Voronoi deformation density (VDD) charges: Assessment of the Mulliken, Bader, Hirshfeld, Weinhold, and VDD methods for charge analysis

We present the Voronoi Deformation Density (VDD) method for computing atomic charges. The VDD method does not explicitly use the basis functions but calculates the amount of electronic density that flows to or from a certain atom due to bond formation by spatial integration of the deformation density over the atomic Voronoi cell. We compare our method to the well-known Mulliken, Hirshfeld, Bader, and Weinhold [Natural Population Analysis (NPA)] charges for a variety of biological, organic, and inorganic molecules. The Mulliken charges are (again) shown to be useless due to heavy basis set dependency, and the Bader charges (and often also the NPA charges) are not realistic, yielding too extreme values that suggest much ionic character even in the case of covalent bonds. The Hirshfeld and VDD charges, which prove to be numerically very similar, are to be recommended because they yield chemically meaningful charges. We stress the need to use spatial integration over an atomic domain to get rid of basis set dependency, and the need to integrate the deformation density in order to obtain a realistic picture of the charge rearrangement upon bonding. An asset of the VDD charges is the transparency of the approach owing to the simple geometric partitioning of space. The deformation density based charges prove to conform to chemical experience.     

Effect of charge inhomogeneity and mobility on colloid aggregation

The aggregation of inhomogeneously charged colloids with the same average charge is analyzed using Monte Carlo simulations. We find aggregation of colloids for sizes in the range 10-200 nm, which is similar to the range in which aggregation is observed in several experiments. The attraction arises from the strongly correlated electrostatic interactions associated with the increase in the counterion density in the region between the particles; this effect is enhanced by the discreteness and mobility of the surface charges. Larger colloids attract more strongly when their surface charges are discrete. We study the aggregation as functions of the surface charge density, counterion valence, and volume fraction.     

Fitting electron densities of molecules

Compact, convenient expressions for the electron density are derived using a fitting functional. Results are given for H₂O, CH₄, HF, NH₃, and PH₃, and show the accuracy obtainable using differing numbers of functions on each center. The definition of an atomic charge using these expressions is discussed and it is shown that the Bader virial-partitioning definition, in which the density is integrated over a volume around each nucleus, leads to convergent results.     

Quantitative comparison of molecular electrostatic potentials for structure-activity studies

An approach for representing, efficiently calculating and comparing discrete three-dimensional molecular electrostatic potentials using a quantitative similarity index (MEP-SI) based on a Carbo-type formalism is presented. A radial-type (MACRA) grid representation is described that provides more efficient storage of MEP information than a cubic grid of similar range, appropriate emphasis, and a convenient means for restricting the comparison of MEP functions to a local molecule region. The MACRA based MEP-SI formalism was used to evaluate the suitability of a variety of approximate methods for efficiently calculating the MEP for use in MEP-SI comparison of dissimilar molecules. The Mulliken charge method was found inadequate, while the method of potential-derived charges (PDCs), with additional charges for lone electron pairs included on sulfur, provided an efficient and sufficiently accurate representation of the MEP for this purpose. Convergence of the MEP-SI with respect to MACRA grid extent and mesh size was demonstrated; the effect of MEP error and different grid point emphasis in the MACRA versus the cubic grid results was investigated, and MEP-SI results were compared for different forms of the SI equation. The methodology proposed in this study provides an efficient and practical means for comparing MEP functions for two molecules and gives discriminating results for a sample series of molecular analogues consistent with expectations.     

Energy decompositions according to physical space partitioning schemes: treatments of the density cumulant

This article is a continuation of our previous paper on schemes of energy decompositions of molecular systems in the real space [D. R. Alcoba et al., J. Chem. Phys. 122, 074102 (2005)] now using correlated state functions. We study, according to physical arguments, the appropriate management of the density cumulant arising from the second-order reduced density matrix at correlated level, whose contributions can be assigned to one-center or to two-center terms in the energy partitioning. Our treatments are applied within two physical space partitioning schemes: the Bader partitioning into atomic basins and the fuzzy atom procedure. The results obtained in selected molecules are analyzed and discussed in detail.     

Efficient solution of Poisson's equation with free boundary conditions

Interpolating scaling functions give a faithful representation of a localized charge distribution by its values on a grid. For such charge distributions, using a fast Fourier method, we obtain highly accurate electrostatic potentials for free boundary conditions at the cost of O(N log N) operations, where N is the number of grid points. Thus, with our approach, free boundary conditions are treated as efficiently as the periodic conditions via plane wave methods.     

