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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

A new upper bound for the growth factor in Gaussian elimination with complete pivoting

Abstract The growth factor in Gaussian elimination measures how large the entries of an LU factorization can be relative to the entries of the original matrix. It is a key parameter in error estimates, and one of the most fundamental topics in numerical analysis. We produce an upper bound of for the growth factor in Gaussian elimination with complete pivoting — the first improvement upon Wilkinson's original 1961 bound of .

Bisain, Ankit [Department of Mathematics Massachus↗
DOE OSTI2024

Determining the N -Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling

The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen–Cooper–Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1987

Ethynylated aromatics as high temperature matrix resins

Difunctional and trifunctional arylacetylenes were used as monomers to form thermoset matrix resin composites. Composites can be hot-pressed at 180 C to react 80 percent of the acetylene groups. Crosslinking is completed by postcuring at 350 C. The postcured resins are thermally stable to nominally 460 C in air. As a result of their high crosslink density, the matrix exhibits brittle failure when uniaxial composites are tested in tension. Failure of both uniaixial tensile and flexural specimens occurs in shear at the fiber-matrix interface. Tensile fracture stresses for 0-deg composites fabricated with 60 v/o Celion 6K graphite fiber were 827 MPa. The strain to failure was 0.5 percent. Composites fabricated with 8 harness satin Celion cloth (Fiberite 1133) and tested in tension also failed in shear at tensile stresses of 413 MPa.

Hurwitz, F. I.↗
DOE OSTI2023

Fission gas retention of densely packed uranium carbonitride tristructural-isotropic fuel particles in a 3D printed SiC matrix

The Transformational Challenge Reactor (TCR) fuel form was designed to contain large, densely packed uranium carbonitride (UCN) tristructural-isotropic (TRISO) fuel particles within a 3D printed SiC matrix, increasing the uranium density compared to conventional TRISO fuel forms and offering full geometric freedom for core design. Here, this work summarizes initial low-burnup, high-power irradiation testing of TCR fuel materials, including loose UCN TRISO particles and integral fuel compacts with ~55% TRISO particles by volume, to evaluate fission gas retention. Fission gasses were fully retained in all loose particle tests and in integral compacts irradiated at low (<250 °C) surface temperatures. Initial testing at higher (~700–750 °C) fuel surface temperatures showed fission gas release (FGR) and complete fracture of three compacts, but no FGR was observed in later high temperature tests (~300–750 °C) of both fueled compacts and loose TRISO particles. Calculated thermal stresses in the failed compacts were far less than the measured strength of the SiC matrix and the stresses in some failed compacts were less than those in compacts that did not show FGR. Thermal stress-induced matrix cracks also would not cause complete fracture because the tensile stresses transition to compression in the higher temperature regions. Therefore, fuel failure was likely not caused by thermal stresses and may have been related to leakage currents from the electrical heaters and erratic fuel surface temperatures that were only observed in the test for which failure was observed. In any case, the matrix cracks propagated through the coatings of TRISO particles located in the high-density matrix regions on the peripheries of the compacts, resulting in measurable fission gas release. The discussion focuses on the importance of understanding matrix density distributions and the particle-matrix interface properties to prevent matrix cracks from causing TRISO particle failures.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
NASA NTRS1995

Prioritization methodology for chemical replacement

Since United States of America federal legislation has required ozone depleting chemicals (class 1 & 2) to be banned from production, The National Aeronautics and Space Administration (NASA) and industry have been required to find other chemicals and methods to replace these target chemicals. This project was initiated as a development of a prioritization methodology suitable for assessing and ranking existing processes for replacement 'urgency.' The methodology was produced in the form of a workbook (NASA Technical Paper 3421). The final workbook contains two tools, one for evaluation and one for prioritization. The two tools are interconnected in that they were developed from one central theme - chemical replacement due to imposed laws and regulations. This workbook provides matrices, detailed explanations of how to use them, and a detailed methodology for prioritization of replacement technology. The main objective is to provide a GUIDELINE to help direct the research for replacement technology. The approach for prioritization called for a system which would result in a numerical rating for the chemicals and processes being assessed. A Quality Function Deployment (QFD) technique was used in order to determine numerical values which would correspond to the concerns raised and their respective importance to the process. This workbook defines the approach and the application of the QFD matrix. This technique: (1) provides a standard database for technology that can be easily reviewed, and (2) provides a standard format for information when requesting resources for further research for chemical replacement technology. Originally, this workbook was to be used for Class 1 and Class 2 chemicals, but it was specifically designed to be flexible enough to be used for any chemical used in a process (if the chemical and/or process needs to be replaced). The methodology consists of comparison matrices (and the smaller comparison components) which allow replacement technology to be quantitatively compared in several categories, and a QFD matrix which allows process/chemical pairs to be rated against one another for importance (using consistent categories). Depending on the need for application, one can choose the part(s) needed or have the methodology completed in its entirety. For example, if a program needs to show the risk of changing a process/chemical one may choose to use part of Matrix A and Matrix C. If a chemical is being used, and the process must be changed; one might use the Process Concerns part of Matrix D for the existing process and all possible replacement processes. If an overall analysis of a program is needed, one may request the QFD to be completed.

