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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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

EvoProtGrad (Directed Evolution for Proteins with Gradients) [SWR-23-48]

A Python package for directed evolution on a protein sequence with gradient-based discrete Markov chain monte carlo (MCMC). Users are able to compose custom models that map sequence to function with pretrained models, including protein language models (PLMs), to guide and constrain search. Our package natively integrates with the HuggingFace platform and supports PLMs from transformers. Our MCMC sampler identifies promising amino acids to mutate via model gradients taken with respect to the input (i.e., sensitivity analysis). We allow users to compose their own custom target function for MCMC by leveraging the Product of Experts MCMC paradigm. Each model is an "expert" that contributes its own knowledge about the protein's fitness landscape to the overall target function. The sampler is designed to be more efficient and effective than brute force and random search while maintaining most of the generality and flexibility. Additional information can be found in the related publication: https://iopscience.iop.org/article/10.1088/2632-2153/accacd

Emami, Patrick↗
DOE OSTI2022

Challenges in Markov Chain Monte Carlo for Bayesian Neural Networks

Markov chain Monte Carlo (MCMC) methods have not been broadly adopted in Bayesian neural networks (BNNs). This paper initially reviews the main challenges in sampling from the parameter posterior of a neural network via MCMC. Such challenges culminate to lack of convergence to the parameter posterior. Nevertheless, this paper shows that a nonconverged Markov chain, generated via MCMC sampling from the parameter space of a neural network, can yield via Bayesian marginalization a valuable posterior predictive distribution of the output of the neural network. Further, classification examples based on multilayer perceptrons showcase highly accurate posterior predictive distributions. The postulate of limited scope for MCMC developments in BNNs is partially valid; an asymptotically exact parameter posterior seems less plausible, yet an accurate posterior predictive distribution is a tenable research avenue.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Multimodal parameter spaces of a complex multi-channel neuron model

One of the most common types of models that helps us to understand neuron behavior is based on the Hodgkin–Huxley ion channel formulation (HH model). A major challenge with inferring parameters in HH models is non-uniqueness: many different sets of ion channel parameter values produce similar outputs for the same input stimulus. Such phenomena result in an objective function that exhibits multiple modes (i.e., multiple local minima). This non-uniqueness of local optimality poses challenges for parameter estimation with many algorithmic optimization techniques. HH models additionally have severe non-linearities resulting in further challenges for inferring parameters in an algorithmic fashion. To address these challenges with a tractable method in high-dimensional parameter spaces, we propose using a particular Markov chain Monte Carlo (MCMC) algorithm, which has the advantage of inferring parameters in a Bayesian framework. The Bayesian approach is designed to be suitable for multimodal solutions to inverse problems. We introduce and demonstrate the method using a three-channel HH model. We then focus on the inference of nine parameters in an eight-channel HH model, which we analyze in detail. We explore how the MCMC algorithm can uncover complex relationships between inferred parameters using five injected current levels. The MCMC method provides as a result a nine-dimensional posterior distribution, which we analyze visually with solution maps or landscapes of the possible parameter sets. The visualized solution maps show new complex structures of the multimodal posteriors, and they allow for selection of locally and globally optimal value sets, and they visually expose parameter sensitivities and regions of higher model robustness. We envision these solution maps as enabling experimentalists to improve the design of future experiments, increase scientific productivity and improve on model structure and ideation when the MCMC algorithm is applied to experimental data.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Constraining Cosmology with Simulation-based inference and Optical Galaxy Cluster Abundance

We test the robustness of simulation-based inference (SBI) in the context of cosmological parameter estimation from galaxy cluster counts and masses in simulated optical datasets. We construct ``simulations'' using analytical models for the galaxy cluster halo mass function (HMF) and for the observed richness (number of observed member galaxies) to train and test the SBI method. We compare the SBI parameter posterior samples to those from an MCMC analysis that uses the same analytical models to construct predictions of the observed data vector. The two methods exhibit comparable performance, with reliable constraints derived for the primary cosmological parameters, ($\Omega_m$ and $\sigma_8$), and richness-mass relation parameters. We also perform out-of-domain tests with observables constructed from galaxy cluster-sized halos in the Quijote simulations. Again, the SBI and MCMC results have comparable posteriors, with similar uncertainties and biases. Unsurprisingly, upon evaluating the SBI method on thousands of simulated data vectors that span the parameter space, SBI exhibits worsened posterior calibration metrics in the out-of-domain application. We note that such calibration tests with MCMC is less computationally feasible and highlight the potential use of SBI to stress-test limitations of analytical models, such as in the use for constructing models for inference with MCMC.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Accelerating Markov Chain Monte Carlo sampling with diffusion models

