Learning Associations between Features and Clusters: An Interpretable Deep Clustering Method
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97 MATHEMATICS AND COMPUTING↗
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The HectoMAP Cluster Survey: Spectroscopically Identified Clusters and their Brightest Cluster Galaxies (BCGs)
We apply a friends-of-friends (FoF) algorithm to identify galaxy clusters and we use the catalog to explore the evolutionary synergy between brightest cluster galaxies (BCGs) and their host clusters. We base the cluster catalog on the dense HectoMAP redshift survey (2000 redshifts deg -2 ). The HectoMAP FoF catalog includes 346 clusters with 10 or more spectroscopic members within the range 0.05 < z < 0.55 and with a median z = 0.29. We list these clusters and their members. We also include central velocity dispersions (σ*, BCG ) for the FoF cluster BCGs, a distinctive feature of the HectoMAP FoF catalog. HectoMAP clusters with higher galaxy number density (80 systems) are all genuine clusters with a strong concentration and a prominent BCG in Subaru/Hyper Suprime-Cam images. The phase-space diagrams show the expected elongation along the line of sight. Lower-density systems include some low reliability systems. We establish a connection between BCGs and their host clusters by demonstrating that σ*, BCG /σ cl decreases as a function of cluster velocity dispersion (σ cl ), in contrast, numerical simulations predict a constant σ*, BCG /σ cl . Sets of clusters at two different redshifts show that BCG evolution in massive systems is slow over the redshift range z < 0.4. The data strongly suggest that minor mergers may play an important role in BCG evolution in clusters with σ cl ≳ 300 km s -1 . For lower mass systems (σ cl < 300 km s -1 ), major mergers may play a significant role. The coordinated evolution of BCGs and their host clusters provides an interesting test of simulations in high-density regions of the universe.
79 ASTRONOMY AND ASTROPHYSICS↗
Preventive Power Outage Estimation Based on a Novel Scenario Clustering Strategy
The increasing occurrence of extreme weather events is challenging power grid operation. For extreme weather events, the system operator is responsible for estimating the power outages and scheduling the restoration resources. This paper proposes an outage evaluation framework to identify the possible unserved load profiles, vulnerable areas, and mobile energy adequacy. The outputs of an outage prediction model tool are used to generate numerous faulted line scenarios. Next, each scenario's nodal unserved load profile is obtained by solving a three-phase restoration model that considers repair crews and mobile energy resources (MERs). Then, a novel scenario clustering strategy is developed to cluster the unserved load profiles into multiple representative profiles which the system operator can focus on. Finally, case studies on a distribution system evaluate the damage caused by an extreme weather event and verify the effectiveness of the proposed scenario clustering strategy.
MATHEMATICS AND COMPUTING,POWER TRANSMISSION AND D↗
An X-ray survey of clusters of galaxies. IV - A survey of southern clusters and a compilation of upper limits for both Abell and southern clusters
The results of the HEAO 1 A-1 X-ray survey of galaxy clusters are reported. X-ray error boxes and intensities are presented for all clusters in the Abell catalog and for the catalog of southern clusters and groups compiled by Duus and Newell (1977). A correlation is derived on the basis of the X-ray luminosity function for 2-6 keV which may be used to calculate the contribution of clusters to the diffuse X-ray background at different energies. The cluster X-ray is estimated to be 9.3 percent (+ 1.9 or - 1.5 percent). Correlations between X-ray luminosity and other cluster properties are exmained, and it is found that the distribution of upper limits may be applied to obtaining a more precise estimate of the average X-ray luminosity of clusters. The Abell richness class and southern cluster concentrations were strongly correlated with X-ray luminosity. Correlations between optical x-ray luminosity and optical radius velocity dispersion, spiral fraction, and radio power are analyzed. The evidence for all these correlations was considered to be weak because of poor scatter in the data.
Kowalski, M. P.↗
Cluster-cluster clustering
The cluster correlation function xi sub c(r) is compared with the particle correlation function, xi(r) in cosmological N-body simulations with a wide range of initial conditions. The experiments include scale-free initial conditions, pancake models with a coherence length in the initial density field, and hybrid models. Three N-body techniques and two cluster-finding algorithms are used. In scale-free models with white noise initial conditions, xi sub c and xi are essentially identical. In scale-free models with more power on large scales, it is found that the amplitude of xi sub c increases with cluster richness; in this case the clusters give a biased estimate of the particle correlations. In the pancake and hybrid models (with n = 0 or 1), xi sub c is steeper than xi, but the cluster correlation length exceeds that of the points by less than a factor of 2, independent of cluster richness. Thus the high amplitude of xi sub c found in studies of rich clusters of galaxies is inconsistent with white noise and pancake models and may indicate a primordial fluctuation spectrum with substantial power on large scales.
