Search

Grid is a free and open-source Python library for constructing numerical grids to integrate, interpolate, and differentiate functions (e.g., molecular properties), with a strong emphasis on facilitating these operations in computational chemistry and conceptual density functional theory. Although designed, maintained, and released as a stand-alone Python library, Grid was originally developed for molecular integration, interpolation, and solving the Poisson equation in the HORTON and ChemTools packages. Grid is designed to be easy to use, extend, and maintain; this is why we use Python and adopt many principles of modern software development, including comprehensive documentation, extensive testing, continuous integration/delivery protocols, and package management. We leverage popular scientific packages, such as NumPy and SciPy, to ensure high efficiency and optimized performance in grid development. This article is the official release note of the Grid library showcasing its unique functionality and scope. [ABSTRACT FROM AUTHOR]

HORTON is a free and open-source electronic-structure package written primarily in Python 3 with some underlying C++ components. While HORTON's development has been mainly directed by the research interests of its leading contributing groups, it is designed to be easily modified, extended, and used by other developers of quantum chemistry methods or post-processing techniques. Most importantly, HORTON adheres to modern principles of software development, including modularity, readability, flexibility, comprehensive documentation, automatic testing, version control, and quality-assurance protocols. This article explains how the principles and structure of HORTON have evolved since we started developing it more than a decade ago. We review the features and functionality of the latest HORTON release (version 2.3) and discuss how HORTON is evolving to support electronic structure theory research for the next decade. [ABSTRACT FROM AUTHOR]

We develop a variational procedure for the iterative Hirshfeld (HI) partitioning scheme. The main practical advantage of having a variational framework is that it provides a formal and straightforward approach for imposing constraints (e.g., fixed charges on certain atoms or molecular fragments) when computing HI atoms and their properties. Unlike many other variants of the Hirshfeld partitioning scheme, HI charges do not arise naturally from the information-theoretic framework, but only as a reverse-engineered construction of the objective function. However, the procedure we use is quite general and could be applied to other problems as well. We also prove that there is always at least one solution to the HI equations, but we could not prove that its self-consistent equations would always converge for any given initial pro-atom charges. Our numerical assessment of the constrained iterative Hirshfeld method shows that it satisfies many desirable traits of atoms in molecules and has the potential to surpass existing approaches for adding constraints when computing atomic properties. [ABSTRACT FROM AUTHOR]

The numerical ill‐conditioning associated with approximating an electron density with a convex sum of Gaussian or Slater‐type functions is overcome by using the (extended) Kullback–Leibler divergence to measure the deviation between the target and approximate density. The optimized densities are non‐negative and normalized, and they are accurate enough to be used in applications related to molecular similarity, the topology of the electron density, and numerical molecular integration. This robust, efficient, and general approach can be used to fit any non‐negative normalized functions (e.g., the kinetic energy density and molecular electron density) to a convex sum of non‐negative basis functions. We present a fixed‐point iteration method for optimizing the Kullback–Leibler divergence and compare it to conventional gradient‐based optimization methods. These algorithms are released through the free and open‐source BFit package, which also includes a L2‐norm squared optimization routine applicable to any square‐integrable scalar function. [ABSTRACT FROM AUTHOR]

Fanpy is a free and open‐source Python library for developing and testing multideterminant wavefunctions and related ab initio methods in electronic structure theory. The main use of Fanpy is to quickly prototype new methods by making it easier to convert the mathematical formulation of a new wavefunction ansätze to a working implementation. Fanpy is designed based on our recently introduced Flexible Ansatz for N‐electron Configuration Interaction (FANCI) framework, where multideterminant wavefunctions are represented by their overlaps with Slater determinants of orthonormal spin‐orbitals. In the simplest case, a new wavefunction ansatz can be implemented by simply writing a function for evaluating its overlap with an arbitrary Slater determinant. Fanpy is modular in both implementation and theory: the wavefunction model, the system's Hamiltonian, and the choice of objective function are all independent modules. This modular structure makes it easy for users to mix and match different methods and for developers to quickly explore new ideas. Fanpy is written purely in Python with standard dependencies, making it accessible for various operating systems. In addition, it adheres to principles of modern software development, including comprehensive documentation, extensive testing, quality assurance, and continuous integration and delivery protocols. This article is considered to be the official release notes for the Fanpy library. [ABSTRACT FROM AUTHOR]