Global fits of physics models require efficient methods for exploring high-dimensional and/or multimodal posterior functions. We introduce a novel method for accelerating Markov Chain Monte Carlo (MCMC) sampling by pairing a Metropolis-Hastings algorithm with a diffusion model that can draw global samples with the aim of approximating the posterior. We briefly review diffusion models in the context of image synthesis before providing a streamlined diffusion model tailored towards low-dimensional data arrays. We then present our adapted Metropolis-Hastings algorithm which combines local proposals with global proposals taken from a diffusion model that is regularly trained on the samples produced during the MCMC run. Our approach leads to a significant reduction in the number of likelihood evaluations required to obtain an accurate representation of the Bayesian posterior across several analytic functions, as well as for a physical example based on a global fit of parton distribution functions. Our method is extensible to other MCMC techniques, and we briefly compare our method to similar approaches based on normalising flows. A code implementation can be found at https://github.com/NickHunt-Smith/MCMC-diffusion.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

SDSS-IV MaStar: Stellar parameter determination with continuum-supplemented full-spectrum fitting

We present a stellar parameter catalog built to accompany the MaStar Stellar Library, which is a comprehensive collection of empirical, medium-resolution stellar spectra. We constructed this parameter catalog by using a multicomponent χ 2 fitting approach to match MaStar spectra to models generated by interpolating the ATLAS9-based BOSZ model spectra. The total χ 2 for a given model is defined as the sum of components constructed to characterize narrow-band features of observed spectra (e.g., absorption lines) and the broadband continuum shape separately. Extinction and systematics due to flux calibration were taken into account in the fitting. The χ 2 distribution for a given region of model space was sampled using a Markov chain Monte Carlo (MCMC) algorithm, the data from which were then used to extract atmospheric parameter estimates (T eff , log g, [Fe/H], and [α/Fe]), their corresponding uncertainties, and direct extinction measurements. Two methods were used to extract parameters and uncertainties: one that accepts the MCMC’s prescribed minimum-χ 2 result, and one that uses Bayesian inference to compute a likelihood-weighted mean from the χ 2 distribution sampled by the MCMC. Results were evaluated for internal consistency using repeat observations where available and by comparing them with external data sets (e.g., APOGEE-2 and Gaia DR2). Our spectral-fitting exercise reveals possible deficiencies in current theoretical model spectra, illustrating the potential power of MaStar spectra for helping to improve the models. This paper represents an update to the parameters that were originally presented with SDSS-IV DR17. The MaStar parameter catalog containing our BestFit results is available on the SDSS-IV DR17 website as part of version 2 of the MaStar stellar parameter value-added catalog.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Star formation rate and stellar mass calibrations based on infrared photometry and their dependence on stellar population age and extinction

