Analytical calculation of the solvent-accessible surface area and its nuclear gradients by stereographic projection: A general approach for molecules,polymers, nanotubes,helices, and surfaces

In this article, we explore an alternative to the analytical Gauss–Bonnet approach for computing the solvent-accessible surface area (SASA) and its nuclear gradients. These two key quantities are required to evaluate the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models. We extend a previously proposed analytical approach for finite systems based on the stereographic projection technique to infinite periodic systems such as polymers, nanotubes, helices, or surfaces and detail its implementation in the Crystal code. We provide the full derivation of the SASA nuclear gradients, and introduce an iterative perturbation scheme of the atomic coordinates to stabilize the gradients calculation for certain difficult symmetric systems. An excellent agreement of computed SASA with reference analytical values is found for finite systems, while the SASA size-extensivity is verified for infinite periodic systems. In addition, correctness of the analytical gradients is confirmed by the excellent agreement obtained with numerical gradients and by the translational invariance achieved, both for finite and infinite periodic systems. Overall therefore, the stereographic projection approach appears as a general, simple, and efficient technique to compute the key quantities required for the calculation of the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models applicable to both finite and infinite periodic systems.     

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.     

Application of MDGRAPE-3, a special purpose board for molecular dynamics simulations, to periodic biomolecular systems

We describe the application of a special purpose board for molecular dynamics simulations, named MDGRAPE-3, to the problem of simulating periodic bio-molecular systems. MDGRAPE-3 is the latest board in a series of hardware accelerators designed to calculate the nonbonding long-range interactions much more rapidly than normal processors. So far, MDGRAPEs were mainly applied to isolated systems, where very many nonbonded interactions were calculated without any distance cutoff. However, in order to regulate the density and pressure during simulations of membrane embedded protein systems, one has to evaluate interactions under periodic boundary conditions. For this purpose, we implemented the Particle-Mesh Ewald (PME) method, and its approximation with distance cutoffs and charge neutrality as proposed by Wolf et al., using MDGRAPE-3. When the two methods were applied to simulations of two periodic biomolecular systems, a single MDGRAPE-3 achieved 30-40 times faster computation times than a single conventional processor did in the both cases. Both methods are shown to have the same molecular structures and dynamics of the systems.     

Transport and Helfand moments in the Lennard-Jones fluid. I. Shear viscosity

The authors propose a new method, the Helfand-moment method, to compute the shear viscosity by equilibrium molecular dynamics in periodic systems. In this method, the shear viscosity is written as an Einstein-type relation in terms of the variance of the so-called Helfand moment. This quantity is modified in order to satisfy systems with periodic boundary conditions usually considered in molecular dynamics. They calculate the shear viscosity in the Lennard-Jones fluid near the triple point thanks to this new technique. They show that the results of the Helfand-moment method are in excellent agreement with the results of the standard Green-Kubo method.     

Values in the Development of Early Periodic Tables

Julius Lothar Meyer, John Newlands, and Dmitrii Mendeleev were amongst the discoverers of the periodic system of the elements. Although their systems are similar enough to be recognised as the precursors for the modern periodic system, they were also different. Here, I argue that many of their differences can be explained in terms of how the chemists emphasised different values in the process of developing their systems. In particular, Newland highlighted the simplicity of his arrangements; Meyer was more careful about the quality of data that gave rise to his system of elements; and Mendeleev sought to make his system more complete. By shedding light as to how the values of simplicity, completeness and carefulness guided the development of early periodic systems, this paper contributes to a broader understanding of how values influence science.     

Classical trajectories for two ring-shaped potentials

The present paper deals with the classical trajectories for two superintegrable systems: a system known in quantum chemistry as the Hartmann system and a system of potential use in quantum chemistry and nuclear physics. Both systems correspond to ring-shaped potentials. They admit two maximally superintegrable systems as limiting cases, viz., the isotropic harmonic oscillator system and the Coulomb–Kepler system in three dimensions. The planarity of the trajectories is studied in a systematic way. In general, the trajectories are quasi-periodic rather than periodic. A constraint condition allows to pass from quasi-periodic motions to periodic ones. When written in a quantum mechanical context, this constraint condition leads to new accidental degeneracies for the two systems studied. © 1992 John Wiley & Sons, Inc.     