Using results from atomic spectroscopy, we show that there are two types of flat-planes conditions. The first type of flat-planes condition occurs when the energy as a function of the number of electrons of each spin, Nα and Nβ, has a derivative discontinuity on a line segment where the number of electrons, Nα + Nβ, is an integer. The second type of flat-planes condition occurs when the energy has a derivative discontinuity on a line segment where the spin polarization, Nα - Nβ, is an integer, but does not have a discontinuity associated with an integer number of electrons. Type 2 flat planes are rare-we observed just 15 type 2 flat-planes conditions out of the 4884 cases we tested-but their mere existence has implications for the design of exchange-correlation energy density functionals. To facilitate the development of functionals that have the correct behavior with respect to both fractional number of electrons and fractional spin polarization, we present a dataset for the chromium atom and its ions that can be used to test new functionals. [ABSTRACT FROM AUTHOR]

One can partition the molecular density into its atomic contributions by minimizing the divergence of the atom-in-molecule densities from their corresponding reference pro-atomic densities, subject to the constraint that the sum of the atom-in-molecule densities is the total molecular density. We expose conditions on the divergence measure that are necessary, and sufficient, to recover the popular Hirshfeld partitioning. Specifically, among all local measures of the divergence between two probability distribution functions, the Hirshfeld partitioning is obtained only for f-divergences. [ABSTRACT FROM AUTHOR]

IOData is a free and open‐source Python library for parsing, storing, and converting various file formats commonly used by quantum chemistry, molecular dynamics, and plane‐wave density‐functional‐theory software programs. In addition, IOData supports a flexible framework for generating input files for various software packages. While designed and released for stand‐alone use, its original purpose was to facilitate the interoperability of various modules in the HORTON and ChemTools software packages with external (third‐party) molecular quantum chemistry and solid‐state density‐functional‐theory packages. IOData is designed to be easy to use, maintain, and extend; this is why we wrote IOData in Python and adopted many principles of modern software development, including comprehensive documentation, extensive testing, continuous integration/delivery protocols, and package management. This article is the official release note of the IOData library. [ABSTRACT FROM AUTHOR]

We discuss an extension of the venerable Parr‐Pearson parabolic charge transfer model that explicitly includes 3rd order effects. We then use this new model to provide new arguments supporting the validity of Parr and Yang's "|Δμ| big is good" (DMB) principle. Additionally, we show how to re‐derive this principle from the Parr‐Pearson parabolic model, using simpler arguments than those presented in our previous proof of the DMB rule. [ABSTRACT FROM AUTHOR]

The paper collects the answers of the authors to the following questions:Is the lack of precision in the definition of many chemical concepts one of the reasons for the coexistence of many partition schemes?Does the adoption of a given partition scheme imply a set of more precise definitions of the underlying chemical concepts?How can one use the results of a partition scheme to improve the clarity of definitions of concepts?Are partition schemes subject to scientific Darwinism? If so, what is the influence of a community's sociological pressure in the "natural selection" process?To what extent does/can/should investigated systems influence the choice of a particular partition scheme?Do we need more focused chemical validation of Energy Decomposition Analysis (EDA) methodology and descriptors/terms in general?Is there any interest in developing common benchmarks and test sets for cross‐validation of methods?Is it possible to contemplate a unified partition scheme (let us call it the "standard model" of partitioning), that is proper for all applications in chemistry, in the foreseeable future or even in principle?In the end, science is about experiments and the real world. Can one, therefore, use any experiment or experimental data be used to favor one partition scheme over another? © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