The stellar mass (M $\star$ ) and the star formation rate (SFR) are among the most important features that characterize galaxies. Measuring these fundamental properties accurately is critical for understanding the present state of galaxies, their history, and future evolution. Infrared (IR) photometry is widely used to measure the M $\star$ and SFR of galaxies because the near-IR traces the continuum emission of the majority of their stellar populations (SPs), and the mid/far-IR traces the dust emission powered by star-forming activity. This work explores the dependence of the IR emission of galaxies on their extinction, and the age of their SPs. It aims to provide accurate and precise IR-photometry SFR and M $\star$ calibrations that account for SP age and extinction while providing quantification of their scatter. We used the CIGALE spectral energy distribution (SED) fitting code to create model SEDs of galaxies with a wide range of star formation histories, dust content, and interstellar medium properties. We fit the relations between M $\star$ and SFR with IR and optical photometry of the model-galaxy SEDs with the Markov chain Monte Carlo (MCMC) method. As an independent confirmation of the MCMC fitting method, we performed a machine-learning random forest (RF) analysis on the same data set. The RF model yields similar results to the MCMC fits, thus validating the latter. This work provides calibrations for the SFR using a combination of the WISE bands 1 and 3, or the JWST NIR-F200W and MIRI-F2100W. It also provides mass-to-light ratio calibrations based on the WISE band-1, the JWST NIR-F200W, and the optical u - r or g - r colors. These calibrations account for the biases attributed to the SP age, while they are given in the form of extinction-dependent and extinction-independent relations. The proposed calibrations show robust estimations while minimizing the scatter and biases throughout a wide range of SFRs and stellar masses. The SFR calibration offers better results, especially in dust-free or passive galaxies where the contributions of old SPs or biases from the lack of dust are significant. Similarly, the M $\star$ calibration yields significantly better results for dusty and high-SFR galaxies where dust emission can otherwise bias the estimations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

DIGS: deep inference of galaxy spectra with neural posterior estimation

Abstract With the advent of billion-galaxy surveys with complex data, the need of the hour is to efficiently model galaxy spectral energy distributions (SEDs) with robust uncertainty quantification. The combination of simulation-based inference (SBI) and amortized neural posterior estimation (NPE) has been successfully used to analyse simulated and real galaxy photometry both precisely and efficiently. In this work, we utilise this combination and build on existing literature to analyse simulated noisy galaxy spectra. Here, we demonstrate a proof-of-concept study of spectra that is (a) an efficient analysis of galaxy SEDs and inference of galaxy parameters with physically interpretable uncertainties; and (b) amortized calculations of posterior distributions of said galaxy parameters at the modest cost of a few galaxy fits with Markov chain Monte Carlo (MCMC) methods. We utilise the SED generator and inference framework Prospector to generate simulated spectra, and train a dataset of 2 × 10 6 spectra (corresponding to a five-parameter SED model) with NPE. We show that SBI—with its combination of fast and amortized posterior estimations—is capable of inferring accurate galaxy stellar masses and metallicities. Our uncertainty constraints are comparable to or moderately weaker than traditional inverse-modelling with Bayesian MCMC methods (e.g. 0.17 and 0.26 dex in stellar mass and metallicity for a given galaxy, respectively). We also find that our inference framework conducts rapid SED inference (0.9–1.2 × 10 5 galaxy spectra via SBI/NPE at the cost of 1 MCMC-based fit). With this work, we set the stage for further work that focuses of SED fitting of galaxy spectra with SBI, in the era of JWST galaxy survey programs and the wide-field Roman Space Telescope spectroscopic surveys.

spectroscopy↗
DOE OSTI2022

Implementation of a practical Markov chain Monte Carlo sampling algorithm in PyBioNetFit

Abstract Summary Bayesian inference in biological modeling commonly relies on Markov chain Monte Carlo (MCMC) sampling of a multidimensional and non-Gaussian posterior distribution that is not analytically tractable. Here, we present the implementation of a practical MCMC method in the open-source software package PyBioNetFit (PyBNF), which is designed to support parameterization of mathematical models for biological systems. The new MCMC method, am, incorporates an adaptive move proposal distribution. For warm starts, sampling can be initiated at a specified location in parameter space and with a multivariate Gaussian proposal distribution defined initially by a specified covariance matrix. Multiple chains can be generated in parallel using a computer cluster. We demonstrate that am can be used to successfully solve real-world Bayesian inference problems, including forecasting of new Coronavirus Disease 2019 case detection with Bayesian quantification of forecast uncertainty. Availability and implementation PyBNF version 1.1.9, the first stable release with am, is available at PyPI and can be installed using the pip package-management system on platforms that have a working installation of Python 3. PyBNF relies on libRoadRunner and BioNetGen for simulations (e.g. numerical integration of ordinary differential equations defined in SBML or BNGL files) and Dask.Distributed for task scheduling on Linux computer clusters. The Python source code can be freely downloaded/cloned from GitHub and used and modified under terms of the BSD-3 license (https://github.com/lanl/pybnf). Online documentation covering installation/usage is available (https://pybnf.readthedocs.io/en/latest/). A tutorial video is available on YouTube (https://www.youtube.com/watch?v=2aRqpqFOiS4&t=63s). Supplementary information Supplementary data are available at Bioinformatics online.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Photometric redshift uncertainties in weak gravitational lensing shear analysis: models and marginalization

