Optimized Tensor-Product Approximation Spaces

This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard Sobolev spaces. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal mappings between Hilbert sequence spaces. We construct operator adapted finite element subspaces with a lower dimension than the standard full-grid spaces. These new approximation spaces preserve the approximation order of the standard full-grid spaces, provided that certain additional regularity assumptions are fulfilled. The form of the approximation spaces is governed by the ratios of the smoothness exponents of the considered classes of functions. We show in which cases the so-called curse of dimensionality can be broken. The theory covers elliptic boundary value problems as well as boundary integral equations. September 17, 1998. Date revised: March 5, 1999. Date accepted: September 20, 1999.     

磁超精细相互作用引起的原子能级分裂及能级图

超精细相互作用是"原子物理学"课程中的重要内容之一.目前在国内外的教材中,著者只给出了有无外磁场时原子核磁偶极超精细相互作用引起的原子能级分裂的表达式和列举了一些原子能级分裂的研究实例.本文详细导出和讨论了无外磁场和有强弱磁场时磁超精细相互作用引起的原子能级分裂的公式,并举例说明如何判断和在能级图上标注这些能级的高低顺序.该文将有助于师生学习理解和真正掌握本部分的课程内容.     

On compact symplectic manifolds with Lie group symmetries

In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.

     

An ab initio study of the structures and energetics of the planar ground and 90°‐twisted excited states of substituted ethylenes

Ab initio molecular orbital calculations are performed on the planar ground states (S₀), the 90°‐twisted triplet (T₁), and pyramidalized singlet (S₁) excited states of ethylene, methaniminium cation (MC), monocyano‐ (MCE), 1,1‐dicyano‐ (DCE), 1,1‐dihydroxy‐ (DHE), and 1,1‐dicyano‐2,2‐dihydroxy (DCHE) ethylenes. Equilibrium geometries are optimized at the Hartree–Fock (HF) level with the 6‐31G* basis set. Electron correlation corrections are estimated by optimizing the HF/6‐31G* geometries at the (U)MP2/6‐31G* level and then by carrying out single‐point calculations at the fourth‐order Møller–Plesset perturbation theory ((U)MP4/6‐311G**//MP2/6‐31G*). The effects of various types of perturbations on the structures, energetics, dipole moments, and state ordering of S₀, S₁, and T₁ are carefully investigated. “Positive” S₁–T₁ splittings are estimated at the HF level for all studied molecules, while “negative” S₁–T₁ splittings are obtained at the MP2 level for MC, DHE, and DCHE. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 242–254, 2001     

High distance Heegaard splittings via fat train tracks

We define fat train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in S³.     

Nuclear spin species,spin statistical weights and rovibronic tunnelling splittings of non-rigid water hexamer

Powerful multinomial generating functions together with the character table of the six-dimensional hyperoctahedral wreath product group S₆[S₂] of the non-rigid water hexamer are employed to obtain the nuclear spin species, nuclear spin multiplets and the total nuclear spin statistical weights of the rovibronic-tunnelling levels for both deuterated and regular forms. The methods are composed of algebraic generating functions involving 531,441 nuclear spin functions for (D₂O)₆ in a group of 46,080 operations and 65 irreducible representations of the S₆[S₂] group. Although the deuterated form of the non-rigid water hexamer possesses nuclear spin populations in 58 of the 65 possible symmetries of tunnelling levels, for the regular water hexamer only 23 of the 65 symmetries are populated with nuclear spin functions. The tunnelling splitting correlations of rovibronic levels of the water hexamer have been obtained from the Saykally's semi-rigid (G₄) model to the fully non-rigid limit (G₄₆₀₈₀). The computed nuclear spin statistical weights for protonated form of (H₂O)₆ call for a reinterpretation of the previous assignment of the observed spectra. The results can also be applied to enumerate the hyperfine patterns using the nuclear spin multiplets and intensities in semi-rigid to fully non-rigid limits.     

