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Articles by Cho et al. ( ChemPhysChem , 2020, 21 , 688–696) and Manz ( RSC Adv. , 2020, 10 , 44121–44148) performed unstandardized and standardized, respectively, principal component analysis (PCA) to study atomic charge assignment methods for molecular systems. Both articles used subsets of atomic charges computed by Cho et al. ; however, the data subsets employed were not strictly identical. Herein, an element by element analysis of this dataset is first performed to compare the spread of charge values across individual chemical elements and charge assignment methods. This reveals an underlying problem with the reported Becke partial atomic charges in this dataset. Due to their unphysical values, these Becke charges were not included in the subsequent PCA. Standardized and unstandardized PCA are performed across two datasets: (i) 19 charge assignment methods having a complete basis set limit and (ii) all 25 charge assignment methods (excluding Becke) for which Cho et al. computed atomic charges. The dataset contained ∼2000 molecules having a total of 29 907 atoms in materials. The following five methods (listed here in alphabetical order) showed the greatest correlation to the first principal component in standardized and unstandardized PCA: DDEC6, Hirshfeld-I, ISA, MBIS, and MBSBickelhaupt (note: MBSBickelhaupt does not appear in the 19 methods dataset). For standardized PCA, the DDEC6 method ranked first followed closely by MBIS. For unstandardized PCA, Hirshfeld-I (19 methods) or MBSBickelhaupt (25 methods) ranked first followed by DDEC6 in second place (both 19 and 25 methods). 

This article studies two kinds of information extracted from statistical correlations between methods for assigning net atomic charges (NACs) in molecules. First, relative charge transfer magnitudes are quantified by performing instant least squares fitting (ILSF) on the NACs reported by Cho et al. ( ChemPhysChem , 2020, 21 , 688–696) across 26 methods applied to ∼2000 molecules. The Hirshfeld and Voronoi deformation density (VDD) methods had the smallest charge transfer magnitudes, while the quantum theory of atoms in molecules (QTAIM) method had the largest charge transfer magnitude. Methods optimized to reproduce the molecular dipole moment ( e.g. , ACP, ADCH, CM5) have smaller charge transfer magnitudes than methods optimized to reproduce the molecular electrostatic potential ( e.g. , CHELPG, HLY, MK, RESP). Several methods had charge transfer magnitudes even larger than the electrostatic potential fitting group. Second, confluence between different charge assignment methods is quantified to identify which charge assignment method produces the best NAC values for predicting via linear correlations the results of 20 charge assignment methods having a complete basis set limit across the dataset of ∼2000 molecules. The DDEC6 NACs were the best such predictor of the entire dataset. Seven confluence principles are introduced explaining why confluent quantitative descriptors offer predictive advantages for modeling a broad range of physical properties and target applications. These confluence principles can be applied in various fields of scientific inquiry. A theory is derived showing confluence is better revealed by standardized statistical analysis ( e.g. , principal components analysis of the correlation matrix and standardized reversible linear regression) than by unstandardized statistical analysis. These confluence principles were used together with other key principles and the scientific method to make assigning atom-in-material properties non-arbitrary. The N@C 60 system provides an unambiguous and non-arbitrary falsifiable test of atomic population analysis methods. The HLY, ISA, MK, and RESP methods failed for this material. 

Databases of experimentally-derived metal–organic framework (MOF) crystal structures are useful for large-scale computational screening to identify which MOFs are best-suited for particular applications. However, these crystal structures must be cleaned to identify and/or correct various artifacts. The recently published 2019 CoRE MOF database (Chung et al. , J. Chem. Eng. Data , 2019, 64 , 5985–5998) reported thousands of experimentally-derived crystal structures that were partially cleaned to remove solvent molecules, to identify hundreds of disordered structures (approximately thirty of those were corrected), and to manually correct approximately 100 structures ( e.g. , adding missing hydrogen atoms). Herein, further cleaning of the 2019 CoRE MOF database is performed to identify structures with misbonded or isolated atoms: (i) structures containing an isolated atom, (ii) structures containing atoms too close together ( i.e. , overlapping atoms), (iii) structures containing a misplaced hydrogen atom, (iv) structures containing an under-bonded carbon atom (which might be caused by missing hydrogen atoms), and (v) structures containing an over-bonded carbon atom. This study should not be viewed as the final cleaning of this database, but rather as progress along the way towards the goal of someday achieving a completely cleaned set of experimentally-derived MOF crystal structures. We performed atom typing for all of the accepted structures to identify those structures that can be parameterized by previously reported forcefield precursors (Chen and Manz, RSC Adv ., 2019, 9 , 36492–36507). We report several forcefield precursors ( e.g. , net atomic charges, atom-in-material polarizabilities, atom-in-material dispersion coefficients, electron cloud parameters, etc. ) for more than five thousand MOFs in the 2019 CoRE MOF database. 