Barnes, J.↗
Ultraviolet studies of O and B stars in the LMC cluster NGC 2100, the SMC cluster NGC 330 and the Galactic cluster NGC 6530
High-resolution and low-resolution IUE spectra of O and B stars in the LMC cluster NGC 2100, the SMC cluster NGC 330, and the young Galactic cluster NGC 6530 are investigated. Temperatures and luminosities are determined. In the LMC and SMC clusters, the most luminous stars are evolved stars on the horizontal supergiant branch, while in NGC 6530 the stars are all still on the main sequence. Extinction laws were determined. They confirm the known differences between LMC and Galactic extinctions. No mass loss was detected for the evolved B stars in the LMC and SMC clusters, while the high-luminosity stars in NGC 6530 show P Cygni profiles.
Boehm-Vitense, E.↗
SPT clusters with DES and HST weak lensing. I. Cluster lensing and Bayesian population modeling of multiwavelength cluster datasets
We present a Bayesian population modeling method to analyze the abundance of galaxy clusters identified by the South Pole Telescope (SPT) with a simultaneous mass calibration using weak gravitational lensing data from the Dark Energy Survey (DES) and the Hubble Space Telescope (HST). We discuss and validate the modeling choices with a particular focus on a robust, weak-lensing-based mass calibration using DES data. For the DES Year 3 data, we report a systematic uncertainty in weak-lensing mass calibration that increases from 1% at z = 0.25 to 10% at z = 0.95 , to which we add 2% in quadrature to account for uncertainties in the impact of baryonic effects. We implement an analysis pipeline that joins the cluster abundance likelihood with a multiobservable likelihood for the Sunyaev-Zel’dovich effect, optical richness, and weak-lensing measurements for each individual cluster. We validate that our analysis pipeline can recover unbiased cosmological constraints by analyzing mocks that closely resemble the cluster sample extracted from the SPT-SZ, SPTpol ECS, and SPTpol 500d surveys and the DES Year 3 and HST-39 weak-lensing datasets. This work represents a crucial prerequisite for the subsequent cosmological analysis of the real dataset.
79 ASTRONOMY AND ASTROPHYSICS↗
Studies of the Virgo Cluster. II - A catalog of 2096 galaxies in the Virgo Cluster area. V - Luminosity functions of Virgo Cluster galaxies
The present catalog of 2096 galaxies within an area of about 140 sq deg approximately centered on the Virgo cluster should be an essentially complete listing of all certain and possible cluster members, independent of morphological type. Cluster membership is essentially decided by galaxy morphology; for giants and the rare class of high surface brightness dwarfs, membership rests on velocity data. While 1277 of the catalog entries are considered members of the Virgo cluster, 574 are possible members and 245 appear to be background Zwicky galaxies. Major-to-minor axis ratios are given for all galaxies brighter than B(T) = 18, as well as for many fainter ones.
Binggeli, B.↗
Dynamics of cD Clusters of Galaxies: Conclusion of a Survey of 25 Abell Clusters - 4
We present the final results of a spectroscopic study of a sample of cD galaxy clusters. The goal of this program has been to study the dynamics of the clusters, with emphasis on determining the nature and frequency of cD galaxies with peculiar velocities. Redshifts measured with the MX Spectrometer have been combined with those obtained from the literature to obtain typically 50 - 150 observed velocities in each of 25 galaxy clusters containing a central cD galaxy. We present a dynamical analysis of the final 11 clusters to be observed in this sample. All 25 clusters are analyzed in a uniform manner to test for the presence of substructure, and to determine peculiar velocities and their statistical significance for the central cD galaxy. These peculiar velocities were used to determine whether or not the central cD galaxy is at rest in the cluster potential well. We find that 30 - 50% of the clusters in our sample possess significant subclustering (depending on the cluster radius used in the analysis), which is in agreement with other studies of non-cD clusters. Hence, the dynamical state of cD clusters is not different than other present-day clusters. After careful study, four of the clusters appear to have a cD galaxy with a significant peculiar velocity. Dressler-Shectman tests indicate that three of these four clusters have statistically significant substructure within 1.5/h(sub 75) Mpc of the cluster center. The dispersion 75 of the cD peculiar velocities is 164 +41/-34 km/s around the mean cluster velocity. This represents a significant detection of peculiar cD velocities, but at a level which is far below the mean velocity dispersion for this sample of clusters. The picture that emerges is one in which cD galaxies are nearly at rest with respect to the cluster potential well, but have small residual velocities due to subcluster mergers.