The optimal P3M algorithm for computing electrostatic energies in periodic systems

We optimize Hockney and Eastwood's particle-particle particle-mesh algorithm to achieve maximal accuracy in the electrostatic energies (instead of forces) in three-dimensional periodic charged systems. To this end we construct an optimal influence function that minimizes the root-mean-square (rms) errors of the energies. As a by-product we derive a new real-space cutoff correction term, give a transparent derivation of the systematic errors in terms of Madelung energies, and provide an accurate analytical estimate for the rms error of the energies. This error estimate is a useful indicator of the accuracy of the computed energies and allows an easy and precise determination of the optimal values of the various parameters in the algorithm (Ewald splitting parameter, mesh size, and charge assignment order).     

Excessive excitation of hydrogen peroxide during oscillatory chemical evolution

The peculiar reaction dynamics of the Bray-Liebhafsky chemical oscillator is connected with the nonequilibrium periodic excitations of hydrogen peroxide embedded in a hydrogen-bonded water network. This was indicated by Raman spectroscopy that showed periodic, isothermal, and excessive excitation of the symmetric vibration of hydrogen peroxide. Since such an excessive excitation should be the result of a specific nonequilibrium energy distribution with the active participation of water, understanding this process can be of considerable importance in other systems in which water is the main constituent.     

On almost periodic processes in impulsive fractional-order competitive systems

In this paper an impulsive model for the dynamics of competitive Lotka–Volterra systems using the Caputo fractional-order derivative is developed. The existence and uniqueness of almost periodic solutions are investigated. Applying the fractional Lyapunov method, we give sufficient conditions for global perfect uniform-asymptotic stability of the almost periodic solution and sufficient conditions to save these qualities at the uncertain case. Since the competitive relationship in ecosystem models is relevant in various contexts, including many population, neural nets or chemical kinetics problems, our results can be applied in the investigation of almost periodic processes in a wide range of competitive systems of diverse interest.     

Ultrasonic Broadband Spectrometry of Liquids A Research Tool in Pure and Applied Chemistry and Chemical Physics

This review is a survey of the many scientific applications of ultrasonic broadbandspectrometry (absorption and velocity measurements with coherent sound waves)in liquids and liquid systems, covering, at present, a frequency range from nearly10 kHz to 10 GHz. Ultrasonic spectrometry has proved to be an almost universalresearch tool in many laboratories, one that is useful for investigation of variouschemical, biochemical, and physicochemical systems. Sound waves traversingliquids induce periodic perturbations in pressure and temperature, which can shiftequilibria, resulting in characteristic sound absorption and velocity dispersionspectra. An analysis of such spectra yields valuable information about thermodynamic and kinetic parameters of the particular system that is often difficult toobtain by other methods. Since such periodic perturbations imposed on the systemare incremental in nearly all cases, the system can be studied under equilibriumconditions. All nonlinear effects (heating, nonconstant fluid compressibility, andothers) are negligible, permitting, for instance, the application of linearized rateequations. In this review, various examples of measured broadband spectra arepresented. Related elementary processes are discussed. Among these are ionicand molecular reactions, including mechanisms of association and complexation,proton transfer, solvation, isomerization, interconversion, side-group rotation,hydrogen-bonding, as well as stacking processes and micelle formation. Specialattention will be given to the extensive research on chemical relaxation.Fundamental early and recent publications are cited and discussed. Many referencesare included with particular emphasis on less well known research and publicationsfrom countries of the former USSR. This review aims at a demonstration of thewidespread applications of modern ultrasonic techniques in many fields ofliquid-state research.     

