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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 793 records · Page 44
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Cholesky-based experimental design for Gaussian process and kernel-based emulation and calibration.

Gaussian processes and other kernel-based methods are used extensively to construct approximations of multivariate data sets. The accuracy of these approximations is dependent on the data used. This paper presents a computationally efficient algorithm to greedily select training samples that minimize the weighted L p error of kernel-based approximations for a given number of data. The method successively generates nested samples, with the goal of minimizing the error in high probability regions of densities specified by users. The algorithm presented is extremely simple and can be implemented using existing pivoted Cholesky factorization methods. Training samples are generated in batches which allows training data to be evaluated (labeled) in parallel. For smooth kernels, the algorithm performs comparably with the greedy integrated variance design but has significantly lower complexity. Numerical experiments demonstrate the efficacy of the approach for bounded, unbounded, multi-modal and non-tensor product densities. We also show how to use the proposed algorithm to efficiently generate surrogates for inferring unknown model parameters from data using Bayesian inference.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Statistical Learning for Nonlinear Model Reduction from Local Simulations of Stochastic and Particle- and Agent-Based Systems

Stochastic physical systems across the sciences that have very high-dimensional state spaces, with a large number of fast degrees of freedom that force direct simulators to proceed by integration steps that are orders of magnitude smaller than events of interests (e.g., particle collisions). Examples range from molecular motion to dynamics of large populations of cells. A grand challenge in the simulation and understanding of such systems is the systematic construction of accurate, interpretable, reduced models, enabling faster simulations, revealing fundamental properties of the dynamics, and predicting phenomena of interest that the original simulator could not reached with sufficient accuracy or within a given computational budget. In this projected we developed novel statistical estimation/machine learning techniques for analyzing and building empirical reduced models for important families of high-dimensional stochastic systems, in particular: - we developed techniques for estimating interaction kernels in interacting particle- and agent-based systems, which are ubiquitous in Physics, Biology and many other sciences, given observed trajectories of the system; - we developed techniques for nonlinear model reduction for high-dimensional stochastic systems that have a small number of unknown, nonlinear slow variables, and a large number of fast modes, that are possibly of large magnitude, given observed short trajectories of the system in the form of bursts of trajectories from different initial conditions; - we developed novel techniques for estimating linear dynamical systems on graphs when both the dynamics and the underlying graph are unknown, and we have a sparse set of space-time observations; - we considered the problem of estimating an unknown nonlinear observation function of a standard process (e.g. Brownian motion), so that we can recognized if an observed dynamics is "just" a nonlinear version of a known dynamics; we also developed benchmarks for learning algorithms aimed at learning and classifying diffusion processes.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1986

Transportability, distributability and rehosting experience with a kernel operating system interface set

For the past two years, PRC has been transporting and installing a software engineering environment framework, the Automated Product control Environment (APCE), at a number of PRC and government sites on a variety of different hardware. The APCE was designed using a layered architecture which is based on a standardized set of interfaces to host system services. This interface set called the APCE Interface Set (AIS), was designed to support many of the same goals as the Common Ada Programming Support Environment (APSE) Interface Set (CAIS). The APCE was developed to provide support for the full software lifecycle. Specific requirements of the APCE design included: automation of labor intensive administrative and logistical tasks: freedom for project team members to use existing tools: maximum transportability for APCE programs, interoperability of APCE database data, and distributability of both processes and data: and maximum performance on a wide variety of operating systems. A brief description is given of the APCE and AIS, a comparison of the AIS and CAIS both in terms of functionality and of philosophy and approach and a presentation of PRC's experience in rehosting AIS and transporting APCE programs and project data. Conclusions are drawn from this experience with respect to both the CAIS efforts and Space Station plans.

