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DOE OSTI2018

User Manual for MC2-3 Gamma Library Generation

This document presents a detailed description of generating the gamma library for MC 2 -3 which includes delayed gammas and betas based on the 94 gamma energy groups. The updated methodologies associated with heating and gamma calculations are introduced, which account for the self-shielding effect on KERMA factors and perform the gamma transport calculations to condense the gamma data from 94 to 21 groups. For these updates, the PreGAMMA code was developed to prepare NJOY inputs and generate delayed gammas and betas, and the GenGAMMA code was updated to process the outputs of PreGAMMA and NJOY. This document briefly discusses verification results of the new gamma library using the ABTR fuel pin and the EBR-II benchmark problem.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2019

Development of mathematical model and simulation tool for the high capacity production of carbon fiber

Weight reduction in the transportation is the most effective way for reducing fuel consumption. Carbon fiber composites have been increasingly used by automotive manufacturers in vehicle nonstructural and structural parts. Larger penetration of carbon fiber-based materials requires very large quantities of fiber at a low price, and of consistently high quality. The advancements in carbon fiber technology that are required to achieve this production scale-up and cost reduction, are very challenging. The most expensive production step is the conversion of textile-grade polyacrylonitrile (PAN) precursor to carbon fiber. The conversion takes place in three main stages: stabilization, low and high temperature carbonization. Most of the research to date has focused on the stabilization of the PAN precursors and oxidized PAN fibers. Comparingly little effort has been dedicated to the investigation of processing conditions during low temperature (LT) carbonization. During this stage of fiber conversion, the largest fiber mass loss occurs, and the fundamental carbon fiber structure and material properties are established. The understanding of the conversion processes in the LT furnace, and the effects of spatial and temporal variations of the LT process parameters on the carbon fiber properties are necessary to identify possible process improvements and cost savings. The objective of this research was to develop computational models for gas flow and energy transport in the gas and fiber phases of the LT furnace, and to couple them with the fiber conversion process models. The model is implemented in the three-dimensional framework that enables analysis of process condition variations in all directions. The formulation is based on modified transport equation to simulate steady state conditions and thus significantly reduce computational effort associated with transient methods. The model is implemented in open source computational fluid dynamics (CFD) code OpenFOAM. The model has been demonstrated on a prototype carbon fiber LT furnace and can be extended to other geometries and process conditions. Extensions to other physical effects, such as accounting evolution of multiple gas species, and fiber tensioning are straightforward. The model uses High Performance Computing (HPC) capabilities of OpenFOAM and can be implemented on multi-processor workstations or mid-size computer clusters common in the industrial setting.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Space Charge Driven Emittance Growth and the Effect of Octupoles in IOTA

The Integrable Optics Test Accelerator (IOTA) at Fermilab is a small machine dedicated to a broad frontier accelerator physics program. An important aspect of this program is to investigate the potential benefits of the resonance free tune spread achievable with integrable optics to store and accelerate high intensity proton beams for which space charge is significant. In this context, a good understanding of proton beam emittance growth and particle loss mechanisms is essential. Assuming nominal design parameters, simulations show that for a bunched beam, the bulk of emittance growth takes place immediately following injection, typically within tens of turns. We attempt to account for this growth using a simplified RMS mismatch theory; some of its limitations and possible improvements are briefly discussed. We then compare theoretical predictions to simulations performed using the PIC code pyORBIT. Further exploring ways to mitigate emittance growth and reduce particle loss, we compare two beam matching strategies: (1) matching at the injection point (2) matching at the center of the nonlinear (octupole) insertion region where $\beta_x = \beta_y$. To observe how nonlinearity affects emittance growth and whether it dominates growth due to mismatch, we track two different distributions. Finally, we explore the potential of using octupoles in a quasi-integrable configuration to mitigate growth using a variety of initial distributions both at reduced and full intensities.

