A test problem for molecular dynamics integrators

We derive a test problem for evaluating the ability of time-steppingmethods to preserve the statistical properties of systems inmolecular dynamics. We consider a family of deterministic systemsconsisting of a finite number of particles interacting on acompact interval. The particles are given random initial conditionsand interact through instantaneous energy- and momentum-conservingcollisions. As the number of particles, the particle density,and the mean particle speed go to infinity, the trajectory ofa tracer particle is shown to converge to a stationary Gaussianstochastic process. We approximate this system by one describedby a system of ordinary differential equations and provide numericalevidence that it converges to the same stochastic process. Wesimulate the latter system with a variety of numerical integrators,including the symplectic Euler method, a fourth-order Runge-Kuttamethod, and an energyconserving step-and-project method. Weassess the methods' ability to recapture the system's limitingstatistics and observe that symplectic Euler performs significantlybetter than the others for comparable computational expense.     

Fluid spacetimes admitting covariantly constant vectors and tensors

Spacetimes admitting a covariantly constant vector and satisfying the Einstein field equations for a perfect fluid, a viscous heat-conducting fluid, or an anisotropic fluid are studied. It is found that the only possible perfect fluid spacetimes are the Einstein static universe and stiff-matter spacetimes with an isolated spatial co-ordinate, while the possible viscous fluid and anisotropic fluid spacetimes, although more abundant than their perfect fluid counterparts, must satisfy a number of strong restrictions. Examples illustrating most of the various possible situations are given. The paper concludes with a study of covariantly constant second-rank tensors in fluid spacetimes; the only possible solutions that do not also admit a covariantly constant vector are restricted to 2+2 spacetimes.     

Lanthanide Shift Reagents. Paper 21. Interaction of Arenes with Silver/Lanthanide Shift Reagent and Comparison with Chromium Tricarbonyl Complexes

Lanthanide shift reagents can bring about dramatic simplifications of highly complex NMR spectra^1,2. The applicability of the commonest of these (the thd and fod chelates of Eu, Pr and Yb) has been restricted to those molecules possessing a suitable N, O or S donor atom, although they may turn up in unusual guises, e.g. in quaternary ammonium compounds³.     

An exact FRW cosmology with radiation and imperfect fluid

An exact solution of the Einstein field equations for a combination of black-body radiation and an imperfect fluid, in which the geometrical background is a flat FRW metric, is presented. The solution exhibits an axial preferred direction along which the material content moves relative to the radiation field, the latter representing the cosmic background radiation. The solution is shown to be in excellent agreement with current observations.     

The synthesis and reactions of 4-(2- and 3-thienyl)-tetrahydroisoquinolines

Tetrahydroisoquinoline derivatives substituted in the 4-position by either a 2- or 3-substituted thiophene ring have been synthesised. Simple electrophilic substitution reactions of these systems take place as expected in the α-position of the thiophene ring. Metalation reactions are more complex and take place at the benzylic 4-position of the tetrahydroisoquinoline nucleus in the case of the 2-substituted thiophene derivatives or at either the thiophene α-positions or the benzylic 4-position depending on the nature of the attacking electrophile in the case of the 3-substituted thiophene system.     

Common solutions of the einstein and Brans-Dicke theories

It is shown that non-trivial solutions common to the vacuum field equations of the Einstein and of the Brans-Dicke theories necessarily representpp-waves and the set of all common solutions is precisely the set of allpp-wave solutions of the Einstein equations. The form of the associated scalar field is found and is shown to be singular when ω相似文献   

Eddington’s statistical theory

The density-particle correlation, described in a previous paper of the series, is used to define the «uncertainty of the physical origin» considered by Eddington, and to relate it to the Hubble constant.     

A new look at FRW cosmologies

The question of whether the material distribution in any cosmological model can be interpreted unambiguously is considered. We suggest that the answer is negative, and discuss problems in theoretical cosmology which may arise from this ambiguity of interpretation.     

Synthesis and characterization of scandium complexes with reduced ligands: Crystal structures of Cp∗ScI2, [Cp∗ScI(bpy)]2, and [Cp∗ScCl(bpy)]2

The synthesis and characterization of complexes containing a Cp∗Sc(R₂bpy) (Cp∗ = pentamethylcyclopentadienyl, bpy = 4,4′-R,R-2,2′-bipyridine, R = H, Me) motif are described. Cp∗ScI₂ (1) was prepared from Cp∗Sc(acac)₂ (acac = acetylacetonate) and AlI₃ (2 equiv) in pentane. Compound 1 reacted with bipyridine and 4,4′-dimethyl-2,2′-bipyridine (dmb) in benzene to yield Cp∗ScI₂(bpy) (3) and Cp∗ScI₂(dmb) (4), respectively. Compound 3 was reduced by alkali metal reductants such as Na/Hg, NaK₂, and K in aromatic solvents to yield [Cp∗ScI(bpy)]₂ (5). The chloride analog of 5, [Cp∗ScCl(bpy)]₂ (7), was prepared from Cp∗ScCl₂ by salt metathesis with Li₂(dme)₂bpy (6) (dme = dimethoxyethane) in toluene. Compounds 1, 5, and 7 have been structurally characterized. Analysis of the bond distances of the bipyridine ligands in 5 and 7, together with infrared and UV/vis spectroscopic data, suggest that the bipyridine ligands in these molecules exist as radical anions. The bipyridine ligands in 5 and 7 are arranged co-facially and are in close proximity (?3.30 Å), suggesting the presence of a π-π interaction.