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Computational studies of the coordination chemistry and bonding of lanthanides have grown in recent decades as the need for understanding the distinct physical, optical, and magnetic properties of these compounds increased. Density functional theory (DFT) methods offer a favorable balance of computational cost and accuracy in lanthanide chemistry and have helped to advance the discovery of novel oxidation states and electronic configurations. This Frontier article examines the scope and limitations of DFT in interpreting structural and spectroscopic data of low-valent lanthanide complexes, elucidating periodic trends, and predicting their properties and reactivity, presented through selected examples. 

The natural determinant reference (NDR) or principal natural determinant is the Slater determinant comprised of the N most strongly occupied natural orbitals of an N-electron state of interest. Unlike the Kohn–Sham (KS) determinant, which yields the exact ground-state density, the NDR only yields the best idempotent approximation to the interacting one-particle reduced density matrix, but it is well-defined in common atom-centered basis sets and is representation-invariant. We show that the under-determination problem of prior attempts to define a ground-state energy functional of the NDR is overcome in a grand-canonical ensemble framework at the zero-temperature limit. The resulting grand potential functional of the NDR ensemble affords the variational determination of the ground state energy, its NDR (ensemble), and select ionization potentials and electron affinities. The NDR functional theory can be viewed as an “exactification” of orbital optimization and empirical generalized KS methods. NDR functionals depending on the noninteracting Hamiltonian do not require troublesome KS-inversion or optimized effective potentials. 

The self-consistent phonon (SCP) method allows one to include anharmonic effects when treating a many-body quantum system at thermal equilibrium. The system is then described by an effective temperature-dependent harmonic Hamiltonian, which can be used to estimate its various dynamic and static properties. In this paper, we combine SCP with ab initio (AI) potential energy evaluation in which case the numerical bottleneck of AI-SCP is the evaluation of Gaussian averages of the AI potential energy and its derivatives. These averages are computed efficiently by the quasi-Monte Carlo method utilizing low-discrepancy sequences leading to a fast convergence with respect to the number, S, of the AI energy evaluations. Moreover, a further substantial (an-order-of-magnitude) improvement in efficiency is achieved once a numerically cheap approximation of the AI potential is available. This is based on using a perturbation theory-like (the two-grid) approach in which it is the average of the difference between the AI and the approximate potential that is computed. The corresponding codes and scripts are provided. 

{"Abstract":["This data set contains 194778 quasireaction subgraphs extracted from CHO transition networks with 2-6 non-hydrogen atoms (CxHyOz, 2 <= x + z <= 6).<\/p>\n\nThe complete table of subgraphs (including file locations) is in CHO-6-atoms-subgraphs.csv file. The subgraphs are in GraphML format (http://graphml.graphdrawing.org) and are compressed using bzip2. All subgraphs are undirected and unweighted. The reactant and product nodes (initial and final) are labeled in the "type" node attribute. The nodes are represented as multi-molecule SMILES strings. The edges are labeled by the reaction rules in SMARTS representation. The forward and backward reading of the SMARTS string should be considered equivalent.<\/p>\n\nThe generation and analysis of this data set is described in\nD. Rappoport, Statistics and Bias-Free Sampling of Reaction Mechanisms from Reaction Network Models, 2023, submitted. Preprint at ChemrXiv, DOI: 10.26434/chemrxiv-2023-wltcr<\/p>\n\nSimulation parameters\n- CHO networks constructed using polar bond break/bond formation rule set for CHO.\n- High-energy nodes were excluded using the following rules:\n  (i) more than 3 rings, (ii) triple and allene bonds in rings, (iii) double bonds at\n  bridge atoms,(iv) double bonds in fused 3-membered rings.\n- Neutral nodes were defined as containing only neutral molecules.\n- Shortest path lengths were determined for all pairs of neutral nodes.\n- Pairs of neutral nodes with shortest-path length > 8 were excluded.\n- Additionally, pairs of neutral nodes connected only by shortest paths passing through\n  additional neutral nodes (reducible paths) were excluded.<\/p>\n\nFor background and additional details, see paper above.<\/p>"],"Other":["This work was supported in part by the National Science Foundation under Grant No. CHE-2227112."]} 