The atoms in molecules associated with the Hirshfeld partitioning minimize the generalized Hellinger-Bhattacharya distance to the reference pro-atom densities. Moreover, the reference proatoms can be chosen by minimizing the distance between the pro-molecule density and the true molecular density. This provides an alternative to both the heuristic "stockholder" and the mathematical information-theoretic interpretations of the Hirshfeld partitioning. These results extend to any member of the family of f-divergences. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • If the wavefunction is sufficiently differentiable, it can be approximated with polynomial cost in the high-accuracy limit. • A strategy for picking configurations is proposed and used to define a selected configuration interaction method called GKCI. • Griebel-Knapek configuration interaction (GKCI) is appropriate for wavefunctions with dominating mixed smoothness. • Errors due to truncating the Slater determinant expansion and the one-electron basis set should be balanced. • GKCI gives promising results when tested for small systems. Abstract Most approaches for solving the electronic Schrödinger equation do not fully exploit the functional-analytic simplicity of the electronic wavefunction. Because of this, the cost of these methods explodes exponentially with increasing electron number, an effect that is often called the curse of dimension. Recent work in mathematics and computer science shows how, by exploiting the smoothness of molecular wavefunctions, one can design methods that achieve the same accuracy as full configuration interaction, but with polynomial cost. The mathematical background of this approach is presented, along with a detailed prescription for identifying the relevant Slater determinants and a demonstration of the method's effectiveness for atomic systems. In the basis-set limit, this truncated configuration interaction method is size-consistent. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • An axiomatic approach for defining and assessing chemical concepts is proposed. • Concepts should be mathematically formulated and assessed against specified axioms. • Ideally, concepts should be defined using physical observables. • The best concept for a given objective is the one whose axioms are most appropriate. Abstract Many concepts that are central to chemical language and thought emerge from the wealth of chemists' historical experience and cannot be precisely defined mathematically from the underlying physics. In such cases, it is useful to take an axiomatic approach: list the chemical, mathematical and computational properties that one desires for a concept to possess, and then find the rigorous (and, if possible, elegant) mathematical formulation of the concept that satisfies those desiderata. This mathematical formulation is most useful if it relies on fundamental quantities—quantum-mechanical observables, reduced density matrices, or the N -electron wavefunction—rather than method-dependent quantities (e.g., orbitals) that are not defined for some computational approaches to the molecular electronic structure problem. This ensures that the pursuit of chemical intuition does not lead one too far from the underlying physics. It also ensures that one can interpret the results of any computational method, even methods (e.g., quantum Monte Carlo) that make no reference to any molecular-orbital or valence-bond model. [ABSTRACT FROM AUTHOR]

We argue that when one divides a molecular property into atom‐in‐a‐molecule contributions, one should perform the division based on the property density of the quantity being partitioned. This is opposition to the normal approach, where the electron density is given a privileged role in defining the properties of atoms‐in‐a‐molecule. Because partitioning each molecular property based on its own property density is inconvenient, we design a reference‐free approach that does not (directly) refer atomic property densities. Specifically, we propose a stockholder partitioning method based on relative influence of a molecule's atomic nuclei on the electrons at a given point in space. The resulting method does not depend on an “arbitrary” choice of reference atoms and it has some favorable properties, including the fact that all of the electron density at an atomic nucleus is assigned to that nucleus and the fact all the atoms in a molecule decay at a uniform asymptotic rate. Unfortunately, the resulting model is not easily applied to spatially degenerate ground states. Furthermore, the practical realizations of this strategy that we tried here gave disappointing numerical results. © 2017 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Many population analysis methods are based on the precept that molecules should be built from fragments (typically atoms) that maximally resemble the isolated fragment. The resulting molecular building blocks are intuitive (because they maximally resemble well-understood systems) and transferable (because if two molecular fragments both resemble an isolated fragment, they necessarily resemble each other). Information theory is one way to measure the deviation between molecular fragments and their isolated counterparts, and it is a way that lends itself to interpretation. For example, one can analyze the relative importance of electron transfer and polarization of the fragments. We present key features, advantages, and disadvantages of the information-theoretic approach. We also codify existing information-theoretic partitioning methods in a way that clarifies the enormous freedom one has within the information-theoretic ansatz. [ABSTRACT FROM AUTHOR]

Atoms in molecules methods that rely on reference promolecular densities typically require that one define, or otherwise determine, the densities of unbound atomic anions. Whereas the isolated atomic polyanions are always physically and computationally unbound, monoanions can be either physically bound but computationally unbound (like the oxygen anion at the Hartree-Fock level of theory), or physically unbound but computationally bound (like the nitrogen anion using many DFT methods with a basis set including diffuse functions). Depending on the level of theory and basis set used, the densities of negatively charged atomic ions can decay very slowly and even be nonmonotonically decreasing. These delocalized anionic densities induce ill-behaved atomic properties for compounds containing highly reduced atoms. To treat the problem of unphysical proatom densities in iterative Hirshfeld methods, we compute the smallest (typically fractional) nuclear charge to bind all electrons, called the effective nuclear charge $$ {Z}_{\mathrm{A}}^{\mathrm{eff}} $$ of an atom A. When $$ {Z}_{\mathrm{A}}^{\mathrm{eff}}>{Z}_{\mathrm{A}} $$ at a given level of theory, the scaled density corresponding to the effective nuclear charge is used as the negatively charged proatom density. This novel approach dramatically improves the computational robustness of the iterative Hirshfeld partitioning scheme. [ABSTRACT FROM AUTHOR]