Explicit ions condensation around strongly charged polyelectrolytes and spherical macroions: the influence of salt concentration and chain linear charge density. Monte Carlo simulations

The condensation of monovalent counterions and trivalent salt particles around strong rigid and flexible polyelectrolyte chains as well as spherical macroions is investigated by Monte Carlo simulations. The results are compared with the condensation theory proposed by Manning. Considering flexible polyelectrolyte chains, the presence of trivalent salt is found to play an important role by promoting chain collapse. The attraction of counterions and salt particles near the polyelectrolyte chains is found to be strongly dependent on the chain linear charge density with a more important condensation at high values. When trivalent salt is added in a solution containing monovalent salt, the trivalent cations progressively replace the monovalent counterions. Ion condensation around flexible chains is also found to be more efficient compared with rigid rods due to monomer rearrangement around counterions and salt cations. In the case of spherical macroions, it is found that a fraction of their bare charge is neutralized by counterions and salt cations. The decrease of the Debye length, and thus the increase of salt concentration, promotes the attraction of counterions and salt particles at the macroion surface. Excluded volume effects are also found to significantly influence the condensation process, which is found to be more important by decreasing the ion size.     

C library for topological study of the electronic charge density

The topological study of the electronic charge density is useful to obtain information about the kinds of bonds (ionic or covalent) and the atom charges on a molecule or crystal. For this study, it is necessary to calculate, at every space point, the electronic density and its electronic density derivatives values up to second order. In this work, a grid‐based method for these calculations is described. The library, implemented for three dimensions, is based on a multidimensional Lagrange interpolation in a regular grid; by differentiating the resulting polynomial, the gradient vector, the Hessian matrix and the Laplacian formulas were obtained for every space point. More complex functions such as the Newton–Raphson method (to find the critical points, where the gradient is null) and the Cash–Karp Runge–Kutta method (used to make the gradient paths) were programmed. As in some crystals, the unit cell has angles different from 90°, the described library includes linear transformations to correct the gradient and Hessian when the grid is distorted (inclined). Functions were also developed to handle grid containing files (grd from DMol® program, CUBE from Gaussian® program and CHGCAR from VASP® program). Each one of these files contains the data for a molecular or crystal electronic property (such as charge density, spin density, electrostatic potential, and others) in a three‐dimensional (3D) grid. The library can be adapted to make the topological study in any regular 3D grid by modifying the code of these functions. © 2012 Wiley Periodicals, Inc.     

Periodic projector augmented wave density functional calculations on the hexachlorobenzene crystal and comparison with the experimental multipolar charge density model

The projector augmented wave (PAW) methodology has been used to calculate a high precision electron density distribution ρ(r) for the hexachlorobenzene crystal phase. Implementing the calculation of the crystallographic structure factors in the VASP code has permitted one to obtain the theoretical multipolar ρ(r). This electron density is compared with both the DFT electron density and the experimental multipolar model obtained from high-resolution X-ray diffraction data. This comparison has been carried out in intra- and intermolecular regions within the framework of the quantum theory of atoms-in-molecules (QTAIM) developed by Bader and co-workers. The characterization of the electron density in both C-Cl and Cl···Cl regions, as well as within the atomic basins, shows similar features for the three models. As a consequence, the observation of charge depletion and charge concentration regions around the halogen nuclei (along the C-Cl bonding axis and in the perpendicular plane, respectively) underlines the nature of halogen bonding in terms of electrophilic and nucleophilic interactions.     

DFT-B3LYP, NPA-, and QTAIM-based study of the physical properties of [M(II)(H2O)2(15-crown-5)] (M = Mn, Fe, Co, Ni, Cu, Zn) complexes

A density functional theory study of the structure of the title compounds with the divalent metal ions in their high-spin ground state, obtained using B3LYP/6-311++G(d,p) in vacuo and in aqueous solution simulated using a polarized continuum medium, is reported for the first time. The modeling reproduces the pseudo pentagonal bipyramidal crystallographic structures very well, including some asymmetry in the equatorial bonds lengths to the crown ether O donors. The very marked asymmetry in the Ni(2+) structure due to a Jahn-Teller distortion of a d(8) system in a D(5h) ligand field is also well reproduced. The gas phase binding energies of the complexes follow the order Mn(2+) < Fe(2+) < Co(2+) < Ni(2+) < Cu(2+) > Zn(2+), in precise agreement with the Irving-William series. Both the NPA and Bader charges show there is ligand-to-metal charge transfer; however, the values obtained from the NPA procedure, unlike those obtained from Bader's quantum theory of molecules approach, do not correlate with the electronegativity of the metal ions, the stabilization energies of the solvated complexes or the ionic radii of the metal ions, and so appear to be less reliable. The nature of the bonding between the ligands and the metal ions has been explored using the topological properties of the electron charge density. The metal-ligand bond distances were found to be exponentially correlated with the electron charge density, its Laplacian, and with its curvature in the direction of the bond path at M-O bond critical points. While the bonding with coordinated H(2)O is predominantly ionic, that to the crown ether donor atoms has some covalent character the extent of which increases across the first transition series. The delocalization indices of M-O bonds in these complexes correlate reasonably well with the electron density and its Laplacian at the bond critical points; this therefore provides a rapid and computationally very efficient way of determining these properties, from which insight into the nature of the bonding can be obtained, obviating the need for time-consuming integration over atomic basins.     