Blumberg, F. C.↗
DOE OSTI2022

Towards Enhancing Coding Productivity for GPU Programming Using Static Graphs

The main contribution of this work is to increase the coding productivity of GPU programming by using the concept of Static Graphs. GPU capabilities have been increasing significantly in terms of performance and memory capacity. However, there are still some problems in terms of scalability and limitations to the amount of work that a GPU can perform at a time. To minimize the overhead associated with the launch of GPU kernels, as well as to maximize the use of GPU capacity, we have combined the new CUDA Graph API with the CUDA programming model (including CUDA math libraries) and the OpenACC programming model. We use as test cases two different, well-known and widely used problems in HPC and AI: the Conjugate Gradient method and the Particle Swarm Optimization. In the first test case (Conjugate Gradient) we focus on the integration of Static Graphs with CUDA. In this case, we are able to significantly outperform the NVIDIA reference code, reaching an acceleration of up to 11x thanks to a better implementation, which can benefit from the new CUDA Graph capabilities. In the second test case (Particle Swarm Optimization), we complement the OpenACC functionality with the use of CUDA Graph, achieving again accelerations of up to one order of magnitude, with average speedups ranging from 2x to 4x, and performance very close to a reference and optimized CUDA code. Our main target is to achieve a higher coding productivity model for GPU programming by using Static Graphs, which provides, in a very transparent way, a better exploitation of the GPU capacity. The combination of using Static Graphs with two of the current most important GPU programming models (CUDA and OpenACC) is able to reduce considerably the execution time w.r.t. the use of CUDA and OpenACC only, achieving accelerations of up to more than one order of magnitude. Finally, we propose an interface to incorporate the concept of Static Graphs into the OpenACC Specifications.

58 GEOSCIENCES↗
DOE OSTI2024

Generalized quantum master equations can improve the accuracy of semiclassical predictions of multitime correlation functions

Multitime quantum correlation functions are central objects in physical science, offering a direct link between the experimental observables and the dynamics of an underlying model. While experiments such as 2D spectroscopy and quantum control can now measure such quantities, the accurate simulation of such responses remains computationally expensive and sometimes impossible, depending on the system’s complexity. A natural tool to employ is the generalized quantum master equation (GQME), which can offer computational savings by extending reference dynamics at a comparatively trivial cost. However, dynamical methods that can tackle chemical systems with atomistic resolution, such as those in the semiclassical hierarchy, often suffer from poor accuracy, limiting the credence one might lend to their results. By combining work on the accuracy-boosting formulation of semiclassical memory kernels with recent work on the multitime GQME, here we show for the first time that one can exploit a multitime semiclassical GQME to dramatically improve both the accuracy of coarse mean-field Ehrenfest dynamics and obtain orders of magnitude efficiency gains.

Chemistry↗
DOE OSTI2020

Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures

Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. Here, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green’s kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal ϵ -2 asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

ICED: An Integrated CGRA Framework Enabling DFVS-Aware Acceleration

oarse-grained reconfigurable arrays (CGRAs) are a promising solution to enable energy-efficient acceleration of applications from different domains. By leveraging reconfiguration at the functional level, they can adapt to significantly different computational patterns. Existing CGRA mapping approaches extract instruction-level parallelism, exploit loop-pipelining opportunities, guarantee the data dependency, and target high throughput of a given loop. However, the recurrence data-dependency in the DFG and the mismatch between required and available computing/communication resources complicate the mapping, and might lead to significant unbalances in the utilization of the CGRA's tiles. This results in wasted power for tiles with low utilization. Applying dynamic voltage and frequency scaling (DVFS) can potentially solve this challenge and improve energy efficiency by adjusting voltage and frequency of different tiles independently. CGRAs have also been successful in accelerating data-dependent streaming applications. However, in these applications, the execution time of each kernel in the pipeline might dynamically vary depending on the characteristics of the input. This also leads to under-utilization of resources for the dynamically changing kernels that do not limit the application throughput. DVFS can also improve energy efficiency for these applications by dynamically changing the voltage and frequency levels of tiles that host non performance-constraining kernels. This paper proposes ICEDTEA -- an integrated DVFS-aware framework to map applications on CGRAs that support power islands. ICEDTEA proposes a CGRA architecture supporting DVFS islands at varying granularity (from a single tile to a group of tiles) and the related DVFS-aware compilation and mapping toolchain. ICEDTEA is the first work that introduces DVFS support for spatio-temporal CGRAs at power-island levels. The experimental evaluation shows that ICEDTEA improves average utilization by 2.3$\times$ and energy-efficiency by 1.32$\times$ over a conventional CGRA. With streaming applications, ICEDTEA improves energy efficiency by 1.12$\times$ over a state-of-the-art CGRA that introduces partial dynamic reconfiguration to adapt to variations in kernels' throughput.