Oegerle, William R.↗
Dark Energy Survey Year 1 Results: Cosmological Constraints from Cluster Abundances, Weak Lensing, and Galaxy Correlations
We present the first joint analysis of cluster abundances and auto or cross-correlations of three cosmic tracer fields: galaxy density, weak gravitational lensing shear, and cluster density split by optical richness. From a joint analysis (4×2pt+N) of cluster abundances, three cluster cross-correlations, and the auto correlations of the galaxy density measured from the first year data of the Dark Energy Survey, we obtain Ωm=0.305-0.038+0.055 and σ8=0.783-0.054+0.064. This result is consistent with constraints from the DES-Y1 galaxy clustering and weak lensing two-point correlation functions for the flat νΛCDM model. Consequently, we combine cluster abundances and all two-point correlations from across all three cosmic tracer fields (6×2pt+N) and find improved constraints on cosmological parameters as well as on the cluster observable-mass scaling relation. This analysis is an important advance in both optical cluster cosmology and multiprobe analyses of upcoming wide imaging surveys.
79 ASTRONOMY AND ASTROPHYSICS↗
The dynamical state of eROSITA clusters and its impact on the brightest cluster galaxy luminosity
The first Spectrum-Roentgen-Gamma (SRG) eROSITA public release contains 12 247 clusters and groups from its first 6 months of operation. We used the offset between the brightest cluster galaxy (BCG) and the X-ray peak ( D BCG − X ) to classify the cluster dynamical state of 3946 galaxy clusters and groups. We aim to investigate the evolution of the merger and relaxed cluster distributions with redshift and mass, and the distributions’ impact on the BCG. We used the X-ray peak from the eROSITA survey and the BCG position from the LS DR10 optical data, which includes the DECam eROSITA Survey optical data, to measure the D BCG − X offset. We modelled the distribution of D BCG − X , in units of R 500 , as the sum of two Rayleigh distributions representing the cluster’s relaxed and disturbed populations, and explored their evolution with redshift and mass. To explore the impact of the cluster’s dynamical state on the BCG luminosity, we separated the main sample according to the dynamical state. We defined clusters as relaxed if D BCG − X < 0.25R500, disturbed if D BCG − X > 0.5R500, and ‘diverse’ if 0.25R500 < D BCG − X < 0.5R500. We find no evolution of the merging fraction with redshift or mass. The width of the relaxed distribution increases with redshift, while the width of the two Rayleigh distributions decreases with mass. The analysis reveals that BCGs in relaxed clusters are brighter than BCGs in both the disturbed and diverse cluster populations. The most significant differences are found for high-mass clusters at higher redshifts. The results suggest that BCGs in low-mass clusters are less centrally bound than those in high-mass systems, irrespective of the dynamical state. Over time, BCGs in relaxed clusters progressively align with the potential centre. This alignment correlates with their luminosity growth relative to BCGs in dynamically disturbed clusters, underscoring the critical role of the cluster’s dynamical state in regulating BCG evolution.