We discuss some strategies for extending recent geminal-based methods to open-shells by replacing the geminal-creation operators with more general composite boson creation operators, and even creation operators that mix fermionic and bosonic components. We also discuss the utility of symmetry-breaking and restoration, but using a projective (not a variational) approach. Both strategies—either together or separately—give a pathway for extending geminals-based methods to open shells, while retaining the computational efficiency and conceptual simplicity of existing geminal product wavefunctions. [ABSTRACT FROM AUTHOR]

Making use of the grand canonical ensemble the derivation of the analytical equations for the chemical potential and the Fukui function in the general case of any number of ground and excited states is presented. The expressions thus obtained allow one to establish that the ensemble of three consecutive ground states that has been usually used to analyze the effects of temperature in these quantities provides a satisfactory description for them at temperatures of chemical interest. Nevertheless, some situations must be considered cautiously, as for example, when the N + k and N + k + 1 (N is the electron number) ground states are (nearly) quasidegenerate or when the first excited state of both the anion and the cation (with respect to the reference state) is very low in energy. Results for the copper atom (with the ground state of Cu+ as the reference state), using some selected ensemble models constituted by several ground and excited states, are presented to show that the very low-lying excited states of some of the copper species are able to contribute to chemical reactivity at relatively low temperatures (∼2000 K). A relevant aspect is that due to its generality, the present approach provides a new way to study the reactivity of the chemical species under extreme conditions. [ABSTRACT FROM AUTHOR]

We present benchmark values for the electronic chemical potential and chemical hardness from reference data for ionization potentials and electron affinities. In cases where the energies needed to compute these quantities are not available, we estimate the ionization potential of the metastable (di)anions by extrapolation along the isoelectronic series, taking care to ensure that the extrapolated data satisfy reasonable intuitive rules to the maximum possible extent. We also propose suitable values for the chemical potential and chemical hardness of zero-electron species. Because the values we report are faithful to the trends in accurate data on atomic energies, we believe that our proposed values for the chemical potential and chemical hardness are ideally suited to conceptual studies of chemical trends across the periodic table. The critical nuclear charge (Z critical) of the isoelectronic series with 2 ＜ N ＜ 96 has also been reported for the first time. [ABSTRACT FROM AUTHOR]

We present explicit formulas for arbitrary-order derivatives of the energy, grand potential, electron density, and higher-order response functions with respect to the number of electrons, and the chemical potential for any smooth and differentiable model of the energy versus the number of electrons. The resulting expressions for global reactivity descriptors (hyperhardnesses and hypersoftnesses), local reactivity descriptors (hyperFukui functions and local hypersoftnesses), and nonlocal response functions are easy to evaluate computationally. Specifically, the explicit formulas for global/local/nonlocal hypersoftnesses of arbitrary order are derived using Bell polynomials. Explicit expressions for global and local hypersoftness indicators up to fifth order are presented. [ABSTRACT FROM AUTHOR]

The ambiguity of the local hardness is resolved by using information theory to select definitions of the local hardness that are as close as possible to a well-defined approximate formula for the local hardness. A condensed local hardness is derived by using the atomic hardnesses as a reference distribution; a pointwise local hardness is derived by using the uniform electron gas as a reference distribution. This information-theoretic condensed local hardness is tested by examining electrophilic attack on some substituted pyridines. [ABSTRACT FROM AUTHOR]

This contribution deals with the subsystem variational procedure within the context of the quantum theory of atoms in positronic molecules (QTAIPM). Before introducing the subsystem energy functional termed as joint subsystem energy functional, a novel notation and the combination strategy are disclosed in detail by restating the positronic subsystem hypervirial theorem. They are employed in proposing the proper subsystem energy functional, the validity of which is checked by various criteria. The zero flux surfaces of the joint density distribution are used to define the topological atoms in the positronic molecules, and they are incorporated into the subsystem variational procedure as proper real space boundary conditions. The variational procedure finally yields the flux of the joint current property density that also appears in the positronic subsystem hypervirial theorem. At every stage, the corresponding equations for the purely electronic systems within the context of the quantum theory of atoms in molecules (QTAIM) are presented to clearly reveal the analogy between these two formalisms and to emphasize the importance of combining the property density distributions in the QTAIPM. The presented material demonstrates the internal consistency of the whole framework and discloses the fact that the QTAIM must be regarded as a variant of the QTAIPM. Furthermore, this formalism promises an extended QTAIM, which is hoped to resolve the issue of molecular structure beyond the clamp nuclei approximation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011 [ABSTRACT FROM AUTHOR]