43 PARTICLE ACCELERATORS↗

DOE OSTI2026

HFIR Steady State Heat Transfer Code (HSSHTC) Statistical Uncertainty Analysis

HSSHTC, the safety basis steady state TH code for HFIR, uses a highly conservative approach in which all input and calculation uncertainties are resolved simultaneously at their most limiting setting. This results in excessive conservatism which does not account for the high unlikelihood of such simultaneous worst-case conditions. The present study explores an alternative approach, BEPU, in which reasonable working assumptions for the probability distribution of each input uncertainty are used to determine a relationship between burnout power margin and core fuel failure probability. This was performed under a philosophy of perturbing uncertainty parameters already defined within the HSSHTC methodology while preserving the HSSHTC calculation approach and solution methodology itself. Based on the assumptions employed in this study, the BEPU approach resulted in a 0.29 increase in burnout power ratio (25 MW increase in burnout power) compared to the latest HSSHTC calculations of C-HFIR-2026-004. The study can be refined in the future by employing fuel fabrication data to provide more realistic input distributions. Future changes to the HSSHTC methodology would potentially allow a more comprehensive treatment of uncertainties which may further increase the burnout power ratio.

Wysocki, Aaron [ORNL] (ORCID:0000000222043779)↗

DOE OSTI2020

SN 2018fif: The Explosion of a Large Red Supergiant Discovered in Its Infancy by the Zwicky Transient Facility

High-cadence transient surveys are able to capture supernovae closer to their first light than ever before. Applying analytical models to such early emission, we can constrain the progenitor stars' properties. Here, we present observations of SN 2018fif (ZTF 18abokyfk). The supernova was discovered close to first light and monitored by the Zwicky Transient Facility (ZTF) and the Neil Gehrels Swift Observatory. Early spectroscopic observations suggest that the progenitor of SN 2018fif was surrounded by relatively small amounts of circumstellar material compared to all previous cases. This particularity, coupled with the high-cadence multiple-band coverage, makes it a good candidate to investigate using shock-cooling models. We employ the SOPRANOS code, an implementation of the model by Sapir & Waxman and its extension to early times by Morag et al. Compared with previous implementations, SOPRANOS has the advantage of including a careful account of the limited temporal validity domain of the shock-cooling model as well as allowing usage of the entirety of the early UV data. We find that the progenitor of SN 2018fif was a large red supergiant with a radius of $R={744.0}_{-128.0}^{+183.0}\,{R}_{\odot }$ and an ejected mass of ${M}_{\mathrm{ej}}={9.3}_{-5.8}^{+0.4}\,{M}_{\odot }$. Our model also gives information on the explosion epoch, the progenitor's inner structure, the shock velocity, and the extinction. The distribution of radii is double-peaked, with smaller radii corresponding to lower values of the extinction, earlier recombination times, and a better match to the early UV data. If these correlations persist in future objects, denser spectroscopic monitoring constraining the time of recombination, as well as accurate UV observations (e.g., with ULTRASAT), will help break the extinction/radius degeneracy and independently determine both.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1973

Decay of far flow field in trailing vortices

A finite-difference machine code is brought to bear on the wake-vortex problem in the quasi-cylindrical boundary-layer approxmation. A turbulent-energy model containing new features is developed that accounts for the major effects disclosed by more advanced models in which the parameters are not yet established. Several puzzles are resolved that arose in previous theoretical investigations of wake vortices.

Baldwin, B. S.↗

NASA NTRS1986

Constitutive modeling for isotropic materials (HOST)

The results of the third year of work on a program which is part of the NASA Hot Section Technology program (HOST) are presented. The goals of this program are: (1) the development of unified constitutive models for rate dependent isotropic materials; and (2) the demonstration of the use of unified models in structural analyses of hot section components of gas turbine engines. The unified models selected for development and evaluation are those of Bodner-Partom and of Walker. A test procedure was developed for assisting the generation of a data base for the Bodner-Partom model using a relatively small number of specimens. This test procedure involved performing a tensile test at a temperature of interest that involves a succession of strain-rate changes. The results for B1900+Hf indicate that material constants related to hardening and thermal recovery can be obtained on the basis of such a procedure. Strain aging, thermal recovery, and unexpected material variations, however, preluded an accurate determination of the strain-rate sensitivity parameter is this exercise. The effects of casting grain size on the constitutive behavior of B1900+Hf were studied and no particular grain size effect was observed. A systematic procedure was also developed for determining the material constants in the Bodner-Partom model. Both the new test procedure and the method for determining material constants were applied to the alternate material, Mar-M247 . Test data including tensile, creep, cyclic and nonproportional biaxial (tension/torsion) loading were collected. Good correlations were obtained between the Bodner-Partom model and experiments. A literature survey was conducted to assess the effects of thermal history on the constitutive behavior of metals. Thermal history effects are expected to be present at temperature regimes where strain aging and change of microstructure are important. Possible modifications to the Bodner-Partom model to account for these effects are outlined. The use of a unified constitutive model for hot section component analyses was demonstrated by applying the Walker model and the MARC finite-element code to a B1900+Hf airfoil problem.