Tan, Cheng↗
DOE OSTI2016

WE-AB-207B-07: Dose Cloud: Generating “Big Data” for Radiation Therapy Treatment Plan Optimization Research

Purpose: To provide a tool to generate large sets of realistic virtual patient geometries and beamlet doses for treatment optimization research. This tool enables countless studies exploring the fundamental interplay between patient geometry, objective functions, weight selections, and achievable dose distributions for various algorithms and modalities. Methods: Generating realistic virtual patient geometries requires a small set of real patient data. We developed a normalized patient shape model (PSM) which captures organ and target contours in a correspondence-preserving manner. Using PSM-processed data, we perform principal component analysis (PCA) to extract major modes of variation from the population. These PCA modes can be shared without exposing patient information. The modes are re-combined with different weights to produce sets of realistic virtual patient contours. Because virtual patients lack imaging information, we developed a shape-based dose calculation (SBD) relying on the assumption that the region inside the body contour is water. SBD utilizes a 2D fluence-convolved scatter kernel, derived from Monte Carlo simulations, and can compute both full dose for a given set of fluence maps, or produce a dose matrix (dose per fluence pixel) for many modalities. Combining the shape model with SBD provides the data needed for treatment plan optimization research. Results: We used PSM to capture organ and target contours for 96 prostate cases, extracted the first 20 PCA modes, and generated 2048 virtual patient shapes by randomly sampling mode scores. Nearly half of the shapes were thrown out for failing anatomical checks, the remaining 1124 were used in computing dose matrices via SBD and a standard 7-beam protocol. As a proof of concept, and to generate data for later study, we performed fluence map optimization emphasizing PTV coverage. Conclusions: We successfully developed and tested a tool for creating customizable sets of virtual patients suitable for large-scale radiation therapy optimization research.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2022

Portable C++ Code that can Look and Feel Like Fortran Code with Yet Another Kernel Launcher (YAKL)

This paper introduces the Yet Another Kernel Launcher (YAKL) C++ portability library, which strives to enable user-level code with the look and feel of Fortran code. The intended audience includes both C++ developers and Fortran developers unfamiliar with C++. The C++ portability approach is briefly explained, YAKL’s main features are described, and code examples are given that demonstrate YAKL’s usage. YAKL fills a niche capability important particularly to scientific applications seeking to port Fortran code quickly to a portable C++ library. YAKL places heavy emphasis on simplicity, readability, and productivity with performance mainly emphasizing Graphics Processing Units (GPUs). Central to YAKL’s ability to allow Fortran-like user-level code are three features: (1) a multi-dimensional Array class that allows Fortran behavior; (2) a limited library of Fortran intrinsic functions; and (3) an efficient pool allocator that transparently enables cheap frequent allocations and deallocations of YAKL Arrays. While YAKL allows Fortran-style code, it also allows Arrays that exhibit C-like behavior as well, including row-major index ordering and lower bounds of “0”. YAKL currently supports CPUs, CPU threading, and Nvidia, AMD, and Intel GPUs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Propagation Pattern for Moment Representation of the Lattice Boltzmann Method

A propagation pattern for the moment representation of the regularized lattice Boltzmann method (LBM) in three dimensions is presented. Using effectively lossless compression, the simulation state is stored as a set of moments of the lattice Boltzmann distribution function, instead of the distribution function itself. An efficient cache-aware propagation pattern for this moment representation has the effect of substantially reducing both the storage and memory bandwidth required for LBM simulations. This article extends recent work with the moment representation by expanding the performance analysis on central processing unit (CPU) architectures, considering how boundary conditions are implemented, and demonstrating the effectiveness of the moment representation on a graphics processing unit (GPU) architecture.