galaxy clusters↗
Improving Cosmological Constraints from Galaxy Cluster Number Counts with CMB-cluster-lensing Data: Results from the SPT-SZ Survey and Forecasts for the Future
Abstract We show the improvement to cosmological constraints from galaxy cluster surveys with the addition of cosmic microwave background (CMB)-cluster lensing data. We explore the cosmological implications of adding mass information from the 3.1 σ detection of gravitational lensing of the CMB by galaxy clusters to the Sunyaev–Zel’dovich (SZ) selected galaxy cluster sample from the 2500 deg 2 SPT-SZ survey and targeted optical and X-ray follow-up data. In the ΛCDM model, the combination of the cluster sample with the Planck power spectrum measurements prefers σ 8 Ω m / 0.3 0.5 = 0.831 ± 0.020 . Adding the cluster data reduces the uncertainty on this quantity by a factor of 1.4, which is unchanged whether the 3.1 σ CMB-cluster lensing measurement is included or not. We then forecast the impact of CMB-cluster lensing measurements with future cluster catalogs. Adding CMB-cluster lensing measurements to the SZ cluster catalog of the ongoing SPT-3G survey is expected to improve the expected constraint on the dark energy equation of state w by a factor of 1.3 to σ ( w ) = 0.19. We find the largest improvements from CMB-cluster lensing measurements to be for σ 8 , where adding CMB-cluster lensing data to the cluster number counts reduces the expected uncertainty on σ 8 by respective factors of 2.4 and 3.6 for SPT-3G and CMB-S4.
79 ASTRONOMY AND ASTROPHYSICS↗
Composition-Dependent Reconstructions of Titanium Oxide Clusters and Cu(111) Support via Cluster-Adatom Interactions
Mass-selected cluster deposition and scanning tunneling microscopy (STM) were used to investigate the surface structure and thermal stability of Ti 3 O y (y = 5, 6) clusters on a Cu(111) surface. STM imaging shows that the substoichiometric Ti 3 O 5 clusters aggregate into large dendritic assemblies at step edges, while the stoichiometric Ti 3 O 6 clusters are highly dispersed as single or small groupings of clusters. Cluster deposition also results in extensive reconstruction of the Cu(111) surface. The Ti 3 O 5 /Cu(111) surface exhibits one Cu-layer deep pits that are ringed by cluster assemblies that nucleate and grow at the perimeter step edges. By comparison, the Ti 3 O 6 /Cu(111) surface consists of irregular step edges and small one Cu-layer islands that are decorated with clusters. The restructuring of the Cu(111) surface is attributed to strong Cu-cluster interactions that release Cu adatoms from the step edges. This mechanism is supported by density functional theory (DFT) calculations, which predict that the Cu adatom attachment to the clusters is energetically favorable. Annealing the surfaces at elevated temperatures (400-700 K) results in higher densities of irregular steps, pits and isolated Cu islands as well as increased cluster aggregation. In conclusion, this work highlights adatom-cluster interactions as another manifestation of strong cluster-support interactions that can significantly alter the nature of the oxide-metal interface.
77 NANOSCIENCE AND NANOTECHNOLOGY↗
Clustering of CODEX clusters
The clustering of galaxy clusters links the spatial nonuniformity of dark matter halos to the growth of the primordial spectrum of perturbations. The amplitude of the clustering signal is widely used to estimate the halo mass of astrophysical objects. The advent of cluster mass calibrations enables using clustering in cosmological studies. Aims. We analyze the autocorrelation function of a large contiguous sample of galaxy clusters, the Constrain Dark Energy with X-ray (CODEX) sample, in which we take particular care of cluster definition. These clusters were X-ray selected using the ROentgen SATellite All-Sky Survey and then identified as galaxy clusters using the code redMaPPer run on the photometry of the Sloan Digital Sky Survey. We develop methods for precisely accounting for the sample selection effects on the clustering and demonstrate their robustness using numerical simulations. Methods. Using the clean CODEX sample, which was obtained by applying a redshift-dependent richness selection, we computed the two-point autocorrelation function of galaxy clusters in the 0.1 < z < 0.3 and 0.3 < z < 0.5 redshift bins. We compared the bias in the measured correlation function with values obtained in numerical simulations using a similar cluster mass range. Results. By fitting a power law, we measured a correlation length r 0 = 18.7 ± 1.1 and slope γ = 1.98 ± 0.14 for the correlation function in the full redshift range. By fixing the other cosmological parameters to their nine-year Wilkinson Microwave Anisotropy Probe values, we reproduced the observed shape of the correlation function under the following cosmological conditions: Ω m0 = $0.22_{-0.03}^{+0.04}$ and S 8 = σ 8 (Ω m0 /0.3)0.5 = $0.85_{-0.08}^{+0.10}$ with estimated additional systematic errors of σ Ω m0 = 0.02 and σS 8 = 0.20. We illustrate the complementarity of clustering constraints by combining them with CODEX cosmological constraints based on the X-ray luminosity function, deriving Ωm = 0.25 ± 0.01 and σ 8 = $0.81_{-0.02}^{+0.01}$ with an estimated additional systematic error of σ Ω m0 = 0.07 and σ σ8 = 0.04. The mass calibration and statistical quality of the mass tracers are the dominant source of uncertainty.