The forbidden rotational Q branch of CH3SiF3: A study in internal rotation

The pure rotational spectrum driven by the small dipole moment produced perpendicular to the symmetry axis by centrifugal distortion has been investigated for CH₃SiF₃ in the ground vibrational state using a Fourier transform waveguide spectrometer. Between 10.9 and 17.0 GHz, four (k + 3 ← k) series in the Q branch have been measured in the lowest torsional state v₆ = 0 for k = 4, 5, 6, and 7 with 54 ? J ? 65. In each transition, the quantum number σ = 0, +1, −1 labelling the different torsional sub-levels is conserved. For given (J,k), splittings from ∼10 to ∼45 MHz have been observed between lines with different values of σ. The global data set includes the anticrossing molecular beam energy differences of [W.L. Meerts, I. Ozier, Chem. Phys. 71 (1982) 401-415] as well as the mm-wave R branch frequencies and (A₁ − A₂) splittings of [P. Dréan, J.-M. Colmont, J. Demaison, L. Dore, C. Degli Esposti, J. Mol. Spectrosc. 176 (1996) 23-27]). A good fit was obtained by varying 15 molecular parameters characterizing the torsion-rotation Hamiltonian H_TR for the vibrational ground state. Because of the strong correlation between two of the quartic torsion-distortion parameters (F_0,3K and D_0,Km) and a redundancy connecting the centrifugal distortion constants, four models were obtained yielding comparable fits. In each case, effective values were determined for the A-rotational constant and the height of the potential hindering the internal rotation. A high precision determination of the structural parameter ρ was made that is the same in all four models. For the off-diagonal quartic centrifugal distortion constant ε₀ and the sextic constants H_0,J, H_0,JK, H_0,KJ, and h_0,3, the differences in the values obtained in the two different reductions used have been explained in terms of the redundancy connecting these parameters. For σ = 0, +1, −1, the energy level pattern for (|k| = 3) is discussed for the case where the pure torsional energy splitting and the matrix elements off-diagonal in k are of comparable magnitude. A method is described of using an R branch study of the resulting σ-splittings for (|k| = 3) to probe the zeroth-order torsional Hamiltonian.     

Unrestricted Hartree-Fock spin density calculations with orthogonalized atomic orbitals on aza and nitroaromatic radical anions

The spin density distributions in some aza and nitroaromatic radical anions have been calculated using Löwdin's orthogonalized basis set of atomic orbitals in the Unrestricted Hartree-Fock method of Amos and Snyder. The present calculations lead to a satisfactory account of proton splittings in these radicals. Least squares analyses correlating the observed ¹⁴N splittings and the spin density results over completely localized nonorthogonal basis have been separately carried out for aza and nitroaromatic radical anions and the sigma-pi parameters thus obtained are discussed and compared with earlier estimates for these quantities. Unlike the earlier results, the present estimate of Q _NN ^N for aza and nitroaromatic radicals are not very much different from each other.     

Finite Nuclei in a Relativistic mean—Field Theory with quadratic Couplings

A quadratic scalar and vector coupling model proposed recently has been applied to finite nuclei.The calculated results are compared with those of the derivative scalar couling(DSC) model and the nonlinear Walecka model,The results show that the spin-orbit splittings are improved considerably by quadratic ouplings in contrast to the DSC model,However,the binding energy per nucleon,rms charge radius,as well as the spin-orbit splittings in the quadratic model are still small compared with those given by the nonlinear Walecka model and the experimental data.     

Theoretical and experimental investigation of (13)C relayed (2)H-(2)H-COSY 2D experiments: application to the analysis of weakly aligned solutes

We describe new NMR 2D experiments denoted DECADENCY for DEuterium CArbon DEuterium Nuclear Correlation spectroscopY dedicated to the analysis of anisotropic deuterium spectra. They belong to the class of X-relayed Y,Y-COSY 2D experiments that was initially explored in the case of a (1)H-X-(1)H fragment (I(X)=1/2) in isotropic medium. DECADENCY 2D experiments permit to correlate the quadrupolar doublets associated with two inequivalent deuterium nuclei in an oriented CD(2) fragment through heteronuclear polarization transfers. Two kinds of pulse sequences are described here using either a double INEPT-type or DEPT-type process. DECADENCY 2D experiments provide an interesting alternative to (2)H-(2)H COSY experiments when the geminal (2)H-(2)H total coupling (scalar and dipolar) is null or too small to provide visible cross-correlation peaks. Such a situation is typically observed for geminal deuteriums in prochiral or chiral molecules dissolved in chiral liquid crystals. The efficiency of these techniques is illustrated using dideuterated prochiral molecules, the phenyl[(2)H(2)]methanol and the 1-chloro[1-(2)H(2)]nonane, both dissolved in organic solutions of poly-gamma-benzyl-l-glutamate. The advantages of each sequence are presented and discussed. It is shown that the relative sign of the quadrupolar doublets can be determined.