42 ENGINEERING↗
DOE OSTI2017

Next generation extended Lagrangian first principles molecular dynamics

Extended Lagrangian Born-Oppenheimer molecular dynamics [Phys. Rev. Lett., 100, 123004 (2008)] is formulated for general Hohenberg-Kohn density functional theory and compared to the extended Lagrangian framework of rst principles molecular dynamics by Car and Parrinello [Phys. Rev. Lett. 55, 2471 (1985)]. It is shown how extended Lagrangian Born-Oppenheimer molecular dynamics overcomes several shortcomings of regular, direct Born-Oppenheimer molecular dynamics, while improving or maintaining important features of Car-Parrinello simulations. The accuracy of the electronic degrees of freedom in extended Lagrangian Born-Oppenheimer molecular dynamics, with respect to the exact Born-Oppenheimer solution, is of second order in the size of the integration time step and of fourth order in the potential energy surface. Improved stability over recent formulations of extended Lagrangian Born-Oppenheimer molecular dynamics is achieved by generalizing the theory to finite temperature ensembles, using fractional occupation numbers in the calculation of the inner-product kernel of the extended harmonic oscillator that appears as a preconditioner in the electronic equations of motion. Materials systems that normally exhibit slow self-consistent field convergence can be simulated using integration time steps of the same order as in direct Born- Oppenheimer molecular dynamics, but without the requirement of an iterative, non-linear electronic ground state optimization prior to the force evaluations and without a systematic drift in the total energy. In combination with proposed low-rank and on-the- y updates of the kernel, this formulation provides an efficient and general framework for quantum based Born-Oppenheimer molecular dynamics simulations.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2022

Imaging Wavepackets in Real-Space & Disentangling Ultrafast Solvation Dynamics with High-Energy Ultrafast X-ray Scattering

The microscopic information of solute and solute-solvent motions can be measured using ultrafast diffuse x-ray scattering, and compared to molecular dynamics simulations, as demonstrated in studies of solvated molecular systems done in X-ray free-electron lasers. However, for the typical photon energies used in these experiments, the photon momentum transfer range was limited to lower Q values where the signals arising from different parts of the studied system overlap significantly. Recently, this limitation was greatly alleviated by extending the photon energy at the Linac Coherent Light-Source (LCLS), significantly increasing the accessible momentum transfer range. This improvement is transformative for ultrafast diffuse x-ray scattering measurements, enabling for the first time to disentangle the solute scattering difference signal, that persists to higher Q values, from the bulk solvent and solute-solvent cross-terms difference signals. In addition, it enhances the prediction ability of modeling and simulation methods such as the hybrid QM/MM approach. The extended Q range also opens the way to resolve in real-space details regarding coherent wavepackets motions beyond their center-of-mass positions. Here, we present the first results on high-energy (18keV) diffuse x-ray scattering in solution, demonstrating high fidelity time-resolved scattering and analysis of the photoexcited model photocatalysts PtPOP ([Pt2(POP)4]4-), and IrDimen ([Ir2(dimen)4]2+) in several solvents. These complexes provide ideal systems for demonstrating the ability of high-Q ultrafast scattering and QM/MM simulations to decompose ultrafast solvation dynamics into specific changes in the solute-solvent pair distribution function with a particular focus on how electronically excited states change the interaction between the solvent and photo-catalytically active metal sites. We introduce an approach to further utilize the high-energy capability and demonstrate a single-shot ultra-wide-angle X-ray Scattering modality using two perpendicular detectors, spanning a scattering angle range of more than 100 degrees, allowing to extend the accessible momentum transfer range up to Q~14 Å-1. We develop a model-free method to invert the x-ray scattering signals and enable the recovery of multiple pair-density motions that happen simultaneously, allowing us to directly measure in real-space nuclear wavepacket motions. . [1] Natan, Adi. "Real-Space Inversion and Super-Resolution of Ultrafast X-ray Scattering using Natural Scattering Kernels." arXiv preprint arXiv:2107.05576 (2021)