A host of important performance properties for metal–organic frameworks (MOFs) and other complex materials can be calculated by modeling statistical ensembles. The principle challenge is to develop accurate and computationally efficient interaction models for these simulations. Two major approaches are (i) ab initio molecular dynamics in which the interaction model is provided by an exchange–correlation theory ( e.g. , DFT + dispersion functional) and (ii) molecular mechanics in which the interaction model is a parameterized classical force field. The first approach requires further development to improve computational speed. The second approach requires further development to automate accurate forcefield parameterization. Because of the extreme chemical diversity across thousands of MOF structures, this problem is still mostly unsolved today. For example, here we show structures in the 2014 CoRE MOF database contain more than 8 thousand different atom types based on first and second neighbors. Our results showed that atom types based on both first and second neighbors adequately capture the chemical environment, but atom types based on only first neighbors do not. For 3056 MOFs, we used density functional theory (DFT) followed by DDEC6 atomic population analysis to extract a host of important forcefield precursors: partial atomic charges; atom-in-material (AIM) C 6 , C 8 , and C 10 dispersion coefficients; AIM dipole and quadrupole moments; various AIM polarizabilities; quantum Drude oscillator parameters; AIM electron cloud parameters; etc. Electrostatic parameters were validated through comparisons to the DFT-computed electrostatic potential. These forcefield precursors should find widespread applications to developing MOF force fields. 

We present two algorithms to compute system-specific polarizabilities and dispersion coefficients such that required memory and computational time scale linearly with increasing number of atoms in the unit cell for large systems. The first algorithm computes the atom-in-material (AIM) static polarizability tensors, force-field polarizabilities, and C 6 , C 8 , C 9 , C 10 dispersion coefficients using the MCLF method. The second algorithm computes the AIM polarizability tensors and C 6 coefficients using the TS-SCS method. Linear-scaling computational cost is achieved using a dipole interaction cutoff length function combined with iterative methods that avoid large dense matrix multiplies and large matrix inversions. For MCLF, Richardson extrapolation of the screening increments is used. For TS-SCS, a failproof conjugate residual (FCR) algorithm is introduced that solves any linear equation system having Hermitian coefficients matrix. These algorithms have mathematically provable stable convergence that resists round-off errors. We parallelized these methods to provide rapid computation on multi-core computers. Excellent parallelization efficiencies were obtained, and adding parallel processors does not significantly increase memory requirements. This enables system-specific polarizabilities and dispersion coefficients to be readily computed for materials containing millions of atoms in the unit cell. The largest example studied herein is an ice crystal containing >2 million atoms in the unit cell. For this material, the FCR algorithm solved a linear equation system containing >6 million rows, 7.57 billion interacting atom pairs, 45.4 billion stored non-negligible matrix components used in each large matrix-vector multiplication, and ∼19 million unknowns per frequency point (>300 million total unknowns). 