79 ASTRONOMY AND ASTROPHYSICS↗
Brightest Cluster Galaxy ellipticity as proxy for halo shape: Orientation bias, assembly bias, and potential selection effects in SZ-selected clusters
The orientation of triaxial galaxy clusters with respect to the line-of-sight is expected to be one of the prime sources of scatter and potential bias in optical observables (e.g., richness and weak-lensing signal) of galaxy clusters. In this work, we use the observed shape of the central Brightest Cluster Galaxy (BCG) as proxy for the orientation along the line-of-sight for clusters selected via the Sunyaev-Zel'dovich (SZ) effect from the South Pole Telescope (SPT) and Atacama Cosmology Telescope (ACT) surveys, matched to optically selected clusters from the Dark Energy Survey Year 3 (DES). We construct two samples of clusters that are designed to be identical in SZ mass estimate and redshift but with the roundest vs. the most elliptical BCGs, which we expect to correspond to BCGs (and clusters) with major axes aligned along the line-of-sight vs. in the plane of the sky, respectively. We find that the optical richness of round-BCG clusters is $\sim 10$% larger than that of elliptical-BCG clusters, in agreement with the expectation from projection effects and presenting the first such detection in data. The density profiles, however, are not in agreement with the expectation from projection effects: the 1-halo term (below $6~h^{-1}\rm{Mpc}$) of both the weak-lensing and galaxy density profiles are the same for the subsamples, contrary to previous studies based on X-ray selected clusters. In the 2-halo regime (above $6~h^{-1}\rm{Mpc}$), we find a significant excess of the elliptical-BCG cluster profiles compared to the round-BCG cluster profiles, which is the opposite of the expectation from numerical simulations. We hypothesize that the intrinsic shape of the BCG reflects not just the orientation angle, but also intrinsic properties of the cluster which can affect both the SZ signal and the amplitude of the 2-halo term.
Srinivasan, Radhakrishnan [SUNY, Stony Brook] (ORC↗
Detection of spatial clustering in the 1000 richest SDSS DR8 redMaPPer clusters with nearest neighbor distributions
ABSTRACT Distances to the k-nearest-neighbor (kNN) data points from volume-filling query points are a sensitive probe of spatial clustering. Here, we present the first application of kNN summary statistics to observational clustering measurement, using the 1000 richest redMaPPer clusters (0.1 ≤ z ≤ 0.3) from the SDSS DR8 catalog. A clustering signal is defined as a difference in the cumulative distribution functions (CDFs) of kNN distances from fixed query points to the observed clusters versus a set of unclustered random points. We find that the k = 1, 2-NN CDFs of redMaPPer deviate significantly from the randoms’ across scales of 35 to 155 Mpc, which is a robust signature of clustering. In addition to kNN, we also measure the two-point correlation function for the same set of redMaPPer clusters versus random points, which shows a noisier and less significant clustering signal within the same radial scales. Quantitatively, the χ2 distribution for both the kNN-CDFs and the two-point correlation function measured on the randoms peak at χ2 ∼ 50 (null hypothesis), whereas the kNN-CDFs (χ2 ∼ 300, p = 1.54 × 10−36) pick up a much more significant clustering signal than the two-point function (χ2 ∼ 100, p = 1.16 × 10−6) when measured on redMaPPer. Finally, the measured 3NN and 4NN CDFs deviate from the predicted k = 3, 4-NN CDFs assuming an ideal Gaussian field, indicating a non-Gaussian clustering signal for redMaPPer clusters, although its origin might not be cosmological due to observational systematics. Therefore, kNN serves as a more sensitive probe of clustering complementary to the two point correlation function, providing a novel approach for constraining cosmology and galaxy–halo connection.