Size ratio effects on interparticle interactions and phase behavior of microsphere-nanoparticle mixtures

We investigate the interparticle interactions and phase behavior of microsphere-nanoparticle mixtures of high charge asymmetry and varying size ratio. In the absence of nanoparticles, negligibly charged microspheres flocculate as a result of van der Waals interactions. Upon addition of a lower critical nanoparticle volume fraction, the microspheres are stabilized by the formation of nanoparticle halos around each microsphere. , A weak attraction between the two species leads to a pronounced enhancement of the effective nanoparticle concentration near the microsphere surface relative to the bulk solution. Above an upper critical nanoparticle volume fraction, the microspheres undergo reentrant gelation. Binary mixtures, in which the effective nanoparticle size is reduced at a fixed microsphere diameter, exhibit a narrow window of stability that ultimately disappears with increasing ionic strength. By contrast, binary mixtures of varying microsphere diameter are stabilized at similar nanoparticle volume fractions and exhibit a broader window of stability with decreasing size ratio. This unexpected observation may arise from the reduced attraction between smaller microspheres because negligible differences in nanoparticle halo formation are observed in these mixtures.     

Molecular density functional theory of solvation: from polar solvents to water

A classical density functional theory approach to solvation in molecular solvent is presented. The solvation properties of an arbitrary solute in a given solvent, both described by a molecular force field, can be obtained by minimization of a position and orientation-dependent free-energy density functional. In the homogeneous reference fluid approximation, limited to two-body correlations, the unknown excess term of the functional approximated by the angular-dependent direct correlation function of the pure solvent. We show that this function can be extracted from a preliminary MD simulation of the pure solvent by computing the angular-dependent pair distribution function and solving subsequently the molecular Ornstein-Zernike equation using a discrete angular representation. The corresponding functional can then be minimized in the presence of an arbitrary solute on a three-dimensional cubic grid for positions and Gauss-Legendre angular grid for orientations to provide the solvation structure and free-energy. This two-step procedure is proved to be much more efficient than direct molecular dynamics simulations combined to thermodynamic integration schemes. The approach is shown to be relevant and accurate for prototype polar solvents such as the Stockmayer solvent or acetonitrile. For water, although correct for neutral or moderately charged solute, it tends to underestimate the tetrahedral solvation structure around H-bonded solutes, such as spherical ions. This can be corrected by introducing suitable three-body correlation terms that restore both an accurate hydration structure and a satisfactory energetics.     

Resonance structures of the amide bond: the advantages of planarity

Delocalization indexes based on magnitudes derived from electron-pair densities are demonstrated to be useful indicators of electron resonance in amides. These indexes, based on the integration of the two-electron density matrix over the atomic basins defined through the zero-flux condition, have been calculated for a series of amides at the B3LYP/6-31+G* level of theory. These quantities, which can be viewed as a measure of the sharing of electrons between atoms, behave in concordance with the traditional resonance model, even though they are integrated in Bader atomic basins. Thus, the use of these quantities overcomes contradictory results from analyses of atomic charges, yet keeps the theoretical appeal of using nonarbitrary atomic partitions and unambiguously defined functions such as densities and pair densities. Moreover, for a large data set consisting of 24 amides plus their corresponding rotational transition states, a linear relation was found between the rotational barrier for the amide and the delocalization index between the nitrogen and oxygen atoms, indicating that this parameter can be used as an ideal physical-chemical indicator of the electron resonance in amides.     

Equilibrium thermodynamics from basin-sampling

We present a "basin-sampling" approach for calculation of the potential energy density of states for classical statistical models. It combines a Wang-Landau-type uniform sampling of local minima and a novel approach for approximating the relative contributions from local minima in terms of the volumes of basins of attraction. We have employed basin-sampling to study phase changes in atomic clusters modeled by the Lennard-Jones potential and for ionic clusters. The approach proves to be efficient for systems involving broken ergodicity and has allowed us to calculate converged heat capacity curves for systems that could previously only be treated using the harmonic superposition approximation. Benchmarks are also provided by comparison with parallel tempering and Wang-Landau simulations, where these proved feasible.     