Selection bias is inevitable in manually curated computational reaction databases but can have a significant impact on generalizability of quantum chemical methods and machine learning models derived from these data sets. Here, we propose quasireaction subgraphs as a discrete, graph-based representation of reaction mechanisms that has a well-defined associated probability space and admits a similarity function using graph kernels. Quasireaction subgraphs are thus well suited for constructing representative or diverse data sets of reactions. Quasireaction subgraphs are defined as subgraphs of a network of formal bond breaks and bond formations (transition network) composed of all shortest paths between reactant and product nodes. However, due to their purely geometric construction, they do not guarantee that the corresponding reaction mechanisms are thermodynamically and kinetically feasible. As a result, a binary classification of feasible (reaction subgraphs) and infeasible (non-reactive subgraphs) must be applied after sampling. In this paper, we describe the construction and properties of quasireaction subgraphs and characterize the statistics of quasireaction subgraphs from CHO transition networks with up to six nonhydrogen atoms. We explore their clustering using Weisfeiler–Lehman graph kernels. 

This dataset contains sequence information, three-dimensional structures (from AlphaFold2 model), and substrate classification labels for 358 short-chain dehydrogenase/reductases (SDRs) and 953 S-adenosylmethionine dependent methyltransferases (SAM-MTases).</p> The aminoacid sequences of these enzymes were obtained from the UniProt Knowledgebase (https://www.uniprot.org). The sets of proteins were obtained by querying using InterPro protein family/domain identifiers corresponding to each family: IPR002347 (SDRs) and IPR029063 (SAM-MTases). The query results were filtered by UniProt annotation score, keeping only those with score above 4-out-of-5, and deduplicated by exact sequence matches.</p> The structures were submitted to the publicly available AlphaFold2 protein structure predictor (J. Jumper et al., Nature, 2021, 596, 583) using the ColabFold notebook (https://colab.research.google.com/github/sokrypton/ColabFold/blob/v1.1-premultimer/batch/AlphaFold2_batch.ipynb, M. Mirdita, S. Ovchinnikov, M. Steinegger, Nature Meth., 2022, 19, 679, https://github.com/sokrypton/ColabFold). The model settings used were  msa_model = MMSeq2(Uniref+Environmental), num_models = 1, use_amber = False, use_templates = True, do_not_overwrite_results = True. The resulting PDB structures are included as ZIP archives</p> The classification labels were obtained from the substrate and product annotations of the enzyme UniProtKB records. Two approaches were used: substrate clustering based on molecular fingerprints and manual substrate type classification. For the substate clustering, Morgan fingerprints were generated for all enzymatic substrates and products with known structures (excluding cofactors) with radius = 3 using RDKit (https://rdkit.org). The fingerprints were projected onto two-dimensional space using the UMAP algorithm (L. McInnes, J. Healy, 2018, arXiv 1802.03426) and Jaccard metric and clustered using k-means. This procedure generated 9 clusters for SDR substrates and 13 clusters for SAM-MTases. The SMILES representations of the substrates are listed in the SDR_substrates_to_cluster_map_2DIMUMAP.csv and SAM_substrates_to_13clusters_map_2DIMUMAP.csv files.</p> The following manually defined classification tasks are included for SDRs: NADP/NAD cofactor classification; phenol substrate, sterol substrate, coenzyme A (CoA) substrate. For SAM-MTases, the manually defined classification tasks are: biopolymer (protein/RNA/DNA) vs. small molecule substrate, phenol subsrates, sterol substrates, nitrogen heterocycle substrates. The SMARTS strings used to define the substrate classes are listed in substructure_search_SMARTS.docx.  </p>