromorphic units

Quantifying hydrologic exchange fluxes (HEFs) at the stream-groundwater interface and their residence time distributions (RTDs) in the subsurface are important for managing the water quality and ecosystem health in dynamic river corridors. However, direct simulating high-spatial resolution HEFs and RTDs can be time-consuming, especially for watershed-scale modeling. Efficient surrogate models linking RTDs to hydromorphic units (HUs) can be alternatives for simulating RTDs in large-scale models. A common concern of these surrogate models, though, is the transferability of the relationship between the RTDs and HUs from one river corridor to another. To address this issue, this work evaluates the HEFs and resulting RTD-HU relationships for two 10-km long river corridors along the Columbia River leveraging a one-way coupled three-dimensional transient surface-subsurface water transport modeling framework we previously developed. Applying such a framework at the two river corridors with similar HUs allows for quantitative comparisons of HEFs and RTDs using both statistical tests and machine learning classification models. Finally, our comparison shows that the similarity and transferability of the RTD-HU relationship is very low for the two investigated river sections, which suggests that devising a general algorithm to estimate RTDs based solely on surface water hydrodynamics and short-distance river channel topography data, as well as HU classification, might be nearly impossible.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Are better combinations of DERs more profitable?: Combinatorial optimization for aggregation of DERs in wholesale electricity markets

Recently, regulatory changes in various countries have enabled the participation of small-scale distributed energy resources (DERs) aggregated in virtual power plants (VPPs) in wholesale electricity markets. The inherent uncertainty and variability of resources comprising VPPs can lead to imbalances between forecasted and metered outputs, potentially resulting in the deficient settlement of generation under imbalance settlement rules. To address this challenge, it is essential to manage variability in the planning phase and uncertainty in the operation phase. Most current research focuses on managing forecasting errors in the operational phase, with insufficient attention given to the planning phase. Here, to bridge this gap, this paper proposes an optimal combination strategy for DERs to maximize the market participation revenue of VPPs by proactively managing variability in the planning phase. To estimate the expected revenue, we conducted analyses for homogeneous and heterogeneous DERs using Monte Carlo simulations and genetic algorithms. Remarkably, the proposed method demonstrated approximately 8 % higher revenue compared to the neighboring group case when considering diversity in DER set configuration with equal proportions of photovoltaics and wind.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Leveraging Existing Assets for Long Duration Energy Storage

Increased renewables penetration to electrical grid is necessary to reduce overall emissions from the electrical power generation sector. Nonetheless, its integration creates challenges to grid operators who must match the power being generated by intermittent renewables and other traditional energy sources with the demand from consumers, while ensuring the reliability and power quality for the entire system. Energy storage has been proposed as an alternative to natural gas peaking plants and a form to deliver excess renewable energy generation at times of peak demand. For energy storage to provide benefits to end customers (energy consumers), it must be reliable, efficient, and cost effective. The Illinois Sustainable Technology Center (ISTC), one of the surveys that integrate the Prairie Research Institute (PRI), aims to develop a Center for Energy Storage at Existing Assets (CESEA) at UIUC with the participation of Waste Pressure Corp and Ecotek Engineering USA LLC. CESEA will focus on LDES systems that can integrate to existing infrastructure in a manner that reduces the initial capital expenditure and demonstrates the ability to repurpose fossil assets that would otherwise become stranded, to serve the energy transition. CESEA aims to leverage UIUC’s unique facilities to validate LDES systems performance at a relevant operating environment. UIUC’s facilities include a 85-MW combined heat and power (CHP) power plant, two (2) solar PV plants totaling over 18 MWdc of installed capacity, an electrical grid along with a substation at transmission and distribution voltages, a 22-mile gas pipeline network operating at two pressure levels, along with steam and chilled water distribution networks. The new LDES systems will connect to the existing UIUC grid through a new test electrical station, which will have the capacity to accommodate additional connections to test new devices and technologies as part of future CESEA R&D activities. The test electrical station will contain meters, instrumentation, and controls to accurately capture data and allow optimization of control algorithms. CESEA will initially focus on technologies that: i) utilize existing equipment or facilities to perform at least one of the process steps in LDES (charging, storage, or discharging), ii) leverage mature or commercially available components or controls, iii) show potential for cost-leadership in 10+ hour storage at a commercial scale. Initial technologies that were identified to meet these criteria include Compressed Gas Energy Storage (CGES), and TES. CGES stores electricity by raising the pressure of a compressible gas inside a control volume and converting the stored energy to electricity via expansion-generation. CGES is a generalization of CAES that covers any working gas (not just air). A successful CGES demo will help to circumvent many challenges faced by CAES (long development times due to site prospecting, high cost of compression and storage, heat recovery management, etc.) by: 1) utilizing existing infrastructure (compressors, pipelines, underground storage or pressure vessels) used in the transportation and storage of industrial gases for LDES charging and storage; 2) deploying over sites already-developed for industrial applications with minor additional work; 3) leveraging the price structure of commercial industrial gas to cover the costs of electricity used during charging. A previous DOE-sponsored conceptual study (DE-FE-0032018) estimated the levelized cost of energy of a 1.1 MW / 17 MWh CGES system at $0.08/kWh, with a commercial 10x scale system cost estimated at <$0.04/kWh (Giardinella, 2022). The pilot-sized system was estimated to avoid up to 2693 tons of CO2/year.

25 ENERGY STORAGE↗
DOE OSTI2024

An Orbital Basis Set for Double Photoionization of Atoms and Molecules

The ab initio theoretical treatment of one-photon double photoionization processes has been limited to atoms and diatomic molecules by the challenges posed by large grid-based representations of the double ionized continuum wave function. To provide a path for extensions to polyatomics, an energy-adapted orbital basis approach is demonstrated that reduces the dimensions of such representations and simultaneously allows larger time steps in time-dependent computational descriptions of double ionization. Additionally, an algorithm that exploits the diagonal nature of the two-electron integrals in the grid basis and dramatically accelerates the transformation between grid and orbital representations is presented. Excellent agreement between the present results and benchmark theoretical calculations is found for H – and Be atoms, as well as the hydrogen molecule, including for the triply differential cross sections that relate the angular distribution and energy sharing of all of the particles in the molecular frame.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