Recent study on counterion-mediated attraction between colloidal particles

Recent experimental results were reviewed. The 1D- and 2D-USAXS studies gave higher orders of Bragg diffraction for single crystals of colloidal silica particles, allowing more accurate determinations of the lattice constant, lattice symmetry, and direction. The closest interparticle spacing thus determined was confirmed to be smaller than the average spacing. The most closely packed planes ((110) planes for bcc) of negatively charged particles were found to be parallel to the likewise negatively charged capillary surface, inconsistently with the accepted double layer interaction theory but consistently with a recent experimental finding of positive adsorption. Shaking caused disruption of the single crystals but newly formed microcrystals retained the lattice constant and the preference of the (110) planes. The liquid-solid-liquid transition, a re-entrant phase transition, was found for silica particles and latex particles at given particle volume fraction and salt concentration, when the charge density of particles was varied. It was demonstrated that the purely repulsive Yukawa potential and the concept of renormalized charge cannot account for the re-entrant behavior. The Monte-Carlo simulation using the Sogami potential, which contains short-range repulsion and long-range attraction, was found to account for the fcc–bcc transition, which was earlier claimed to be explainable only by the Yukawa potential. Furthermore, the homogeneous-inhomogeneous phase transition and void formation could be accounted for by the simulation using the Sogami potential; the Yukawa potential could not reproduce the experiments. Attention was drawn to the experimental conditions in direct measurements of interparticle forces; only short interparticle distance and low charge density particles were covered, which make it practically impossible to detect the long-range counterion-mediated attraction. It is hoped that, by technical improvements, these shortcomings may be made up and quantitative argument become possible on the attraction.     

Isotropic periodic sum: a method for the calculation of long-range interactions

This work presents an accurate and efficient approach to the calculation of long-range interactions for molecular modeling and simulation. This method defines a local region for each particle and describes the remaining region as images of the local region statistically distributed in an isotropic and periodic way, which we call isotropic periodic images. Different from lattice sum methods that sum over discrete lattice images generated by periodic boundary conditions, this method sums over the isotropic periodic images to calculate long-range interactions, and is referred to as the isotropic periodic sum (IPS) method. The IPS method is not a lattice sum method and eliminates the need for a reciprocal space sum. Several analytic solutions of IPS for commonly used potentials are presented. It is demonstrated that the IPS method produces results very similar to that of Ewald summation, but with three major advantages, (1) it eliminates unwanted symmetry artifacts raised from periodic boundary conditions, (2) it can be applied to potentials of any functional form and to fully and partially homogenous systems as well as finite systems, and (3) it is more computationally efficient and can be easily parallelized for multiprocessor computers. Therefore, this method provides a general approach to an efficient calculation of long-range interactions for various kinds of molecular systems.     

Efficient and accurate three-dimensional Poisson solver for surface problems

We present a method that gives highly accurate electrostatic potentials for systems where we have periodic boundary conditions in two spatial directions but free boundary conditions in the third direction. These boundary conditions are needed for all kinds of surface problems. Our method has an O(N log N) computational cost, where N is the number of grid points, with a very small prefactor. This Poisson solver is primarily intended for real space methods where the charge density and the potential are given on a uniform grid.     

On the connections between the quantum theory of atoms in molecules (QTAIM) and density functional theory (DFT): a letter from Richard F. W. Bader to Lou Massa

A 31-year-old letter from Professor Richard F. W. Bader to Professor Lou Massa outlining the connections between the quantum theory of atoms in molecules (QTAIM) and density functional theory (DFT) especially with regard to the first Hohenberg-Kohn theorem is brought to light. This connection has not often been the topic of such a focused review by Bader and is presented here for the first time. The scientific importance of this letter is, in the opinion of the presenter, as timely today as it was back then in 1986. In Bader’s own opening words: “... that if I sent you a summary of what I think are the important connections between our work and density functional theory, ...”. He then takes us in a grand tour of the foundations of QTAIM culminating into the antecedents of a paper he later published with Professor Pierre Becker, whereby the Hohenberg-Kohn theorem is shown to operate at the level of an atom-in-a-molecule. Bader closes his letter by suggesting to Massa: “Study these two charge distributions – they are proof of the theorem of Hohenberg and Kohn”. By that Bader meant that when the charge distributions of two atoms or groups are identical within a given precision, then the kinetic and total energy contributions of these atoms to the corresponding molecular quantities are also identical. It is revealing to follow the intellectual threads weaved by Bader which provides us with a glimpse of his thought processes and intuition that guided him to some of his key discoveries. The lucidity, rigor, and clarity characteristic of Bader and the informality of style of a letter makes it of pedagogic and historic interest.

     

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.