Cruit, Wendy↗
NASA NTRS1991

Multiphase complete exchange on a circuit switched hypercube

On a distributed memory parallel computer, the complete exchange (all-to-all personalized) communication pattern requires each of n processors to send a different block of data to each of the remaining n - 1 processors. This pattern is at the heart of many important algorithms, most notably the matrix transpose. For a circuit switched hypercube of dimension d(n = 2(sup d)), two algorithms for achieving complete exchange are known. These are (1) the Standard Exchange approach that employs d transmissions of size 2(sup d-1) blocks each and is useful for small block sizes, and (2) the Optimal Circuit Switched algorithm that employs 2(sup d) - 1 transmissions of 1 block each and is best for large block sizes. A unified multiphase algorithm is described that includes these two algorithms as special cases. The complete exchange on a hypercube of dimension d and block size m is achieved by carrying out k partial exchange on subcubes of dimension d(sub i) Sigma(sup k)(sub i=1) d(sub i) = d and effective block size m(sub i) = m2(sup d-di). When k = d and all d(sub i) = 1, this corresponds to algorithm (1) above. For the case of k = 1 and d(sub i) = d, this becomes the circuit switched algorithm (2). Changing the subcube dimensions d, varies the effective block size and permits a compromise between the data permutation and block transmission overhead of (1) and the startup overhead of (2). For a hypercube of dimension d, the number of possible combinations of subcubes is p(d), the number of partitions of the integer d. This is an exponential but very slowly growing function and it is feasible over these partitions to discover the best combination for a given message size. The approach was analyzed for, and implemented on, the Intel iPSC-860 circuit switched hypercube. Measurements show good agreement with predictions and demonstrate that the multiphase approach can substantially improve performance for block sizes in the 0 to 160 byte range. This range, which corresponds to 0 to 40 floating point numbers per processor, is commonly encountered in practical numeric applications. The multiphase technique is applicable to all circuit-switched hypercubes that use the common e-cube routing strategy.

Bokhari, Shahid H.↗
DOE OSTI2026

PET waste- and bio-derived imine vitrimers for shape-memory, intrinsic flame-retardant, and recyclable carbon fiber composites

Developing circular multifunctional vitrimers and carbon fiber–reinforced polymers (CFRPs) that are simultaneously recyclable, mechanically robust, and intrinsically flame retardant remains a major challenge. Here, in this study, we report multifunctional vitrimers and their carbon fiber–reinforced vitrimer (CFRV) composites, where the vitrimer design integrates closed-loop recyclability, enhanced interfacial adhesion, and intrinsic flame retardancy within a single materials platform. The vitrimer matrix is synthesized from post-consumer polyethylene terephthalate (PET) waste and a vanillin-derived phosphorus-containing crosslinker, forming an imine-based network. The resulting vitrimer resin exhibits high tensile strength, thermal healability, repeated reprocessability, programmable shape memory, and rapid chemical depolymerization under mild conditions. Amine-functionalized carbon fibers significantly improve fiber–matrix interfacial bonding, yielding CFRVs with tensile strengths up to 789 MPa and complete recovery of structurally intact fibers after chemical recycling. The phosphorus-rich aromatic network further imparts intrinsic flame retardancy, enabling self-extinguishing behavior without external additives. This work advances a materials design paradigm for next-generation multifunctional, sustainable vitrimers and CFRVs, while simultaneously addressing the recycling challenges associated with both plastic and CFRP waste.

