Low-rank canonical-tensor decomposition of potential energy surfaces: application to grid-based diagrammatic vibrational Green's function theory

ABSTRACT

A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm^?1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.     

Nilpotent bases for distributions and control systems

Let V(M) be the Lie algebra (infinite dimensional) of real analytic vector fields on the n-dimensional manifold M. Necessary conditions that a real analytic k-dimensional distibution on M have a local basis which generates a nilpotent subalgebra of V(M) are derived. Two methods for sufficient conditions are given, the first depending on the existence of a solution to a system of partial differential equations, the second using Darboux's theorem to give a computable test for an (n ? 1)-dimensional distribution. A nonlinear control system in which the control variables appear linearly can be transformed into an orbit equivalent system whose describing vector fields generate a nilpotent algebra if the distribution generated by the original describing vector fields admits a nilpotent basis. When this is the case, local analysis of the control system is greatly simplified.     

The correlation of proton affinities with atomic charges and electronegativities for the group 14 to 17 hydrides

Proton affinities for hydrides of formula $\mathrm{AH}^{-}_{n-1}$ containing the elements A from the second to the fifth period of the periodic table and groups 14 to 17 are predicted at the Hartree–Fock, MP2 and B3LYP levels of theory employing both core potential basis sets and the 3‐21G basis set. The core potential methods perform well when compared with all electron calculations using the 3‐21++G** basis set. The proton affinities of the hydrides containing elements from groups 15 and 16 of the periodic table are more accurate than those with elements from groups 14 and 17. A cancellation of errors appears to occur more completely if the protonated and nonprotonated molecules contain both bond and lone pairs before and after the protonation reaction. Proton affinities correlate nearly linearly with the atomic charges on the hydrogen atoms when these charges are determined by the generalized atomic polar tensor (GAPT) method. This tendency can be associated, in principle, with the group electronegativities as introduced by Iczkowski and Margrave. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1119–1131, 2000     

Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: a comparison of simulation techniques

Over the last number of years several simulation methods have been introduced to study rare events such as nucleation. In this paper we examine the crystal nucleation rate of hard spheres using three such numerical techniques: molecular dynamics, forward flux sampling, and a Bennett-Chandler-type theory where the nucleation barrier is determined using umbrella sampling simulations. The resulting nucleation rates are compared with the experimental rates of Harland and van Megen [Phys. Rev. E 55, 3054 (1997)], Sinn et al. [Prog. Colloid Polym. Sci. 118, 266 (2001)], Sch?tzel and Ackerson [Phys. Rev. E 48, 3766 (1993)], and the predicted rates for monodisperse and 5% polydisperse hard spheres of Auer and Frenkel [Nature 409, 1020 (2001)]. When the rates are examined in units of the long-time diffusion coefficient, we find agreement between all the theoretically predicted nucleation rates, however, the experimental results display a markedly different behavior for low supersaturation. Additionally, we examined the precritical nuclei arising in the molecular dynamics, forward flux sampling, and umbrella sampling simulations. The structure of the nuclei appears independent of the simulation method, and in all cases, the nuclei contains on average significantly more face-centered-cubic ordered particles than hexagonal-close-packed ordered particles.     

Trivalent‐Intermediate Valent Cerium Ordering in Ce2RuZn4

The new intermetallic cerium compound Ce₂RuZn₄ was synthesized from the elements in a sealed tantalum tube in a water‐cooled sample chamber of an induction furnace. Ce₂RuZn₄ crystallizes with a new structure type: P4/nmm, Z = 2, a = 719.6(1), c = 520.2(1) pm, wR2 = 0.0816, 273 F² values and 15 variables. The structure contains two crystallographically independent cerium atoms: Ce1 with CN 16 (12 Zn + 4 Ce) and Ce2 with CN 14 (2 Ru + 8 Zn + 4 Ce). Based on the interatomic distances the two sites can be assigned to trivalent Ce1 and intermediate valent Ce2. The trivalent‐intermediate valent cerium ordering is underlined by magnetic susceptibility measurements. Ce₂RuZn₄ shows modified Curie‐Weiss behaviour in the temperature range 10–290 K with an experimental magnetic moment of 2.57(1) μ_B per formula unit. Thus only half of the cerium atoms are trivalent in Ce₂RuZn₄. A remarkable feature of the Ce₂RuZn₄ structure are short Ce2–Ru distances of 260 pm. The crystal chemistry of Ce₂RuZn₄ is discussed.     

Determination of Cesium on Nickel Hexacyanoferrate‐Aluminum Oxide Modified Electrodes

This report investigates the electrochemical behavior of hexacyanoferrate casted on Ni‐Al₂O₃ modified electrodes and the preconcentration and detection of cesium ions on such films. It also studies the morphology and the composition of these surfaces. The film was grown on a glassy carbon (GC) surface. Five consecutive voltammetric cycles applied within 0.0 V and ?1.6 V at a scan rate of 10 mV/s were enough to cast the film. Scanning electron microscopy (SEM) analyses showed a homogeneous, porous but broken surface of the film. Its composition was studied by X‐ray diffraction (XRD). The presence of NiHCFe was confirmed by Fourier transformed infrared spectroscopy (FT‐IR). The Cs⁺ preconcentration from diluted solutions was accomplished in 90 s, under a negative potential of ?0.20 V applied to the modified working electrode. The detection of cesium has a good sensitivity and a wide linear interval (10^?8 and 10^?12 mol L^?1). Even so, the limit of detection calculated was extremely low (2×10^?16 mol L^?1), cesium concentrations lower than 10^?12 mol L^?1 gave signals with no analytical significance. However, to our knowledge, this is the lowest level of cesium ever detected by an electroanalytical technique.     

Self-assembly of a colloidal interstitial solid with tunable sublattice doping

We determine the phase diagram of a binary mixture of small and large hard spheres with a size ratio of 0.3 using free-energy calculations in Monte Carlo simulations. We find a stable binary fluid phase, a pure face-centered-cubic (fcc) crystal phase of the small spheres, and binary crystal structures with LS and LS(6) stoichiometries. Surprisingly, we demonstrate theoretically and experimentally the stability of a novel interstitial solid solution in binary hard-sphere mixtures, which is constructed by filling the octahedral holes of an fcc crystal of large spheres with small spheres. We find that the fraction of octahedral holes filled with a small sphere can be completely tuned from 0 to 1. Additionally, we study the hopping of the small spheres between neighboring octahedral holes, and interestingly, we find that the diffusion increases upon increasing the density of small spheres.     

Anzahl der Zerlegungen einer ganzen,rationalen Zahl in Summanden,II

Ohne Zusammenfassung