Polarizabilities and London dispersion forces are important to many chemical processes. Force fields for classical atomistic simulations can be constructed using atom-in-material polarizabilities and C n ( n = 6, 8, 9, 10…) dispersion coefficients. This article addresses the key question of how to efficiently assign these parameters to constituent atoms in a material so that properties of the whole material are better reproduced. We develop a new set of scaling laws and computational algorithms (called MCLF) to do this in an accurate and computationally efficient manner across diverse material types. We introduce a conduction limit upper bound and m -scaling to describe the different behaviors of surface and buried atoms. We validate MCLF by comparing results to high-level benchmarks for isolated neutral and charged atoms, diverse diatomic molecules, various polyatomic molecules ( e.g. , polyacenes, fullerenes, and small organic and inorganic molecules), and dense solids (including metallic, covalent, and ionic). We also present results for the HIV reverse transcriptase enzyme complexed with an inhibitor molecule. MCLF provides the non-directionally screened polarizabilities required to construct force fields, the directionally-screened static polarizability tensor components and eigenvalues, and environmentally screened C 6 coefficients. Overall, MCLF has improved accuracy compared to the TS-SCS method. For TS-SCS, we compared charge partitioning methods and show DDEC6 partitioning yields more accurate results than Hirshfeld partitioning. MCLF also gives approximations for C 8 , C 9 , and C 10 dispersion coefficients and quantum Drude oscillator parameters. This method should find widespread applications to parameterize classical force fields and density functional theory (DFT) + dispersion methods. 

Bond order quantifies the number of electrons dressed-exchanged between two atoms in a material and is important for understanding many chemical properties. Diatomic molecules are the smallest molecules possessing chemical bonds and play key roles in atmospheric chemistry, biochemistry, lab chemistry, and chemical manufacturing. Here we quantum-mechanically calculate bond orders for 288 diatomic molecules and ions. For homodiatomics, we show bond orders correlate to bond energies for elements within the same chemical group. We quantify and discuss how semicore electrons weaken bond orders for elements having diffuse semicore electrons. Lots of chemistry is effected by this. We introduce a first-principles method to represent orbital-independent bond order as a sum of orbital-dependent bond order components. This bond order component analysis (BOCA) applies to any spin-orbitals that are unitary transformations of the natural spin-orbitals, with or without periodic boundary conditions, and to non-magnetic and (collinear or non-collinear) magnetic materials. We use this BOCA to study all period 2 homodiatomics plus Mo 2 , Cr 2 , ClO, ClO − , and Mo 2 (acetate) 4 . Using Manz's bond order equation with DDEC6 partitioning, the Mo–Mo bond order was 4.12 in Mo 2 and 1.46 in Mo 2 (acetate) 4 with a sum of bond orders for each Mo atom of ∼4. Our study informs both chemistry research and education. As a learning aid, we introduce an analogy between bond orders in materials and message transmission in computer networks. We also introduce the first working quantitative heuristic model for all period 2 homodiatomic bond orders. This heuristic model incorporates s–p mixing to give heuristic bond orders of ¾ (Be 2 ), 1¾ (B 2 ), 2¾ (C 2 ), and whole number bond orders for the remaining period 2 homodiatomics. 

The DDEC6 method is one of the most accurate and broadly applicable atomic population analysis methods. It works for a broad range of periodic and non-periodic materials with no magnetism, collinear magnetism, and non-collinear magnetism irrespective of the basis set type. First, we show DDEC6 charge partitioning to assign net atomic charges corresponds to solving a series of 14 Lagrangians in order. Then, we provide flow diagrams for overall DDEC6 analysis, spin partitioning, and bond order calculations. We wrote an OpenMP parallelized Fortran code to provide efficient computations. We show that by storing large arrays as shared variables in cache line friendly order, memory requirements are independent of the number of parallel computing cores and false sharing is minimized. We show that both total memory required and the computational time scale linearly with increasing numbers of atoms in the unit cell. Using the presently chosen uniform grids, computational times of ∼9 to 94 seconds per atom were required to perform DDEC6 analysis on a single computing core in an Intel Xeon E5 multi-processor unit. Parallelization efficiencies were usually >50% for computations performed on 2 to 16 cores of a cache coherent node. As examples we study a B-DNA decamer, nickel metal, supercells of hexagonal ice crystals, six X@C 60 endohedral fullerene complexes, a water dimer, a Mn 12 -acetate single molecule magnet exhibiting collinear magnetism, a Fe 4 O 12 N 4 C 40 H 52 single molecule magnet exhibiting non-collinear magnetism, and several spin states of an ozone molecule. Efficient parallel computation was achieved for systems containing as few as one and as many as >8000 atoms in a unit cell. We varied many calculation factors ( e.g. , grid spacing, code design, thread arrangement, etc. ) and report their effects on calculation speed and precision. We make recommendations for excellent performance.