ABSTRACT Recovering credible cosmological parameter constraints in a weak lensing shear analysis requires an accurate model that can be used to marginalize over nuisance parameters describing potential sources of systematic uncertainty, such as the uncertainties on the sample redshift distribution n(z). Due to the challenge of running Markov chain Monte Carlo (MCMC) in the high-dimensional parameter spaces in which the n(z) uncertainties may be parametrized, it is common practice to simplify the n(z) parametrization or combine MCMC chains that each have a fixed n(z) resampled from the n(z) uncertainties. In this work, we propose a statistically principled Bayesian resampling approach for marginalizing over the n(z) uncertainty using multiple MCMC chains. We self-consistently compare the new method to existing ones from the literature in the context of a forecasted cosmic shear analysis for the HSC three-year shape catalogue, and find that these methods recover statistically consistent error bars for the cosmological parameter constraints for predicted HSC three-year analysis, implying that using the most computationally efficient of the approaches is appropriate. However, we find that for data sets with the constraining power of the full HSC survey data set (and, by implication, those upcoming surveys with even tighter constraints), the choice of method for marginalizing over n(z) uncertainty among the several methods from the literature may modify the 1σ uncertainties on Ωm–S8 constraints by ∼4 per cent, and a careful model selection is needed to ensure credible parameter intervals.

Zhang, Tianqing (ORCID:000000025596198X)↗
DOE OSTI2026

Determination of proton PDF uncertainties with Markov chain Monte Carlo

We present an analysis of parton distribution functions (PDFs) of the proton using Markov chain Monte Carlo (MCMC) methods. The MCMC approach naturally implements Bayes’ theorem and, thus, provides a means to directly sample the underlying probability distribution—in this case, the probability distribution of the PDF parameters. This allows for a straightforward propagation of the resulting uncertainties into any PDF-dependent observable, preserving their simple probabilistic interpretation. In our analysis we include a broad set of deep inelastic scattering data from HERA, BCDMS and NMC experiments along with the Drell-Yan, 𝑊 and 𝑍 boson data from LHC and Tevatron experiments, which combined with theoretical calculations at next-to-next-to-leading order in QCD allow for realistic determination of PDFs. The main focus of this analysis is to explore alternative methods for PDF uncertainty estimation that are more firmly grounded in statistical principles. We show that the flexibility of the Bayes framework, allowing one, e.g., to account for non-Gaussianity or inconsistencies of datasets, is crucial to extract realistic uncertainties when such assumptions are not fulfilled. We also demonstrate that MCMC allows one to determine the Δ⁢𝜒 2 value corresponding to a given confidence level in the sample, which can, in turn, be used as a statistically well-founded tolerance criterion used in the Hessian method, thus addressing one of its main long-standing drawbacks.

Risse, Peter Clemens [Universität Münster (Germany↗
DOE OSTI2026

Determination of nuclear PDFs using Markov chain Monte Carlo methods

Global QCD analyses of nuclear parton distribution functions (nPDFs) have traditionally relied on the Hessian method for uncertainty estimation. However, the inherent Gaussian approximation and reliance on local curvature often prove insufficient for nPDF fits, which are frequently characterized by limited data constraints and non-Gaussian likelihoods. In this paper, we present the first nPDF determination based on Markov Chain Monte Carlo (MCMC) techniques, implemented within the nCTEQ framework using an adaptive Metropolis-Hastings algorithm. The MCMC approach enables a direct mapping of the posterior distribution and reveals a highly nontrivial parameter-space structure, including multiple modes and pronounced non-Gaussian behavior, particularly for the valence PDFs. We perform the first single-nucleus global analysis of lead PDFs using exclusively lead data and compare it to a multi-nuclei fit employing a standard analytic A dependence. The inclusion of lighter nuclei reduces quark uncertainties and modifies the shape of the lead PDFs, while leaving the gluon distribution largely unaffected. A complementary Hessian analysis exposes systematic limitations of the Gaussian approximation. Our results demonstrate that MCMC methods provide a more reliable framework for uncertainty quantification in nPDF determinations.