Challenges for the Periodic Systems of Elements: Chemical,Historical and Mathematical Perspectives

We celebrate 150 years of periodic systems that reached their maturity in the 1860s. They began as pedagogical efforts to project corpuses of substances on the similarity and order relationships of the chemical elements. However, these elements are not the canned substances wrongly displayed in many periodic tables, but rather the abstract preserved entities in compound transformations. We celebrate the systems, rather than their tables or ultimate table. The periodic law, we argue, is not an all-encompassing achievement, as it does not apply to every property of all elements and compounds. Periodic systems have been generalised as ordered hypergraphs, which solves the long-lasting question on the mathematical structure of the systems. In this essay, it is shown that these hypergraphs may solve current issues such as order reversals in super-heavy elements and lack of system predictive power. We discuss research in extending the limits of the systems in the super-heavy-atom region and draw attention to other limits: the antimatter region and the limit arising from compounds under extreme conditions. As systems depend on the known chemical substances (chemical space) and such a space grows exponentially, we wonder whether systems still aim at projecting knowledge of compounds on the relationships among the elements. We claim that systems are not based on compounds anymore, rather on 20th century projections of the 1860s systems of elements on systems of atoms. These projections bring about oversimplifications based on entities far from being related to compounds. A linked oversimplification is the myth of vertical group similarity, which raises questions on the approaches to locate new elements in the system. Finally, we propose bringing back chemistry to the systems by exploring similarity and order relationships of elements using the current information of the chemical space. We ponder whether 19th century periodic systems are still there or whether they have faded away, leaving us with an empty 150^th celebration.     

Condensed phase ionic polarizabilities from plane wave density functional theory calculations

A method is presented to allow the calculation of the dipole polarizabilities of ions and molecules in a condensed-phase coordination environment. These values will be useful for understanding the optical properties of materials and for developing simulation potentials which incorporate polarization effects. The reported values are derived from plane wave density functional theory calculations, though the method itself will apply to first-principles calculations on periodic systems more generally. After reporting results of test calculations on atoms to validate the procedure, values for the polarizabilities of the oxide ion and various cations in a range of materials are reported and compared with experimental information as well as previous theoretical results.     

Development of the cyclic cluster model formalism for Kohn-Sham auxiliary density functional theory methods

The development of the cyclic cluster model (CCM) formalism for Kohn-Sham auxiliary density functional theory (KS-ADFT) methods is presented. The CCM is a direct space approach for the calculation of perfect and defective systems under periodic boundary conditions. Translational symmetry is introduced in the CCM by integral weighting. A consistent weighting scheme for all two-center and three-center interactions appearing in the KS-ADFT method is presented. For the first time, an approach for the numerical integration of the exchange-correlation potential within the cyclic cluster formalism is derived. The presented KS-ADFT CCM implementation was applied to covalent periodic systems. The results of cyclic and molecular cluster model (MCM) calculations for trans-polyacetylene, graphene, and diamond are discussed as examples for systems periodic in one, two, and three dimensions, respectively. All structures were optimized. It is shown that the CCM results represent the results of MCM calculations in the limit of infinite molecular clusters. By analyzing the electronic structure, we demonstrate that the symmetry of the corresponding periodic systems is retained in CCM calculations. The obtained geometric and electronic structures are compared with available data from the literature.     

From the plateau problem to periodic minimal surfaces in lipids,surfactants and diblock copolymers

A novel method is presented for generating periodic surfaces. Such periodic surfaces appear in all systems which are characterized by internal interfaces and which additionally exhibit ordering. One example are systems of AB diblock copolymers, where the internal interfaces are formed by the chemical bonds between the A and B blocks. In these systems at least two bicontinuous phases are formed: the ordered bicontinuous double diamond phase and the gyroid phase. In these phases the ordered domains of A monomers and B monomers are separated by a periodic interface of the same symmetry as the phases themselves. Here we present a novel method for the generation of such periodic surfaces based on the simple Landau-Ginzburg model of microemulsions. We test the method on four known minimal periodic surfaces, find two new surfaces of cubic symmetry, and show how to obtain periodic surfaces of high genus and n-tuply continuous phases (n > 2). So far only bicontinuous (n = 2) phases have been known. We point out that the Landau model used here should be generic for all systems characterized by internal interfaces, including the diblock copolymer systems.     