Natan, Adi↗
DOE OSTI2018

Evaluating cloud feedbacks and rapid responses in the ACCESS model

A cloud feedback diagnostic package is implemented in the Australian Community Climate and Earth-System Simulator General Circulation Model, based on the methodology of “cloud radiative kernels.” Using separate increased sea surface temperature and CO 2 experiments, both the “rapid response” cloud contribution to forcing and temperature-mediated cloud feedbacks are analyzed. Under increased temperature and CO 2 changes, temperature-mediated cloud radiative feedback dominates over the rapid response in the final radiative response. Cloud feedback is positive in both long and short wave, with short wave dominating global values. Contributing most to this are low to mid-level clouds, of medium-to-high optical thickness. As a means of illustration of the methodology, a number of key parameters related to clouds, precipitation, and convection that are typically used in “tuning” in the model are modified. These changes result in substantial impacts on the model's current climate, but only modest changes to rapid response and feedbacks occur globally, regionally, and as a function of cloud optical thickness and height. This limited set of experiments shows that cloud adjustments and feedbacks in this model are robust under these changes, lending confidence that both model climate change projections and the conclusions of attribution studies are not overly sensitive to such parameterization tuning. Of course, a considerably larger set of experiments would be needed to demonstrate that feedbacks and rapid response are robust under the wider set of tuning adjustments commonly undertaken.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Using Flory–Huggins-informed human-in-the-loop Bayesian optimization to map the phase diagram of polymer blends

Mapping the phase diagram of polymer blends is an essential step in controlling the structure–property relationship of polymer-based materials. However, traditional grid-based approaches are inefficient and rely on subjective judgements for terminating the experimental campaign. Artificial intelligence-guided experimentation offers a compelling alternative, especially when data-driven decision-making is interfaced with established polymer thermodynamics to improve efficiency and interpretability. Here, we introduce a physics-informed Bayesian optimization approach to guide the mapping of the phase diagram of a model blend containing poly(methyl methacrylate) and poly(styrene-ran-acrylonitrile). Physical information is derived from a Flory–Huggins representation of the spinodal curve, which is integrated into the Bayesian optimization process as a structured prior mean that acts as a soft constraint. Implemented as a human-in-the-loop workflow, the approach leverages optical imaging of film cloudiness with iterative Gaussian process surrogate modeling and a parameter selection decision policy to identify the composition-temperature conditions for sequential iterations. Convergence of kernel and Flory–Huggins-based hyperparameters provided a stopping criterion, ensuring an objective and interpretable termination of the experimental campaign. The framework recovered the known lower critical solution temperature (∼160 °C), while increasing material efficiency through targeted sampling. This work establishes a proof-of-concept for the application of Bayesian optimization workflows to study polymer blend miscibility.

36 MATERIALS SCIENCE↗
NASA NTRS1986

Numerical solution of random singular integral equation appearing in crack problems

The solution of several elasticity problems, and particularly crack problems, can be reduced to the solution of one-dimensional singular integral equations with a Cauchy-type kernel or to a system of uncoupled singular integral equations. Here a method for the numerical solution of random singular integral equations of Cauchy type is presented. The solution technique involves a Chebyshev series approximation, the coefficients of which are the solutions of a system of random linear equations. This method is applied to the problem of periodic array of straight cracks inside an infinite isotropic elastic medium and subjected to a nonuniform pressure distribution along the crack edges. The statistical properties of the random solution are evaluated numerically, and the random solution is used to determine the values of the stress-intensity factors at the crack tips. The error, expressed as the difference between the mean of the random solution and the deterministic solution, is established. Values of stress-intensity factors at the crack tip for different random input functions are presented.

