Coulomb potentials in two and three dimensions under periodic boundary conditions

A method to sum over logarithmic potential in two-dimensions (2D) and Coulomb potential in three dimensions (3D) with periodic boundary conditions in all directions is given. We consider the most general form of unit cells, the rhombic cell in 2D and the triclinic cell in 3D. For the 3D case, this paper presents a generalization of Sperb's work [R. Sperb, Mol. Simulation 22, 199 (1999)]. The expressions derived in this work converge extremely fast in all region of the simulation cell. We also obtain results for slab geometry. Furthermore, self-energies for both 2D as well as 3D cases are derived. Our general formulas can be employed to obtain Madelung constants for periodic structures.     

Electrostatic potentials in periodic boundary conditions

We discuss the effects of using periodic boundary conditions on the statistical mechanics of particles with electrostatic interactions. For the system of particles with electrostatic interactions plus periodic boundary conditions we develop an effective interaction which may be used with ordinary boundary conditions. The lattice sums which arise from the periodic boundary conditions contain divergences. We use a variant of Ewald's method to evaluate the sums which makes the handling of the divergences fairly simple. We discuss the effect of the effective interaction on the thermodynamics of the system.     

Analytical Hartree–Fock gradients with respect to the cell parameter: systems periodic in one and two dimensions

Analytical Hartree–Fock gradients with respect to the cell parameter have been implemented in the electronic structure code CRYSTAL, for the case of one- and two-dimensional periodicity. As in most molecular codes, Gaussian type orbitals are used to express the wavefunction. Examples demonstrate that the gradients have a good.     

Electrostatic molecular potentials in hydrogen-bonded systems

A study is made of the modifications of the electrostatic molecular potential brought about by hydrogen bonding both in the hydrogen-bond region itself and in the external regions of the proton-acceptor and proton-donor molecules. Systems up to four successive units in a chain of donor-acceptors are considered. The possibility of obtaining a satisfactory picture of the global potential by a simple superposition of the potentials of the individual units is evaluated.     

Monodisperse foams in one to three dimensions

Most major recent advances in the physics of monodisperse foams are centred around the ability to generate them with excellent control of bubble sizes down to a few micrometers thanks to the development of appropriate micro- and millifluidic techniques. As a natural consequence, monodisperse liquid and solid foams are playing an increasingly important role in fundamental research and in the development of industrial applications. In this review, we will address the different properties of monodisperse foams, comparing them to the more standard polydisperse foams.     

Integrin clustering in two and three dimensions

Integrins are transmembrane proteins that allow cells to bind to their external environment. They are the primary regulators of cell-matrix interactions, with direct roles in cell motility and signaling, which in turn regulate numerous physiological processes. Under common experimental conditions, integrins tend to cluster for sturdy and effective binding to extracellular matrix molecules. These clusters often evolve into focal adhesions, which regulate downstream signaling. However, integrin clusters are more pronounced and have longer lifetimes in two-dimensional assays than in more realistic three-dimensional environments. While a number of models and theoretical approaches have focused on integrin binding and diffusion, the reasons for the differences between two- and three-dimensional clustering have remained elusive. In this study, we model an individual cluster attached to a two-dimensional collagen film and attached to collagen fibers of various sizes in three-dimensional matrices. We then discuss how our results explain differences in size and lifetime, and how they hint at reasons for other differences between the two environments. Further, we make predictions regarding the stability of clusters based on different overall intracellular conditions. Our results show good agreement with experiments and provide a quantitative basis for understanding how matrix dimensionality and structure regulate integrin behavior in environments that mimic in vivo conditions.     

Electrostatic potentials of strained systems: nitrocyclopropane

The presence of an NO₂ substituent is found to eliminate the negative electrostatic potentials associated with the “bent” CC bonds cyclopropane. This is presumably due in large part to a rearrangement of electronics charge caused by the nitro group, as is shown by electronic density difference plots. Bond path calculations reveal bent bonds in nitrocyclopropane that are very similar to those in cycloprophane.     