Bio-derived crosslinker↗
NASA NTRS1985

Long-life high performance fuel cell program

A multihundred kilowatt Regenerative Fuel Cell for use in a space station is envisioned. Three 0.508 sq ft (471.9 cm) active area multicell stacks were assembled and endurance tested. The long term performance stability of the platinum on carbon catalyst configuration suitability of the lightweight graphite electrolyte reservoir plate, the stability of the free standing butyl bonded potassium titanate matrix structure, and the long life potential of a hybrid polysulfone cell edge frame construction were demonstrated. A 18,000 hour demonstration test of multicell stack to a continuous cyclical load profile was conducted. A total of 12,000 cycles was completed, confirming the ability of the alkaline fuel cell to operate to a load profile simulating Regenerative Fuel Cell operation. An orbiter production hydrogen recirculation pump employed in support of the cyclical load profile test completed 13,000 hours of maintenance free operation. Laboratory endurance tests demonstrated the suitability of the butyl bonded potassium matrix, perforated nickel foil electrode substrates, and carbon ribbed substrate anode for use in the alkaline fuel cell. Corrosion testing of materials at 250 F (121.1 C) in 42% wgt. potassium identified ceria, zirconia, strontium titanate, strontium zirconate and lithium cobaltate as candidate matrix materials.

Martin, R. E.↗
DOE OSTI2020

Discontinuous Galerkin discretization for quantum simulation of chemistry

All-electron electronic structure methods based on the linear combination of atomic orbitals method with Gaussian basis set discretization offer a well established, compact representation that forms much of the foundation of modern correlated quantum chemistry calculations—on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block-diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O(N 4.5 ) in molecular orbital representations to O(N 2.6 ) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1–2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Evaluation of a pair-based, joint-likelihood association approach for regional infrasound event identification

SUMMARY A Bayesian framework for the association of infrasonic detections is presented and evaluated for analysis at regional propagation scales. A pair-based, joint-likelihood association approach is developed that identifies events by computing the probability that individual detection pairs are attributable to a hypothetical common source and applying hierarchical clustering to identify events from the pair-based analysis. The framework is based on a Bayesian formulation introduced for infrasonic source localization and utilizes the propagation models developed for that application with modifications to improve the numerical efficiency of the analysis. Clustering analysis is completed using hierarchical analysis via weighted linkage for a non-Euclidean distance matrix defined by the negative log-joint-likelihood values. The method is evaluated using regional synthetic data with propagation distances of hundreds of kilometres in order to study the sensitivity of the method to uncertainties and errors in backazimuth and time of arrival. The method is found to be robust and stable for typical uncertainties, able to effectively distinguish noise detections within the data set from those in events, and can be made numerically efficient due to its ease of parallelization.

58 GEOSCIENCES↗
NASA NTRS1999

Carbon Fiber Reinforced/Silicon Carbide Turbine Blisk Testing in the SIMPLEX Turbopump

A program designed to implement a ceramic matrix composite integrally bladed disk (blisk) into rocket engine style turbomachinery has successfully completed testing. The Marshall Space Flight Center (MSFC) program, utilizing the MSFC turbomachinery design, analysis, and testing capabilities along with materials development capabilities from both Glenn Research Center (GRC) and MSFC, has tested two carbon fiber reinforced silicon carbide blisks in the Simplex Turbopump at MSFC's Test Stand 500. One blisk contained a polar woven fiber preform, while the second blisk tested utilized a quasi-isotropic preform. Vhile earlier papers have chronicled the program's design, material testing, and torque testing efforts, this paper focuses on the testing of the blisks in the Simplex turbopump. Emphasis will be placed on the actual condition of the blisks before and after the testing test program design methodology, and conclusions that can be drawn from the test data and blisk final conditions. The program performed three separate test series. The first series was needed to validate that the Simplex turbopump was correctly re-built following a major incident to the turbopump. The turbopump had two major differences from the original design. The most obvious difference was the sleeve required throughout the bore of the main housing. The second major difference was modifications to the pump diffuser to improve performance. Several areas were burnt during the incident and were either repaired by weld repair (pump inlet housing) or simply smoothed out (turbine nozzle discharge). The test series was designed to weed out any turbopump design and manufacturing flaws or fatigue issues prior to putting the C/SiC blisks into it. The second and third series were the C/SiC blisk test series. The primary goal of these series was to expose the blisks to as much fatigue causing dynamic stress as possible to examine the material's capability. Initially, the test plan was to put equal time on the two blisks, however, as the test series progressed, the funding allowed additional testing to occur. The additional test time was placed on the polar weave blisk. The total test time accrued on the polar blisk was 2550 seconds with 860 seconds near the turbopump design speed of 25,000 rpm. This testing included 6 tests / 775 seconds pumping liquid nitrogen and 7 tests / 1775 seconds pumping liquid oxygen. The drive gas for all of the tests was gaseous nitrogen due to the lack of hot gas source for the Simplex turbopump. The quasi-isotropic blisk was tested for XX total tests and XXXX seconds with X tests/XXXX seconds pumping liquid nitrogen and X tests/XXXX seconds pumping liquid \oxygen.During the test series, the blisks were inspected following each test. Inspections initially were viewed from the downstream side of the blisks only. Midway through the testing, a method of borescoping the leading edges of the blades was devised, and subsequently, both sides of the blades were inspected following each test. The leading and trailing edges of the polar blisk held up better than the quasi-isotropic blisk. This was a known possibility due to the varying fiber direction in the blades as the rectangular preform weave is cut in a circular pattern. The surprising fact about the testing was that there was no measurable performance loss due to the inaccuracies in the blade manufacturing in the C/SiC blisks, the surface roughness C/SiC of the blades, or the loss of the material in the polar blisk. A performance shift was seen in the quasi-isotropic blisk as portions of the leading and trailing edges were lost. After the testing was completed, detailed inspections of the blisks were performed. The largest surprise was the polar blisk had a obvious crack in a single blade that was located nearly midspan which was not detected in test. The crack ran completely through the blade circumferenciary and through the radial length of the blade. However, the crack does not appear to extend into the blisk hub. Although the cause of the crack is still under investigation, the material appears to be tolerant of this crack, and other hairline cracks discovered under higher magnification. This bodies well for eventual use of this material in actual flight turbopumps where monolithic fracture toughness issues limit its use.