Sambandham, M.↗
DOE OSTI2024

LibERI—A portable and performant multi-GPU accelerated library for electron repulsion integrals via OpenMP offloading and standard language parallelism

A portable and performant graphics processing unit (GPU)-accelerated library for electron repulsion integral (ERI) evaluation, named LibERI, has been developed and implemented via directive-based (e.g., OpenMP and OpenACC) and standard language parallelism (e.g., Fortran DO CONCURRENT). Offloaded ERIs consist of integrals over low and high contraction s, p, and d functions using the rotated-axis and Rys quadrature methods. GPU codes are factorized based on previous developments with two layers of integral screening and quartet presorting. In this work, the density screening is moved to the GPU to enhance the computational efficacy for large molecular systems. Here, the L-shells in the Pople basis set are also separated into pure S and P shells to increase the ERI homogeneity and reduce atomic operations and the memory footprint. LibERI is compatible with any quantum chemistry drivers supporting the MolSSI Driver Interface. Benchmark calculations of LibERI interfaced with the GAMESS software package were carried out on various GPU architectures and molecular systems. The results show that the LibERI performance is comparable to other state-of-the-art GPU-accelerated codes (e.g., TeraChem and GMSHPC) and, in some cases, outperforms conventionally developed ERI CUDA kernels (e.g., QUICK) while fully maintaining portability.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1991

Linear and nonlinear dynamic analysis by boundary element method

An advanced implementation of the direct boundary element method (BEM) applicable to free-vibration, periodic (steady-state) vibration and linear and nonlinear transient dynamic problems involving two and three-dimensional isotropic solids of arbitrary shape is presented. Interior, exterior, and half-space problems can all be solved by the present formulation. For the free-vibration analysis, a new real variable BEM formulation is presented which solves the free-vibration problem in the form of algebraic equations (formed from the static kernels) and needs only surface discretization. In the area of time-domain transient analysis, the BEM is well suited because it gives an implicit formulation. Although the integral formulations are elegant, because of the complexity of the formulation it has never been implemented in exact form. In the present work, linear and nonlinear time domain transient analysis for three-dimensional solids has been implemented in a general and complete manner. The formulation and implementation of the nonlinear, transient, dynamic analysis presented here is the first ever in the field of boundary element analysis. Almost all the existing formulation of BEM in dynamics use the constant variation of the variables in space and time which is very unrealistic for engineering problems and, in some cases, it leads to unacceptably inaccurate results. In the present work, linear and quadratic isoparametric boundary elements are used for discretization of geometry and functional variations in space. In addition, higher order variations in time are used. These methods of analysis are applicable to piecewise-homogeneous materials, such that not only problems of the layered media and the soil-structure interaction can be analyzed but also a large problem can be solved by the usual sub-structuring technique. The analyses have been incorporated in a versatile, general-purpose computer program. Some numerical problems are solved and, through comparisons with available analytical and numerical results, the stability and high accuracy of these dynamic analysis techniques are established.

Ahmad, Shahid↗
DOE OSTI2020

Formation energy puzzle in intermetallic alloys: Random phase approximation fails to predict accurate formation energies

We performed density functional calculations to estimate the formation energies of intermetallic alloys. We used two semilocal approximations, the generalized gradient approximation (GGA) by Perdew-Burke- Ernzerhof (PBE) and the strongly constrained and appropriately normed (SCAN) meta-GGA. In addition, we utilized two nonlocal DFT functionals, the hybrid HSE06, and the state-of-the-art random phase approximation (RPA). The nonlocal functionals such as HSE06 and RPA yield accurate formation energies of binary alloys with completely-filled d-band metals, where semilocal functionals underperform. The accuracy at the nonlocal functionals is greatly reduced when a partially-filled d-band metal is present in an alloy, while PBE-GGA outperforms in these cases. We show that the accurate prediction of formation energies by any DFT method depends on its ability to predict the accurate electronic properties, e.g., valence d-band contribution to the density of states (DOS). The SCAN meta-GGA often corrects the PBE-DOS, however, it does not provide accurate formation energies compared to PBE. This is assumed to be due to the lack of proper error cancellation that should be expected due to the similar bulk nature of both alloys and their constituents, which may improve with the modification of meta-GGA ingredients. RPA yields too negative formation energies of alloys with partially-filled d-band metals. RPA results can be corrected by restoring the exchange-correlation kernel, thereby improving the short-range electron-electron correlation in metallic densities.

36 MATERIALS SCIENCE↗