79 ASTRONOMY AND ASTROPHYSICS↗
Characterizing the surface compositions of supported bimetallic PtSn clusters: Effects of cluster-support interactions and surface adsorbates
PtSn bimetallic clusters on TiO2(110) and highly oriented pyrolytic graphite (HOPG) surfaces have been characterized by scanning tunneling microscopy, low energy ion scattering (LEIS), Xray photoelectron spectroscopy, and temperature programmed desorption (TPD); density functional theory (DFT) calculations have also been performed to better understand adsorption of CO and D2 on the PtSn surfaces. On TiO2 at coverages of 2 ML of Pt and 2 ML of Sn, exclusively bimetallic clusters are formed for both orders of deposition because clusters of the first metal completely cover the surface such that all atoms of the second metal are incorporated into the existing clusters. In contrast, on HOPG, the high mobility and weak cluster-support interactions on HOPG result in much larger 2 ML monometallic clusters (~30 Å high) that do not completely cover the surface, and deposition of the second metal produces larger clusters as well as smaller ones. Despite the difference in cluster morphologies for the different orders of deposition and supports, the LEIS experiments demonstrate that in all cases, the PtSn clusters are rich in Sn at the surface, as expected based on the lower surface free energy for Sn compared to Pt. Furthermore, the +0.2 eV shift in the Sn(3d5/2) binding energy observed on all surfaces in the presence of Pt is consistent with PtSn alloy formation. Deposition of 2 ML of Sn on TiO2 produces two-dimensional clusters with oxidation of Sn and reduction of titania at the clustersupport interface, but addition of Pt to the Sn clusters causes Sn to diffuse away from this interface, leaving Sn in the metallic state. TPD experiments on 2 ML Pt/TiO2 with increasing coverages of Sn show that the number of adsorption sites for D2 sharply decreases to nearly zero at 0.5 ML, while CO adsorption decreases to zero only at much higher Sn coverages of 2 ML. DFT studies for Sn modified Pt surfaces and bulk structures demonstrate that for CO adsorption at low Sn coverages (<0.25 ML), the strong Pt-CO interactions induce diffusion of Pt to the cluster surface and the formation of a bulk Pt3Sn alloy, whereas D2 adsorption does not lead to interactions with the Pt surface that are strong enough to induce alloy formation. A single Sn adatom prevents D2 adsorption on four neighboring Pt atoms via site-blocking and the donation of electron density to Pt.
Li, Fangliang↗
H-cluster Intermediates and Catalytic Properties of Clostridium pasteurianum [FeFe]-Hydrogenase III
[FeFe]-Hydrogenases are structurally diverse enzymes that catalyze reversible H2 activation at a catalytic cofactor or H-cluster. The H-cluster is a [4Fe-4S] cubane linked by a cysteine thiolate to a diiron subsite containing unique CO, CN-, and dithiomethylamine ligands. The established H-cluster resting state of [4Fe-4S]2+-[FeII-FeI], or Hox, functions in H2 binding and oxidation, or by proton-coupled reduction initiates H2 evolution. In contrast, in Clostridium pasteurianum [FeFe]-hydrogenase III (CpIII) the resting state of the H-cluster is fully oxidized, [4Fe-4S]2+-[FeII-FeII], or Hox+1. To begin to understand if Hox+1 has a role in the mechanism of CpIII, we determined the spectroscopic and redox properties of CpIII H-cluster states under catalytic conditions. CpIII poised in Hox+1 and either equilibrated under 1 atm of H2 or reduced with sodium dithionite, resulted in a mixture of reduced states including Hox (Em8 = -407 mV), Htrans-like [4Fe-4S]+-[FeII-FeII] (Em8 = -418 mV), Hred [4Fe-4S]+-[FeII-FeI], and HredH+ [4Fe-4S]2+-[FeI-FeI] (Em8 = -455-480 mV). Under H2 the population of the Htrans-like state was >20-fold higher than Hox, implicating a role in CpIII catalysis. Unlike other enzymes, there was no spectral evidence of fully reduced states, such as HsredH+ ([4Fe-4S]+-[FeI-FeI]) or Hhyd ([4Fe-4S]+-[FeII-FeII]-H-). Thus, while the H-cluster states of CpIII encompass most of the catalytic intermediates, it is either unable to form HsredH+ and Hhyd, or these states are highly destabilized in CpIII. Thus, these results demonstrate that catalytic intermediates of reduced CpIII differ from the typical intermediates of other catalytic [FeFe]-hydrogenases and may explain the catalytic preference for H2 production.
08 HYDROGEN↗