Genge, Gary G.↗
NASA NTRS2008

Part Marking and Identification Materials' for MISSE

The Materials on International Space Station Experiment (MISSE) is being conducted with funding from NASA and the U.S. Department of Defense, in order to evaluate candidate materials and processes for flight hardware. MISSE modules include test specimens used to validate NASA technical standards for part markings exposed to harsh environments in low-Earth orbit and space, including: atomic oxygen, ultraviolet radiation, thermal vacuum cycling, and meteoroid and orbital debris impact. Marked test specimens are evaluated and then mounted in a passive experiment container (PEC) that is affixed to an exterior surface on the International Space Station (ISS). They are exposed to atomic oxygen and/or ultraviolet radiation for a year or more before being retrieved and reevaluated. Criteria include percent contrast, axial uniformity, print growth, error correction, and overall grade. MISSE 1 and 2 (2001-2005), MISSE 3 and 4 (2006-2007), and MISSE 5 (2005-2006) have been completed to date. Acceptable results were found for test specimens marked with Data Matrix(TradeMark) symbols by Intermec Inc. and Robotic Vision Systems Inc using: laser bonding, vacuum arc vapor deposition, gas assisted laser etch, chemical etch, mechanical dot peening, laser shot peening, laser etching, and laser induced surface improvement. MISSE 6 (2008-2009) is exposing specimens marked by DataLase(Registed TradeMark), Chemico technologies Inc., Intermec Inc., and tesa with laser-markable paint, nanocode tags, DataLase and tesa laser markings, and anodized metal labels.

Roxby, Donald↗
NASA NTRS2004

Low Emissions Alternative Power (LEAP) Project Office Business Team of the Aeropropulsion Research Program Office (ARPO) Org. 0140

The program for which I am working at this summer is Propulsion and Power/Low Emissions Alternative Power (P&P/LEAP). It invests in a fundamental TRL 1-6 research and technology portfolio that will enable the future of: Alternative fuels and/or alternative propulsion systems, non-combustion (electric) propulsion systems. P&P/LEAP will identify and capitalize on the highest potential concepts generated both internal and external to the Agency. During my 2004 summer at NASA Glenn Research Center, I worked with my mentor Barbara Mader, in the Project Office with the Business Team completing various tasks for the project and personnel. The LEAP project is a highly matrixed organization. The Project Office is responsible for the goals advocacy and dollar (budget) of the LEAP project. The objectives of the LEAP Project are to discover new energy sources and develop unconventional engines and power systems directed towards greatly reduced emissions, enable new vehicle concepts for public mobility, new science missions and national security. The Propulsion and PowerLow Emissions Alternative Power directly supports the environmental, mobility, national security objectives of the Vehicle Systems Program and the Aeronautics Technology Theme. Technology deliverables include the demonstration through integrated ground tests, a constant volume combustor in an engine system, and UAV/small transport aircraft all electric power system. My mentor serves as a key member of the management team for the Aeropropulsion Research Program Office (ARPO). She has represented the office on numerous occasions, and is a member of a number of center-wide panels/teams, such as the Space management Committee and is chair to the Business Process Consolidation Team. She is responsible for the overall coordination of resources for the Propulsion and Power Project - from advocacy to implementation. The goal for my summer at NASA was to document processes and archive program documents from the past years. I used the computer and office machines, and also worked with personnel in setting up a Cost Estimation Plan. I gained office experience in Word, Excel, and Power Point, with the completion of a variety of tasks. I made spreadsheets that pertained to the budget plan for Journey to Tomorrow, to name a few I have supported the office by tracking resource information: including programmatic travel, project budget at the center level to budgets for individual research sub-projects and grants. I also assisted the Program Support Office in their duties including, representing the office on numerous occasions on center-wide team/panels, such as the Space management committee, IFMP Budget Formulation, Journey to Tomorrow Committee, and the Vehicle Systems Program Business Process Team.