We present a new, nonarbitrary, internally consistent, and unambiguous framework for spin-polarized conceptual density-functional theory (SP-DFT). We explicitly characterize the convex hull of energy, as a function of the number of electrons and their spin, as the only accessible ground states in spin-polarized density functional theory. Then, we construct continuous linear and quadratic models for the energy. The nondifferentiable linear model exactly captures the simplicial geometry of the complex hull about the point of interest and gives exact representations for the conceptual DFT reactivity indicators. The continuous quadratic energy model is the paraboloid of maximum curvature, which most tightly encloses the point of interest and neighboring vertices. The quadratic model is invariant to the choice of coordinate system (i.e., { N , S } vs { N α , N β }) and reduces to a sensible formulation of spin-free conceptual DFT in the appropriate limit. Using the quadratic model, we generalize the Parr functions { P + ( r ), P - ( r )} (and their derivatives with respect to number of electrons) to this new spin-polarized framework, integrating the Parr function concept into the context of (spin-polarized) conceptual DFT, and extending it to include higher-order effects.

Accurate potential energy models of proteins must describe the many different types of noncovalent interactions that contribute to a protein's stability and structure. Pi-pi contacts are ubiquitous structural motifs in all proteins, occurring between aromatic and nonaromatic residues and play a nontrivial role in protein folding and in the formation of biomolecular condensates. Guided by a geometric criterion for isolating pi-pi contacts from classical molecular dynamics simulations of proteins, we use quantum mechanical energy decomposition analysis to determine the molecular interactions that stabilize different pi-pi contact motifs. We find that neutral pi-pi interactions in proteins are dominated by Pauli repulsion and London dispersion rather than repulsive quadrupole electrostatics, which is central to the textbook Hunter-Sanders model. This results in a notable lack of variability in the interaction profiles of neutral pi-pi contacts even with extreme changes in the dielectric medium, explaining the prevalence of pi-stacked arrangements in and between proteins. We also find interactions involving pi-containing anions and cations to be extremely malleable, interacting like neutral pi-pi contacts in polar media and like typical ion-pi interactions in nonpolar environments. Like-charged pairs such as arginine-arginine contacts are particularly sensitive to the polarity of their immediate surroundings and exhibit canonical pi-pi stacking behavior only if the interaction is mediated by environmental effects, such as aqueous solvation.

In the first paper of this series, the authors derived an expression for the interaction energy between two reagents in terms of the chemical reactivity indicators that can be derived from density functional perturbation theory. While negative interaction energies can explain reactivity, reactivity is often more simply explained using the "|dμ| big is good" rule or the maximum hardness principle. Expressions for the change in chemical potential ( μ ) and hardness when two reagents interact are derived. A partial justification for the maximum hardness principle is that the terms that appear in the interaction energy expression often reappear in the expression for the interaction hardness, but with opposite sign., Competing Interests: The reviewer (WT) declared a past co-authorship with the author (CC) to the handling editor., (Copyright © 2022 Miranda-Quintana, Heidar-Zadeh, Fias, Chapman, Liu, Morell, Gómez, Cárdenas and Ayers.)

Reactivity descriptors indicate where a reagent is most reactive and how it is most likely to react. However, a reaction will only occur when the reagent encounters a suitable reaction partner. Determining whether a pair of reagents is well-matched requires developing reactivity rules that depend on both reagents. This can be achieved using the expression for the minimum-interaction-energy obtained from the density functional reactivity theory. Different terms in this expression will be dominant in different circumstances; depending on which terms control the reactivity, different reactivity indicators will be preferred., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2022 Miranda-Quintana, Heidar-Zadeh, Fias, Chapman, Liu, Morell, Gómez, Cárdenas and Ayers.)

We report a new deep learning message passing network that takes inspiration from Newton's equations of motion to learn interatomic potentials and forces. With the advantage of directional information from trainable force vectors, and physics-infused operators that are inspired by Newtonian physics, the entire model remains rotationally equivariant, and many-body interactions are inferred by more interpretable physical features. We test NewtonNet on the prediction of several reactive and non-reactive high quality ab initio data sets including single small molecules, a large set of chemically diverse molecules, and methane and hydrogen combustion reactions, achieving state-of-the-art test performance on energies and forces with far greater data and computational efficiency than other deep learning models., Competing Interests: The authors declare no competing interests., (This journal is © The Royal Society of Chemistry.)