Chan, Kwai S.↗

NASA NTRS1990

Thermal instabilities in protogalactic clouds

The means by which a protogalaxy can fragment to form the first generation of stars and globular clusters remains an important problem in astrophysics. Gravitational instabilities grow on timescales too long to drive fragmentation before the background density grows by many orders of magnitude (see Murray and Lin 1989a, and references therein). Thermal instability provides a much more likely mechanism. After its initial collapse, a protogalactic cloud is expected to be shock heated to its virial temperature approx. 10(exp 6) K. Cooling by H and He+ below 10(exp 6) K has a negative slope, so that the cloud is subject to strong thermal instabilities. Density enhancements may then grow rapidly, fragmenting the protogalaxy as it cools to lower temperatures. The role of dynamical effects upon the growth of perturbations is considered here. The method used is similar to that used in Murray and Lin (1989a; see also the Erratum to appear September 15), which examined the growth of thermal instabilities with a one-dimensional Lagrangian hydrodynamics code, written for spherical symmetry. Perturbed regions therefore take the form of shells. The dynamical variables are integrated explicitly, while the temperature, ionization fraction, and molecular fraction are integrated implicitly, and account is taken for non-equilibrium values of these quantities.

Murray, S. D.↗

NASA NTRS1992

Supersonic full-potential methods for missile body analysis

Accounts are presented of representative applications to missile bodies of arbitrary shape of methods based on the steady form of the full potential equation. The NCOREL and SIMP full-potential codes are compared, and their results are evaluated for the cases of an arrow wing and a wing-body configuration. Attention is given to the effect of cross-sectional and longitudinal geometries. Comparisons of surface pressure and longitudinal force and moment data for circular and elliptic bodies have shown that the full-potential methods yielded excellent results in attached-flow conditions. Results are presented for a conical star body, waveriders, the Shuttle Orbiter, and a highly swept wing-body cruising at Mach 4.

Pittman, James L.↗

NASA NTRS1996

Propellant Chemistry for CFD Applications

Current concepts for reusable launch vehicle design have created renewed interest in the use of RP-1 fuels for high pressure and tri-propellant propulsion systems. Such designs require the use of an analytical technology that accurately accounts for the effects of real fluid properties, combustion of large hydrocarbon fuel modules, and the possibility of soot formation. These effects are inadequately treated in current computational fluid dynamic (CFD) codes used for propulsion system analyses. The objective of this investigation is to provide an accurate analytical description of hydrocarbon combustion thermodynamics and kinetics that is sufficiently computationally efficient to be a practical design tool when used with CFD codes such as the FDNS code. A rigorous description of real fluid properties for RP-1 and its combustion products will be derived from the literature and from experiments conducted in this investigation. Upon the establishment of such a description, the fluid description will be simplified by using the minimum of empiricism necessary to maintain accurate combustion analyses and including such empirical models into an appropriate CFD code. An additional benefit of this approach is that the real fluid properties analysis simplifies the introduction of the effects of droplet sprays into the combustion model. Typical species compositions of RP-1 have been identified, surrogate fuels have been established for analyses, and combustion and sooting reaction kinetics models have been developed. Methods for predicting the necessary real fluid properties have been developed and essential experiments have been designed. Verification studies are in progress, and preliminary results from these studies will be presented. The approach has been determined to be feasible, and upon its completion the required methodology for accurate performance and heat transfer CFD analyses for high pressure, tri-propellant propulsion systems will be available.