Derakhshanian, N. [Institute of Nuclear Physics Po↗
DOE OSTI2023

Bayesian Cloud Property Retrievals from ARM Active and Passive Measurements

The optimum use of the continuous measurements of thermodynamics, radiation, aerosols, clouds and precipitation from the DOE Atmospheric Radiation Measurement (ARM) program is key to achieve the DOE Atmospheric System Research (ASR)’s objectives. One of the key mission requirements is to retrieve cloud and precipitation properties, as well as vertical motion parameters, along the vertical cross- section defined by the profiling active sensors. Such retrievals are challenging to perform continuously in the entire spectrum of cloud and precipitation conditions due to the large natural microphysical and dynamical variability, the often-limited information content in the measurements, and the lack of proper characterization of measurement quality and uncertainty. Today, the acquisition of new remote and in-situ sensors by the ARM program creates opportunities to address the microphysical retrieval problem by exploiting new, more robust retrieval techniques and integrating various scattered advancements in both sensor techniques and retrieval algorithms. During this project, we constructed a robust Bayesian Markov chain Monte Carlo (MCMC) cloud property retrieval algorithm that includes a state of the art radar forward model. Our MCMC-based retrieval produces both the best estimate of height-resolved cloud and precipitation properties in the radar profile, as well as an estimate of the in-cloud vertical motion and turbulence. In addition, the MCMC algorithm automatically produces robust and flexible estimates of retrieval uncertainty. We tested the algorithm on several synthetic cloud profiles obtained from large eddy simulation (LES) models with bin-resolved microphysics.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Gradient-informed Hamiltonian Monte Carlo for multicomponent CALPHAD model optimization and uncertainty quantification

CALPHAD model parameter optimization is inherently challenging due to non-smooth objective functions, high-dimensional parameter spaces, and the need for uncertainty quantification (UQ). Traditional weighted nonlinear least squares approaches are computationally efficient but local, whereas black-box global optimizers and ensemble Markov Chain Monte Carlo (MCMC) methods provide broader exploration at substantial computational cost. The objective of this work is to combine the global exploration capability of gradient-informed Hamiltonian Monte Carlo – specifically the No-U-Turn Sampler (NUTS) – with local deterministic refinement using BFGS to efficiently optimize multicomponent CALPHAD models with minimal manual intervention. Analytic gradients are computed via the Jansson derivative framework. The methodology is demonstrated on the Cr—Fe binary system and extended to the Cr—Fe—Ni ternary system with 32 degrees of freedom. For Cr—Fe, NUTS achieves comparable or superior optimality relative to ensemble MCMC while requiring over an order-of-magnitude fewer likelihood evaluations. Parameter uncertainties are quantified through NUTS sampling and propagated to thermodynamic observables using local expansion, demonstrating a novel modular approach that combines binary and ternary parameter subsets without requiring global relaxation. These results establish gradient-informed exploration as a scalable strategy for multicomponent CALPHAD optimization and provide a practical route towards efficient higher-order database development with quantified uncertainty.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Joint estimation of biogeochemical model parameters from multiple experiments: A bayesian approach applied to mercury methylation

Here, to characterize complex biogeochemical systems, results from multiple experiments, where each targets a specific subprocess, are commonly combined. The resulting datasets are interpreted through the calibration of biogeochemical models for process inference and predictions. Commonly used calibration approaches of fitting datasets from individual experiments to subprocess models one at a time is prone to missing information shared between datasets and incomplete uncertainty propagation. We propose a Bayesian joint-fitting scheme addressing the above-mentioned concerns by jointly fitting all the available datasets, thus calibrating the entire biogeochemical model in one go using Markov Chain Monte Carlo (MCMC). The identification of null spaces in the parameter distributions from MCMC guided the simplification of certain subprocess models. For example, fast kinetic sorption was replaced by equilibrium sorption, and Monod demethylation was replaced by first-order demethylation. Joint fitting of datasets resulted in complete uncertainty propagation with parameter estimates informed by all available data.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Bayes_Opt-SWMM: A Gaussian process-based Bayesian optimization tool for real-time flood modeling with SWMM