Buttler, Jennifer A.↗
NASA NTRS1999

Deformation Behaviors of HIPped Foil Compared with Those of Sheet Titanium Alloys

Micromechanics-based modeling of composite material behaviors requires an accurate assessment of the constituent properties and behaviors. For the specific case of continuous-fiber-reinforced metal matrix composites (MMC's) manufactured from a foil/fiber/foil process, much emphasis has been placed on characterizing foil-based matrix materials that have been fabricated in the same way as the composite. Such materials are believed to yield mechanical properties and behaviors that are representative of the matrix constituent within the composite (in situ matrix). Therefore, these materials are desired for micromechanics modeling input. Unfortunately, such foils are extremely expensive to fabricate and procure because of the labor-intensive rolling process needed to produce them. As a potential solution to this problem that would maintain appropriately representative in situ properties, the matrix constituent could be characterized with sheet-based materials, which are considerably less expensive to manufacture than foils, are more readily procured, and result in fewer plies to obtain a desired panel thickness. The critical question is, however, does the consolidated sheet material exhibit the same properties and behaviors as do the consolidated foils? Researchers at NASA Lewis Research Center's Life Prediction Branch completed a detailed experimental investigation to answer this question for three titanium alloys commonly used in metal matrix composite form.

Castelli, Michael G.↗
NASA NTRS1986

Concurrent Cholesky factorization of positive definite banded Hermitian matrices

First, the Cholesky factorization is extended to cover uniformly partitioned banded positive definite matrices of rank n which may be real symmetric or Hermitian. Then, two stratagems are given for the use of the algorithm in concurrent machines where the number of processing elements is less than required to factor the matrix in as few serial steps as possible, and where uniformly high efficiency is expected from all processing elements. Expressions are given for the efficiency factor e appearing in the speed-up expression q = eN, and these are specialized for the N node hypercube machine as a function of partition size s, the number N of processing elements of the hypercube machine, and the cost mu of interelement transmission relative to computation. It is shown that the efficiency factor e is inversely proportional to mu/s, and that e is almost independent of N when N is large and mu/s = 0. The task is completed in n/s serial steps with no limit on n. The half bandwidth b of the matrix is 2 Ns.

Utku, S.↗
DOE OSTI2021

Advanced Microscopy Report on UCO Fuel Kernels from Selected AGR-1 and AGR-2 Experiments

A variety of neutron irradiation experiments were completed at Idaho National Laboratory (INL) on tristructural isotropic (TRISO) coated particles contained in a graphitic matrix as part of the Advanced Gas Reactor (AGR) fuel development and qualification program. Although the initial advanced microscopy and microanalysis studies on AGR-1 particles focused predominantly on the SiC layer?s role as the main fission-product containment, studies were further expanded to evaluate UCO kernels of selected AGR-1 and AGR-2 coated particles. Fuel kernels tested in the AGR-1 and AGR-2 experiments is composed of a mixture of UO2 and UC, with a nominal as-fabricated stoichiometry of UO1.37 C0.35 and UO1.43 C0.39, respectively. These as-fabricated kernel-stoichiometry values are determined at the completion of kernel fabrication and do not account for any changes that will occur during the subsequent coating and compacting processes.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2022

A comparison between the one- and two-step spin-orbit coupling approaches based on the ab initio Density Matrix Renormalization Group

The efficient and reliable treatment of both spin-orbit coupling (SOC) and electron correlation is essential for understanding f-element chemistry. Here, we analyze two approaches to the problem, the one-step approach where both effects are treated simultaneously, and the two-step state interaction approach. We report an implementation of the ab initio density matrix renormalization group(DMRG) with a one-step treatment of the SOC effect which can be compared to prior two-step treatments on an equal footing. Using a dysprosium octahedral complex and bridged dimer as benchmark systems, we identify characteristics of problems where the one-step approach is beneficial for obtaining the low-energy spectrum.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