Farmer, R. C.↗

NASA NTRS2016

Using SpaceWire Time Codes for Spacecraft Time Synchronization

This paper describes how SpaceWire Time Codes can be used for synchronizing time within various subsystems of a spacecraft as well as, maintaining a common time reference needed for coordinating operations within a spacecraft. The algorithms to account for inaccuracies in the time distribution method were based on the NASA-4009 Space Telecommunication Radio System (STRS) standard [1], which defined an interface for synchronizing clocks running at different tick rates and tick resolutions.

SpaceWire Time Distribution Protocol↗

NASA NTRS2020

SN 2018fif: The Explosion of a Large Red Supergiant Discovered in its Infancy by the Zwicky Transient Facility

High-cadence transient surveys are able to capture supernovae closer to their first light than ever before. Applying analytical models to such early emission, we can constrain the progenitor stars' properties. In this paper, we present observations of SN 2018fif (ZTF 18abokyfk). The supernova was discovered close to first light and monitored by the Zwicky Transient Facility (ZTF) and the Neil Gehrels Swift Observatory. Early spectroscopic observations suggest that the progenitor of SN 2018fif was surrounded by relatively small amounts of circumstellar material compared to all previous cases. This particularity, coupled with the high-cadence multiple-band coverage, makes it a good candidate to investigate using shock-cooling models. We employ the SOPRANOS code, an implementation of the model by Sapir & Waxman and its extension to early times by Morag et al. Compared with previous implementations, SOPRANOS has the advantage of including a careful account of the limited temporal validity domain of the shock-cooling model as well as allowing usage of the entirety of the early UV data. We find that the progenitor of SN 2018fif was a large red supergiant with a radius of R = 744.0 (sup +183.0) (sub -128.0) R⨀ and an ejected mass of M(sub ej) = 9.3 (sup +0.4)(sub -5.8) M⨀. Our model also gives information on the explosion epoch, the progenitor's inner structure, the shock velocity, and the extinction. The distribution of radii is double-peaked, with smaller radii corresponding to lower values of the extinction, earlier recombination times, and a better match to the early UV data. If these correlations persist in future objects, denser spectroscopic monitoring constraining the time of recombination, as well as accurate UV observations (e.g., with ULTRASAT), will help break the extinction/radius degeneracy and independently determine both.

Supernovae↗

NASA NTRS2022

An Investigation Into Some Important Aspects of Droplet Breakup/Vaporization Behavior Caused By a Gas Flow (Both Nonreacting & Detonative Combustion) in a Shock Tube

As a part of the rotating detonation engine (RDE) technology enablement project at NASA Glenn Research Center (GRC), an effort was undertaken to extend our current computational capabilities of OpenNCC in some important ways with the implementation of a modeling approach to account for the droplet breakup caused by a shock-induced gas motion, & a vaporization model valid over a wide range of pressure conditions encountered in multiphase detonation. With the modified code, a study was undertaken to investigate the individual droplet behavior followed by the passage of a shock front. The study is carried out by tracking a sparse group of droplets to gain some understanding of shock induced droplet behavior under various shock strengths & fuel injector conditions. The study also looks into the effect of randomization involved in determining the droplet breakup outcomes. Over a wide range of sub-critical conditions examined, larger droplets are observed to undergo significant changes in droplet behavior following their breakup. However, smaller droplets (10 µm or less ) remain unaffected by any shock induced breakup. In a follow-on work, we investigated the impact of shock and droplet interaction in a detonation study involving both gaseous as well as gas/liquid (droplet clouds) fuel/air stoichiometric mixtures in a simple 3D shock-tube configuration. The droplet clouds are made up of different initial droplet sizes of either 6, 10, or 30 µm. We also investigated the individual droplet behavior followed by the passage of a detonation front. The results represent conditions that lead to both overdriven and C-J (ChapmanJouguet) detonations. Under both test conditions, most of the droplet vaporization is completed within a short distance (duration) behind the detonation front & well within the region of complete combustion observed in a corresponding equivalent gas-phase fuel/air mixture. The impact of the shock-induced droplet breakup is found to be significant in the calculations involving the 30-µm droplets. Subsequent to the breakup, the drop sizes vary from 1 to 10 µm. Another factor that contributed to the observed rapid vaporization is the result of vaporization taking place under supercritical conditions. The overall detonation properties of various droplet clouds (made up of different initial sizes) are similar to those observed in a corresponding gaseous fuel/air mixture. In the calculation involving a gaseous fuel, the calculated C-J detonation velocity is 1822 m/s involving Jet-A/air and φ = 1. In the overdriven detonation, it is 2044 m/s. In the calculations involving droplet clouds, the corresponding detonation velocities are lower. The impact of increased droplet size is primarily seen in a higher reduction in the detonation velocity.