Real-time flood model plays a pivotal role in averting urban flood damage, particularly when there is minimal lead time for preparatory measures. However, urban flood modeling in real-time often contends with inherent uncertainties arising from input data uncertainty and parameter ambiguities. Here this study introduces a real-time calibration (RTC) tool called Bayes_Opt-SWMM, specifically tailored for real-time urban flood modeling and uncertainty optimization. This tool leverages the Gaussian process-based Bayesian optimization algorithm and interfaces seamlessly with the Stormwater Management Model (SWMM). It integrates real-time model forcing data and flood monitoring collected through sensors and gauges which are strategically placed within critical locations of urban drainage systems. Our approach hinges on the Surrogate Model based Uncertainty Optimization (SMUO) concept, providing an avenue for enhancing real-time flood modeling. Bayes_Opt-SWMM runs the optimization process using a surrogate model called Gaussian Process emulator with two inference methods: (1) the Gaussian Process (GP) model and (2) Markov Chain Monte Carlo (MCMC) algorithm in GP model (GP_MCMC). Furthermore, three acquisition functions, namely Expected Improvement (EI), Maximum Probability of Improvement (MPI), and Lower Confidence Bound (LCB), facilitate optimal parameter fitting within the surrogate models. The efficiency of GP-based surrogate models in learning SWMM model parameters, leads to an improved uncertainty quantification and accelerated real-time flood modeling in urban areas. Overall, Bayes_Opt-SWMM emerges as a cost-effective and valuable tool for real-time flood modeling and monitoring, with significant potential for managing intelligent storm water systems in urban environments.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

GenAI4UQ: A software for forward and inverse uncertainty quantification using conditional generative AI

We introduce GenAI4UQ, a software package for forward and inverse uncertainty quantification in model calibration, parameter estimation, and ensemble forecasting. GenAI4UQ leverages a generative AI-based conditional modeling framework to address limitations of traditional inverse modeling techniques, such as Markov Chain Monte Carlo (MCMC) methods. By replacing computationally intensive iterative processes with a direct, learned mapping, GenAI4UQ enables efficient calibration of input parameters and generation of predictions directly from observations. The software supports rapid ensemble forecasting with robust uncertainty quantification while maintaining computational and storage efficiency. Built-in auto-tuning of hyperparameters simplifies model training, ensuring accessibility for users with varying expertise. Its versatile conditional generative framework is applicable across diverse scientific domains. While GenAI4UQ offers significant advantages in flexibility and efficiency, users should interpret its uncertainty estimates with caution in data-sparse scenarios, as the model may overestimate uncertainty—an effect common to all surrogate-based approaches including MCMC with surrogate models. Despite this, GenAI4UQ transforms inverse modeling by providing a fast, reliable, and user-friendly solution. It empowers researchers and practitioners to quickly estimate parameter distributions and generate model predictions for new observations, facilitating efficient decision-making and advancing the state of uncertainty quantification in computational modeling.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Electrical Conductivities and Association Constants in Dilute Aqueous NdCl 3 Solutions from 298 to 523 K along an Isobar of 25 MPa

Here, electrical measurements were performed in dilute aqueous NdCl 3 solutions from 298 to 523 K along the 25 MPa isobar to obtain limiting conductivities and association constants. The specific conductivity data were estimated using a continuous flow cell and a Markov Chain Monte Carlo (MCMC) correction algorithm. The limiting conductivities for the salts in water were derived by regressing the mean spherical approximation model and speciation analyses based on the MCMC algorithm and the deep earth water model. The limiting conductivities derived from the experimental data were correlated using the Zimmerman-Arcis-Tremaine correlation and agreed well with a predictive correlation proposed by Smolyakov, Anderko, and Lencka. Only the first association constant between neodymium and chloride could be derived at low temperatures (<373 K) due to the apparent large statistical uncertainty of the second association constant. Above 373 K, both association constants could be derived and show a reasonable agreement with Migdisov and Williams-Jones and Gammons et al.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