The electrostatic potential (ESP) is a powerful property for understanding and predicting electrostatic charge distributions that drive interactions between molecules. In this study, we compare various charge partitioning schemes including fitted charges, density-based quantum mechanical (QM) partitioning schemes, charge equilibration methods, and our recently introduced coarse-grained electron model, C-GeM, to describe the ESP for protein systems. When benchmarked against high quality density functional theory calculations of the ESP for tripeptides and the crambin protein, we find that the C-GeM model is of comparable accuracy to ab initio charge partitioning methods, but with orders of magnitude improvement in computational efficiency since it does not require either the electron density or the electrostatic potential as input.

Thirty years ago, Parr and Yang postulated that favorable chemical processes are associated with large changes in the electronic chemical potential or, equivalently, the electronegativity. They called this the "|Δμ| big is good" rule and noted that if the rule could be justified, then it "would constitute a validation of frontier theory from first principles." We provide a simple and insightful justification for the "|Δμ| big is good" rule, with special emphasis on electron-transfer reactions. Furthermore, we show that it implies Pearson's hard/soft acid/base principle mathematically and confirm this result with numerical examples. This supports Parr's intuition that many other reactivity precepts arise as corollaries to the more fundamental "|Δμ| big is good" rule. In all of this, it is essential to consider the intensive nature of the chemical potential.

When one defines the energy of a molecule with a noninteger number of electrons by interpolation of the energy values for integer-charged states, the interpolated electron density, Fukui function, and higher-order derivatives of the density are generally not normalized correctly. The necessary and sufficient condition for consistent energy interpolation models is that the corresponding interpolated electron density is correctly normalized to the number of electrons. A necessary, but not sufficient, condition for correct normalization is that the energy interpolant be a linear function of the reference energies. Consistent with this general rule, polynomial interpolation models and, in particular, the quadratic E vs N model popularized by Parr and Pearson, do give normalized densities and density derivatives. Interestingly, an interpolation model based on the square root of the electron number also satisfies the normalization constraints. We also derive consistent least-norm interpolation models. In contrast to these models, the popular rational and exponential forms for E vs N do not give normalized electron densities and density derivatives.

Atomic partial charges appear in the Coulomb term of many force-field models and can be derived from electronic structure calculations with a myriad of atoms-in-molecules (AIM) methods. More advanced models have also been proposed, using the distributed nature of the electron cloud and atomic multipoles. In this work, an electrostatic force field is defined through a concise approximation of the electron density, for which the Coulomb interaction is trivially evaluated. This approximate "pro-density" is expanded in a minimal basis of atom-centered s-type Slater density functions, whose parameters are optimized by minimizing the Kullback-Leibler divergence of the pro-density from a reference electron density, e.g., obtained from an electronic structure calculation. The proposed method, Minimal Basis Iterative Stockholder (MBIS), is a variant of the Hirshfeld AIM method, but it can also be used as a density-fitting technique. An iterative algorithm to refine the pro-density is easily implemented with a linear-scaling computational cost, enabling applications to supramolecular systems. The benefits of the MBIS method are demonstrated with systematic applications to molecular databases and extended models of condensed phases. A comparison to 14 other AIM methods shows its effectiveness when modeling electrostatic interactions. MBIS is also suitable for rescaling atomic polarizabilities in the Tkatchenko-Scheffler scheme for dispersion interactions.

Using results from atomic spectroscopy, we show that there are two types of flat-planes conditions. The first type of flat-planes condition occurs when the energy as a function of the number of electrons of each spin, Nα and Nβ, has a derivative discontinuity on a line segment where the number of electrons, Nα + Nβ, is an integer. The second type of flat-planes condition occurs when the energy has a derivative discontinuity on a line segment where the spin polarization, Nα - Nβ, is an integer, but does not have a discontinuity associated with an integer number of electrons. Type 2 flat planes are rare-we observed just 15 type 2 flat-planes conditions out of the 4884 cases we tested-but their mere existence has implications for the design of exchange-correlation energy density functionals. To facilitate the development of functionals that have the correct behavior with respect to both fractional number of electrons and fractional spin polarization, we present a dataset for the chromium atom and its ions that can be used to test new functionals.