shock/droplet interaction↗

NASA NTRS2022

Discovery of Chlorophyll d : Isolation and Characterization of a Far-Red Cyanobacterium from the Original Site of Manning and Strain (1943) at Moss Beach, California

We have isolated a chlorophyll-d-containing cyanobacterium from the intertidal field site at Moss Beach, on the coast of Central California, USA, where Manning and Strain (1943) originally discovered this far-red chlorophyll. Here, we present the cyanobacterium’s environmental description, culturing procedure, pigment composition, ultrastructure, and full genome sequence. Among cultures of far-red cyanobacteria obtained from red algae from the same site, this strain was an epiphyte on a brown macroalgae. Its Q y in vivo absorbance peak is centered at 704–705 nm, the shortest wavelength observed thus far among the various known Acaryochloris strains. Its Chl a /Chl d ratio was 0.01, with Chl d accounting for 99% of the total Chl d and Chl a mass. TEM imagery indicates the absence of phycobilisomes, corroborated by both pigment spectra and genome analysis. The Moss Beach strain codes for only a single set of genes for producing allophycocyanin. Genomic sequencing yielded a 7.25 Mbp circular chromosome and 10 circular plasmids ranging from 16 kbp to 394 kbp. We have determined that this strain shares high similarity with strain S15, an epiphyte of red algae, while its distinct gene complement and ecological niche suggest that this strain could be the closest known relative to the original Chl d source of Manning and Strain (1943). The Moss Beach strain is designated Acaryochloris sp. (marina) strain Moss Beach.

chlorophyll d↗

NASA NTRS2020

REQUIEM-2D Methodology: Spatially Resolved Stellar Populations of Massive Lensed Quiescent Galaxies from Hubble Space Telescope 2D Grism Spectroscopy

We present a novel Bayesian methodology to jointly model photometry and deep Hubble Space Telescope 2D grism spectroscopy of high-redshift galaxies. Our requiem2d code measures both unresolved and resolved stellar populations, ages, and star formation histories (SFHs) for the ongoing REsolving QUIEscent Magnified (REQUIEM) Galaxies Survey, which targets strong gravitationally lensed quiescent galaxies at z ∼ 2. We test the accuracy of requiem2d using a simulated sample of massive galaxies at z ∼ 2 from the Illustris cosmological simulation and find that we recover the general trends in SFH and median stellar ages. We further present a pilot study for the REQUIEM Galaxies Survey: MRG-S0851, a quintuply imaged, massive (log M ⁎ / M ʘ = 11.02 ± 0.04) red galaxy at z = 1.883 ± 0.001. With an estimated gravitational magnification of µ = 5.7 - 0.2 + 0.4, we sample the stellar populations on 0.6 kpc physical size bins. The global mass-weighted median age is constrained to be 1.8 - 0.2 + 0.3 Gyr, and our spatially resolved analysis reveals that MRG-S0851 has a flat age gradient in the inner 3 kpc core after taking into account the subtle effects of dust and metallicity on age measurements, favoring an early formation scenario. The analysis for the full REQUIEM-2D sample will be presented in a forthcoming paper with a beta release of the requiem2d code.

