Net atomic charges (NACs) are widely used in all chemical sciences to concisely summarize key information about the partitioning of electrons among atoms in materials. The objective of this article is to develop an atomic population analysis method that is suitable to be used as a default method in quantum chemistry programs irrespective of the kind of basis sets employed. To address this challenge, we introduce a new atoms-in-materials method with the following nine properties: (1) exactly one electron distribution is assigned to each atom, (2) core electrons are assigned to the correct host atom, (3) NACs are formally independent of the basis set type because they are functionals of the total electron distribution, (4) the assigned atomic electron distributions give an efficiently converging polyatomic multipole expansion, (5) the assigned NACs usually follow Pauling scale electronegativity trends, (6) NACs for a particular element have good transferability among different conformations that are equivalently bonded, (7) the assigned NACs are chemically consistent with the assigned atomic spin moments, (8) the method has predictably rapid and robust convergence to a unique solution, and (9) the computational cost of charge partitioning scales linearly with increasing system size. We study numerous materials as examples: (a) a series of endohedral C₆₀ complexes, (b) high-pressure compressed sodium chloride crystals with unusual stoichiometries, (c) metal–organic frameworks, (d) large and small molecules, (e) organometallic complexes, (f) various solids, and (g) solid surfaces. Due to non-nuclear attractors, Bader's quantum chemical topology could not assign NACs for some of these materials. We show for the first time that the Iterative Hirshfeld and DDEC3 methods do not always converge to a unique solution independent of the initial guess, and this sometimes causes those methods to assign dramatically different NACs on symmetry-equivalent atoms. By using a fixed number of charge partitioning steps with well-defined reference ion charges, the DDEC6 method avoids this problem by always converging to a unique solution. These characteristics make the DDEC6 method ideally suited for use as a default charge assignment method in quantum chemistry programs.

Net atomic charges (NACs) are widely used throughout the chemical sciences to concisely summarize key information about charge transfer between atoms in materials. The vast majority of NAC definitions proposed to date are unsuitable for describing the wide range of material types encountered across the chemical sciences. In this article, we show the DDEC6 method reproduces important chemical, theoretical, and experimental properties across an extremely broad range of material types including small and large molecules, organometallics, nanoclusters, porous solids, nonporous solids, and solid surfaces. Some important comparisons we make are: (a) correlations between various NAC models and spectroscopically measured core-electron binding energy shifts for Ti-, Fe-, and Mo-containing solids, (b) comparisons between DDEC6 and experimentally extracted NACs for formamide and natrolite, (c) comparisons of accuracy of different NAC methods for reproducing the electrostatic potential surrounding a material across one and multiple system conformations, (d) comparisons between calculated and chemically expected electron transfer trends for atoms in numerous dense solids, solid surfaces, and molecules, (e) an assessment of NAC transferability between three crystal phases of the diisopropylammonium bromide organic ferroelectric, and (f) comparisons between DDEC6 and polarized neutron diffraction atomic spin moments for the Mn₁₂-acetate single-molecule magnet. We find the DDEC6 NACs are ideally suited for constructing flexible force-fields and give reasonable agreement with force-fields commonly used to simulate biomolecules and water. We find the DDEC6 method is more accurate than the DDEC3 method for analyzing a broad range of materials. This broad applicability to periodic and non-periodic materials irrespective of the basis set type makes the DDEC6 method suited for use as a default atomic population analysis method in quantum chemistry programs.

Developing a comprehensive method to compute bond orders is a problem that has eluded chemists since Lewis's pioneering work on chemical bonding a century ago. Here, a computationally efficient method solving this problem is introduced and demonstrated for diverse materials including elements from each chemical group and period. The method is applied to non-magnetic, collinear magnetic, and non-collinear magnetic materials with localized or delocalized bonding electrons. Examples studied include the stretched O₂ molecule, 26 diatomic molecules, 3d and 5d transition metal solids, periodic materials with 1 to 8748 atoms per unit cell, a biomolecule, a hypercoordinate molecule, an electron deficient molecule, hydrogen bound systems, transition states, Lewis acid–base complexes, aromatic compounds, magnetic systems, ionic materials, dispersion bound systems, nanostructures, and other materials. From near-zero to high-order bonds were studied. Both the bond orders and the sum of bond orders for each atom are accurate across various bonding types: metallic, covalent, polar-covalent, ionic, aromatic, dative, hypercoordinate, electron deficient multi-centered, agostic, and hydrogen bonding. The method yields similar results for correlated wavefunction and density functional theory inputs and for different S_Z values of a spin multiplet. The method requires only the electron and spin magnetization density distributions as input and has a computational cost scaling linearly with increasing number of atoms in the unit cell. No prior approach is as general. The method does not apply to electrides, highly time-dependent states, some extremely high-energy excited states, and nuclear reactions.

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@C₆₀ endohedral fullerene complexes, a water dimer, a Mn₁₂-acetate single molecule magnet exhibiting collinear magnetism, a Fe₄O₁₂N₄C₄₀H₅₂ 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.