Systems Simplicity

A simple system is a system who has no proper ideals. We prove that every simple  system  $\mathcal{J}$ have one of the following assertion: \begin{description}   \item[$(1)$] $\mathcal{J}$  is $\mathfrak{h}-$irreducible.   \item[$(2)$] $\mathcal{J}=\mathcal{J}_1\bigoplus\widetilde{\mathcal{J}_1}$ is the direct summation of two $\mathfrak{h}-$invariant   and $\mathfrak{h}-$irreducible subsystems.   \end{description}

Fast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. III. Application to crystal structures via the Ewald and direct summation methods

The previously reported exact potential and multipole moment (EP/MM) method for fast and accurate evaluation of the intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density [Volkov, Koritsanszky & Coppens (2004). Chem. Phys. Lett. 391, 170–175; Nguyen, Kisiel & Volkov (2018). Acta Cryst. A74, 524–536; Nguyen & Volkov (2019). Acta Cryst. A75, 448–464] is extended to the calculation of electrostatic interaction energies in molecular crystals using two newly developed implementations: (i) the Ewald summation (ES), which includes interactions up to the hexadecapolar level and the EP correction to account for short-range electron-density penetration effects, and (ii) the enhanced EP/MM-based direct summation (DS), which at sufficiently large intermolecular separations replaces the atomic multipole moment approximation to the electrostatic energy with that based on the molecular multipole moments. As in the previous study [Nguyen, Kisiel & Volkov (2018). Acta Cryst. A74, 524–536], the EP electron repulsion integral is evaluated analytically using the Löwdin α-function approach. The resulting techniques, incorporated in the XDPROP module of the software package XD2016, have been tested on several small-molecule crystal systems (benzene, L-dopa, paracetamol, amino acids etc.) and the crystal structure of a 181-atom decapeptide molecule (Z = 4) using electron densities constructed via the University at Buffalo Aspherical Pseudoatom Databank [Volkov, Li, Koritsanszky & Coppens (2004). J. Phys. Chem. A, 108, 4283–4300]. Using a 2015 2.8 GHz Intel Xeon E3-1505M v5 computer processor, a 64-bit implementation of the Löwdin α-function and one of the higher optimization levels in the GNU Fortran compiler, the ES method evaluates the electrostatic interaction energy with a numerical precision of at least 10−5 kJ mol−1 in under 6 s for any of the tested small-molecule crystal structures, and in 48.5 s for the decapeptide structure. The DS approach is competitive in terms of precision and speed with the ES technique only for crystal structures of small molecules that do not carry a large molecular dipole moment. The electron-density penetration effects, correctly accounted for by the two described methods, contribute 28–64% to the total electrostatic interaction energy in the examined systems, and thus cannot be neglected.

GraviDy: a modular, GPU-based, direct-summation N-body code

The direct-summation of N gravitational forces is a complex problem for which there is no analytical solution. Dense stellar systems such as galactic nuclei and stellar clusters are the loci of different interesting problems. In this work we present a new GPU, direct-summation N-body integrator written from scratch and based on the Hermite scheme. The first release of the code consists of the Hermite integrator for a system of N bodies with softening. We find an acceleration factor of about ≈ 90 of the GPU version in a single node as compared to the Serial-Single-CPU one. We additionally investigate the impact of using softening in the dynamics of a dense cluster. We study how it affects the two body relaxation, as compared with another code, NBODY6, which uses KS regularization, so as to understand the role of softening in the evolution of the system. This initial release is the first step towards more and more realistic scenarios, starting for a proper treatment for binary evolution, close encounters and the role of a massive black hole.