Three closely related dibenzazepine carboxylic acids: hydrogen‐bonded aggregation in one,two and three dimensions

In the structure of (6R*,11R*)‐5‐acetyl‐11‐ethyl‐6,11‐dihydro‐5H‐dibenzo[b,e]azepine‐6‐carboxylic acid, C₁₉H₁₉NO₃, (I), the molecules are linked into sheets by a combination of O—H...O and C—H...O hydrogen bonds; in the structure of the monomethyl analogue (6RS,11SR)‐5‐acetyl‐11‐ethyl‐2‐methyl‐6,11‐dihydro‐5H‐dibenzo[b,e]azepine‐6‐carboxylic acid, C₂₀H₂₁NO₃, (II), the molecules are linked into simple C(7) chains by O—H...O hydrogen bonds; and in the structure of the dimethyl analogue (6RS,11SR)‐5‐acetyl‐11‐ethyl‐1,3‐dimethyl‐6,11‐dihydro‐5H‐dibenzo[b,e]azepine‐6‐carboxylic acid, C₂₁H₂₃NO₃, (III), a combination of O—H...O, C—H...O and C—H...π(arene) hydrogen bonds links the molecules into a three‐dimensional framework structure. None of these structures exhibits the R₂²(8) dimer motif characteristic of simple carboxylic acids.     

Analytical Hartree–Fock gradients with respect to the cell parameter for systems periodic in three dimensions

Analytical Hartree–Fock gradients with respect to the cell parameter have been implemented in the electronic structure code CRYSTAL, for the case of three-dimensional periodicity. The code is based on Gaussian-type orbitals, and the summation of the Coulomb energy is performed with the Ewald method. It is shown that a high accuracy of the cell gradient can be achieved.     

3‐Acetylanilinium bromide,nitrate and dihydrogen phosphate: hydrogen‐bonding motifs in one,two and three dimensions

In the title compounds, namely 3‐acetylanilinium bromide, C₈H₁₀NO⁺·Br⁻, (I), 3‐acetylanilinium nitrate, C₈H₁₀NO⁺·NO₃⁻, (II), and 3‐acetylanilinium dihydrogen phosphate, C₈H₁₀NO⁺·H₂PO₄⁻, (III), each asymmetric unit contains a discrete cation, with a protonated amino group, and an anion. In the crystal structure of (I), the ions are connected via N—H...Br and N—H...O hydrogen bonds into a chain of spiro‐fused R²₂(14) and R²₄(8) rings. In compound (II), the non‐H atoms of the cation all lie on a mirror plane in the space group Pnma, while the nitrate ion lies across a mirror plane. The crystal structures of compounds (II) and (III) are characterized by hydrogen‐bonded networks in two and three dimensions, respectively. The ions in (II) are connected via N—H...O hydrogen bonds, with three characteristic graph‐set motifs, viz.C²₂(6), R²₁(4) and R⁴₆(14). The ions in (III) are connected via N—H...O and O—H...O hydrogen bonds, with five characteristic graph‐set motifs, viz.D, C(4), C¹₂(4), R³₃(10) and R⁴₄(12). The significance of this study lies in its illustration of the differences between the supramolecular aggregations in the bromide, nitrate and dihydrogen phosphate salts of a small organic molecule. The different geometry of the counter‐ions and their different potential for hydrogen‐bond formation result in markedly different hydrogen‐bonding arrangements.     

Three-dimensional EPR imaging with one spectral and two spatial dimensions

Three-dimensional EPR imaging with one spectral dimension and two spatial dimensions is demonstrated. An image can be displayed as two-dimensional spatial-spatial maps of spin density or of EPR spectral intensity at a particular value of magnetic field. The spectral dimension gives the hyperfine-split EPR spectrum at each point in the spatial-spatial plane.     

Superfluidity with and without a condensate in two and three dimensions

The recent study of He and You [P.-S. He and W.-L. You, Int. J. Mod. Phys. B26, 1250107 (2012)] on the relation between superfluid and condensate densities in a two-dimensional weakly interacting Boson system assumed a δ(x???x′) interaction between two particles at sites x and x′. The main purpose of this article is to point out the sensitivity of the conclusions of He and You to the choice of interaction. Thus, it is known for logarithmic interactions that one can have superfluidity without a condensate. Finally, some related ideas are mentioned in relation to superfluidity in liquid ⁴ He at exceedingly low temperatures.     

Electrostatic potentials and covalent radii

We begin with a brief overview of the electrostatic potential V(r) as a fundamental determinant of the properties of systems of electrons and nuclei. The minimum of V(r) along the internuclear axis between two bonded atoms is a natural and physically meaningful boundary point, at which the electrostatic forces of the two nuclei upon an element of charge exactly cancel. We propose that the distances from nuclei to V(r) bond minima provide the basis for a well-defined set of covalent radii. Density functional calculations at the B3PW91/6-311+G** level were carried out for 59 molecules to locate the V(r) minima in 95 bonds and use these as the basis for determining single- and multiple-bond covalent radii for eight first- and second-row atoms plus hydrogen. It was found to be unrealistic to assign a single covalent radius to each atom; different values are needed for bonds to first- and second-row atoms, as well as to hydrogen. Using these results, we are able to predict the bond lengths of 33 single and multiple bonds with average errors of less than 0.04 A relative to experimental data.     