Investigation of the stability of oligonucleotides and oligodinucleotides

The mutually-consistent-field (MCF) method together with pseudo polarization tensors has been employed to investigate in detail the intra- and inter-strand interactions in B-DNA helices. Results are reported for all four periodic homo-oligonucleotides showing that these configurations are stable only in the case of adenine and thymine. All the oligodinucleotides studied are stable, but the role of the different contributions, e.g. the two intrastrand and the interstrand interactions is dependent on the nucleotide bases. In the case of periodic guanine-cytosine base pairs, the interstrand energy is dominant, whereas in the corresponding alternating systems all three contributions are attractive. The stability of adenine-thymine systems is less sensitive with respect to periodic and alternating compositions.     

On the electronic structure and conduction properties of aperiodic DNA and proteins. III. The band structures of periodic polypeptides

In this paper the calculation of the energy band structure of periodic polypeptides using the ab initio Hartree-Fock crystal orbital method is described. Results are discussed for the twenty homopolypeptides in the β-pleated sheet configuration and for several periodic systems assuming the α-helix structure. The negative factor method in its matrix block form is used to provide density of states curve for periodic two-component polypeptides. The resulting properties of the band structure suggest that homopolypeptides and periodic more-component polyamino acids are in themselves insulators, but may become weak semiconductors if free charge carriers are generated in their valence or conduction bands, respectively, through charge transfer.     

Design of a reaction field using a linear‐combination‐based isotropic periodic sum method

In our previous study (Takahashi et al., J. Chem. Theory Comput. 2012, 8, 4503), we developed the linear‐combination‐based isotropic periodic sum (LIPS) method. The LIPS method is based on the extended isotropic periodic sum theory that produces a ubiquitous interaction potential function to estimate homogeneous and heterogeneous systems. The LIPS theory also provides the procedure to design a periodic reaction field. To demonstrate this, in the present work, a novel reaction field of the LIPS method was developed. The novel reaction field was labeled LIPS‐SW, because it provides an interaction potential function with a shape that resembles that of the switch function method. To evaluate the ability of the LIPS‐SW method to describe in homogeneous and heterogeneous systems, we carried out molecular dynamics (MD) simulations of bulk water and water–vapor interfacial systems using the LIPS‐SW method. The results of these simulations show that the LIPS‐SW method gives higher accuracy than the conventional interaction potential function of the LIPS method. The accuracy of simulating water–vapor interfacial systems was greatly improved, while that of bulk water systems was maintained using the LIPS‐SW method. We conclude that the LIPS‐SW method shows great potential for high‐accuracy, high‐performance computing to allow large scale MD simulations. © 2014 Wiley Periodicals, Inc.     

Kronecker-product periodic systems of small gas-phase molecules and the search for order in atomic ensembles of any phase

The periodic law, manifested in the chart of the elements, is so fundamental in chemistry and related areas of physics that the question arises "Might periodicity among molecules also be embodied in a periodic system?" This review paper details how a particular periodic system of gas-phase diatomic molecules, allowing for the forecasting of thousands of new data, was developed. It can include ionized and even quarked-nuclei molecules and it coincides with locality (averaging) and the additivity found in some data; it has interesting vector properties, and it may be related in challenging ways to partial order. The review then explains how periodic systems for triatomic and four-atomic species are evolving along a similar path. The systems rest largely upon exhaustive comparisons of tabulated data, relate to some extent to the octet rule, and include reducible representations of the dynamic group SO(4) in higher spaces. Finally, the paper shows how periodicity can be quantified in data for larger molecules. Data for properties of homologous or substituted molecules, in any phase, are quantified with a vector index, and the index for one set can be transformed into that for another set.