Strong gravitational lensing↗

NASA NTRS2013

The Collection 6 'dark-target' MODIS Aerosol Products

Aerosol retrieval algorithms are applied to Moderate resolution Imaging Spectroradiometer (MODIS) sensors on both Terra and Aqua, creating two streams of decade-plus aerosol information. Products of aerosol optical depth (AOD) and aerosol size are used for many applications, but the primary concern is that these global products are comprehensive and consistent enough for use in climate studies. One of our major customers is the international modeling comparison study known as AEROCOM, which relies on the MODIS data as a benchmark. In order to keep up with the needs of AEROCOM and other MODIS data users, while utilizing new science and tools, we have improved the algorithms and products. The code, and the associated products, will be known as Collection 6 (C6). While not a major overhaul from the previous Collection 5 (C5) version, there are enough changes that there are significant impacts to the products and their interpretation. In its entirety, the C6 algorithm is comprised of three sub-algorithms for retrieving aerosol properties over different surfaces: These include the dark-target DT algorithms to retrieve over (1) ocean and (2) vegetated-dark-soiled land, plus the (3) Deep Blue (DB) algorithm, originally developed to retrieve over desert-arid land. Focusing on the two DT algorithms, we have updated assumptions for central wavelengths, Rayleigh optical depths and gas (H2O, O3, CO2, etc.) absorption corrections, while relaxing the solar zenith angle limit (up to 84) to increase pole-ward coverage. For DT-land, we have updated the cloud mask to allow heavy smoke retrievals, fine-tuned the assignments for aerosol type as function of season location, corrected bugs in the Quality Assurance (QA) logic, and added diagnostic parameters such as topographic altitude. For DT-ocean, improvements include a revised cloud mask for thin-cirrus detection, inclusion of wind speed dependence in the retrieval, updates to logic of QA Confidence flag (QAC) assignment, and additions of important diagnostic information. At the same time as we have introduced algorithm changes, we have also accounted for upstream changes including: new instrument calibration, revised land-sea masking, and changed cloud masking. Upstream changes also impact the coverage and global statistics of the retrieved AOD. Although our responsibility is to the DT code and products, we have also added a product that merges DT and DB product over semi-arid land surfaces to provide a more gap-free dataset, primarily for visualization purposes. Preliminary validation shows that compared to surface-based sunphotometer data, the C6, Level 2 (along swath) DT-products compare at least as well as those from C5. C6 will include new diagnostic information about clouds in the aerosol field, including an aerosol cloud mask at 500 m resolution, and calculations of the distance to the nearest cloud from clear pixels. Finally, we have revised the strategy for aggregating and averaging the Level 2 (swath) data to become Level 3 (gridded) data. All together, the changes to the DT algorithms will result in reduced global AOD (by 0.02) over ocean and increased AOD (by 0.02) over land, along with changes in spatial coverage. Changes in calibration will have more impact to Terras time series, especially over land. This will result in a significant reduction in artificial differences in the Terra and Aqua datasets, and will stabilize the MODIS data as a target for AEROCOM studie

Aerosol retrieval algorithms↗

DOE OSTI2021

The ezSpectra suite: An easy‐to‐use toolkit for spectroscopy modeling

Abstract A molecule's spectrum encodes information about its structure and electronic properties. It is a unique fingerprint that can serve as a molecular ID. Quantum chemistry calculations provide key ingredients for interpreting spectra, but modeling the spectra rarely ends there; it requires additional steps that entail combined treatments of electronic and nuclear degrees of freedom and account for specifics of the experimental setup (light energy, polarization, averaging over molecular orientations, temperature, etc.). This Software Focus article describes the ezSpectra suite, which currently comprises two stand‐alone open‐source codes: ezFCF and ezDyson . ezFCF calculates Franck–Condon factors, which yield vibrational progressions for polyatomic molecules, within the double‐harmonic approximation. ezDyson calculates absolute cross‐sections for photodetachment/photoionization processes and photoelectron angular distributions using Dyson orbitals computed by a quantum chemistry program. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Theoretical and Physical Chemistry > Spectroscopy Software > Simulation Methods

Gozem, Samer↗

DOE OSTI2022

Approximate bounds and temperature dependence of adiabatic connection integrands for the uniform electron gas

Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. Methods that account for temperature dependence, such as Mermin–Kohn–Sham finite-temperature density functional theory and free energy density functional theory, are now employed with more regularity and available in many standard code packages. However, approximations from zero-temperature density functional theory are still often used in temperature-dependent simulations using thermally weighted electronic densities as an input to exchange–correlation functional approximations, a practice known to miss temperature-dependent effects in the exchange–correlation free energy of these systems. Here, in this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange–correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb–Oxford inequality, which provides a lower bound for the ground-state exchange–correlation energy, bounds for the exchange–correlation at finite temperatures are approximated for various degrees of electronic correlation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