On higher order variational analysis in one and three dimensions for soft boundaries

For some problems in liquid crystal physics we need to use the Euler equation and the corresponding boundary equation in the three-dimensional case with soft boundaries. As a further complication the free energy expression, which should be minimized, might contain some second-order and third-order derivatives. These higher-order derivatives will cause the spatial derivatives of the boundary normal to appear in the boundary equation. Explicit formulae are given for the Euler equation and the corresponding surface equations for a general case. As an example, the theory is applied to nematic liquid crystals, where the general Euler equations and surface molecular fields are derived, including the effects of an imposed electric field.     

Similarity analysis in two and three dimensions using lattice animals and polycubes

The quantification and analysis of molecular similarity are fundamental problems of both theoretical and applied chemistry. The continuum similarity problem of planar domains with Jordan curve boundaries can be discretized and quantified using interior filling animals (square cell configurations). A similar approach is applicable to the continuum similarity problem of formal molecular bodies enclosed by contour surfaces, where interior filling polycubes provide a method for discretization and quantification of molecular similarity in three dimensions. This technique leads to resolution based similarity measures (RBSMs), suitable for automatic, non-visual evaluation of the degree of similarity between shapes of general objects, in particular, of molecular charge distributions, or fused sphere Van der Waals surfaces. Using the framework of the RBSM method, the polycube method of chirality quantification is extended to the quantification of approximate symmetry of molecular electron distributions.     

Multiple morphologies of amphiphilic diblock copolymer micelles in two and three dimensions

We show that in aqueous solution, diblock copolymers, where one block is hydrophobic and the other hydrophilic can undergo self-assembly in three dimensions in a manner similar to small molecule amphiphiles. In addition, two dimensional self-assembly has been studied at the air-water interface. We describe the various morphologies which have been observed in these systems and the parameters which we can use to tailor them.     

Tetrametallomethanes containing one,two, or three group IV metal atoms and boron

The propanediol methanetetraboronic ester, C(BO₂C₃H₆)₄, provides much better properties than the previously used methyl ester, C[B(OMe)₂]₄, for the synthesis of new tetrametallomethanes. Treatment with butyllithium yields the lithium salt LiC(BO₂C₃H₆)₃, which reacts with Ph₃MCl to form Ph₃MC(BO₂C₃H₆)₃, where M is Ge, Sn, or Pb. Repetition of these processes leads to (Ph₃M)₂C(BO₂C₃H₆)₂, where the Group IV metals M may be the same of different. The series was extended to (Ph₃Sn)₃CBO₂C₃H₆, but the last boron atom resisted removal. Iodination of the appropriate lithium salt intermediates yielded Ph₃SnCl(BO₂C₃H₆)₂ and (Ph₃Sn)₂CIBO₂C₃H₆.     

Self-assembled monolayers derived from alkoxyphenylethanethiols having one, two, and three pendant chains

This article describes the design, synthesis, and study of alkoxyphenylethanethiol-based adsorbates with one (R1ArMT), two (R2ArMT), and three (R3ArMT) pendant octadecyloxy chains substituted at the 4-, 3,5-, and 3,4,5-positions, respectively, of the phenylethanethiol group. These adsorbates are being developed for use in the preparation of compositionally versatile "mixed" self-assembled monolayer (SAM) coatings. The resultant SAMs were characterized by ellipsometry, contact angle goniometry, polarization modulation infrared reflection-absorption spectroscopy (PM-IRRAS), and X-ray photoelectron spectroscopy (XPS). The studies revealed that R1ArMT generates a well-ordered monolayer film, while R2ArMT and R3ArMT generate monolayer films with diminished conformational order in which the degree of crystallinity decreases as follows: C18 ～ R1ArMT > R3ArMT > R2ArMT. In addition, comparison of the molecular and chain packing densities of SAMs derived from these new adsorbates reveals that the R2ArMT and R3ArMT adsorbates give rise to SAMs with reduced chain tilt and smaller surface area per chain when compared to the SAMs derived from C18 and R1ArMT.