Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

In this paper, various implementations of the analytic Hartree–Fock and hybrid density functional energy second derivatives are studied. An approximation-free four-centre implementation is presented, and its accuracy is rigorously analysed in terms of self-consistent field (SCF), coupled-perturbed SCF (CP-SCF) convergence and prescreening criteria. The CP-SCF residual norm convergence threshold turns out to be the most important of these. Final choices of convergence thresholds are made such that an accuracy of the vibrational frequencies of better than 5 cm^?1 compared to the numerical noise-free results is obtained, even for the highly sensitive low frequencies (<100–200 cm^?1). The effects of the choice of numerical grid for density functional exchange–correlation integrations are studied and various weight derivative schemes are analysed in detail. In the second step of the work, approximations are introduced in order to speed up the computation without compromising its accuracy. To this end, the accuracy and efficiency of the resolution of identity approximation for the Coulomb terms and the semi-numerical chain of spheres approximation to the exchange terms are carefully analysed. It is shown that the largest performance improvements are realised if either Hartree–Fock exchange is absent (pure density functionals) and otherwise, if the exchange terms in the CP-SCF step of the calculation are approximated by the COSX method in conjunction with a small integration grid. Default values for all the involved truncation parameters are suggested. For vancomycine (176 atoms and 3593 basis functions), the RIJCOSX Hessian calculation with the B3LYP functional and the def2-TZVP basis set takes ～3 days using 16 Intel® Xeon® 2.60GHz processors with the COSX algorithm having a net parallelisation scaling of 11.9 which is at least ～20 times faster than the calculation without the RIJCOSX approximation.     

Hexamethylphosphoramide Compounds: II. The Structure of a Complex Containing Two Different Coordination Polyhedra for Two Independent Neodymium(III) Ions

Abstract

The crystal structure of the complex of formula {|Nd(NCS)₃ (HMPA)₃‖ |Nd(NCS)₃ (HMPA)₄|} was determined by three-dimensional X-ray diffraction methods and refined anisotropically to a R=0.040. The compound crystallizes in the trigonal system, space group R3 (No 146), with a=19.947(3), b=19.947(3), c=20.106(3) Å, α=β=90, γ=120°, V=6928(4) ų, M=1891.4, Z=3, D_c=1.360 g cm^?3, λ(MoKα)=0.71073 Å, μ=1.4 cm^?1, F (000)=2922.01. There are two independent Nd³⁺ ions located in the three fold axis. One of them, located at the origin (000) is coordinated to nitrogen atoms of three symmetry related NCS^? anions which are below the (x y) plane (negative z) and also to the oxygen atoms of three symmetry related HMPA groups above that plane. The coordination polyhedron is a slightly distorted octahedron. The other Nd³⁺ ion is located at (00,0.5025(1)). It is coordinated to the nitrogen atoms of three symmetry related NCS^? anions above the (xy 1/2) plane and to oxygen atoms of three symmetry related HMPA groups below that plane. Another HMPA group has the O and P atoms located on the three-fold axis, above the (xy 1/2) plane. The coordination number is in this case 7 and the polyhedron is a capped trigonal antiprism. The mean distances are: Nd-N=2.44 Å, Nd-0=2.35 Å and Nd-N=2.52 Å, Nd-0=2.36 Å for the octahedron and antiprism configurations, respectively. (CNPq, FAPESP, FINEP)     

Site-Specific Differences in Proteasome-Dependent Degradation of Monoubiquitinated α-Synuclein

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Analysis of PCBs in soils and sediments by microwave-assisted extraction,headspace-SPME and high resolution gas chromatography with ion-trap tandem mass spectrometry

A procedure for the determination of seven indicator PCBs in soils and sediments using microwave-assisted extraction (MAE) and headspace solid-phase microextraction (HS-SPME) prior to GC-MS/MS is described. Optimization of the HS-SPME was carried out for the most important parameters such as extraction time, sample volume and temperature. The adopted methodology has reduced consumption of organic solvents and analysis runtime. Under the optimized conditions, the method detection limit ranged from 0.6 to 1?ng/g when 5?g of sample was extracted, the precision on real samples ranged from 4 to 21% and the recovery from 69 to 104%. The proposed method, which included the analysis of a certified reference material in its validation procedure, can be extended to several other PCBs and used in the monitoring of soil or sediments for the presence of PCBs.     

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relations, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of n particles in the complex plane, hence to a categorification of the Knizhnik–Zamolodchikov connection. We discuss infinitesimal 2-braidings in a certain monoidal 2-category naturally assigned to every differential crossed module, leading to the notion of a symmetric quasi-invariant tensor in a differential crossed module. Finally, we prove that symmetric quasi-invariant tensors exist in the differential crossed module associated to Wagemann's version of the String Lie-2-algebra. As a corollary, we obtain a more conceptual proof of the flatness of a previously constructed categorified Knizhnik–Zamolodchikov connection with values in the String Lie-2-algebra.