Equivalent rotations associated with the permutation inversion group revisited: symmetry projection of the rovibrational functions of methane

In this work the analysis of the equivalent rotations from the permutation inversion group formalism is revisited. We emphasize that explicit knowledge of changes in the Euler angles are not required in order to determine the transformation that a given symmetry operation causes to the rotational functions when dealing with the permutation inversion group formalism. Indeed, matrix elements of the equivalent rotations are provided by a single Wigner's D ^(j)(R) function. Taking advantage of this, we propose a symmetry projection approach to build the rovibrational functions of methane. This approach focuses on the relevance of the isomorphism between permutations and equivalent rotations. In our method, symmetry adapted functions are obtained by simultaneous diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner's D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits us to obtain in a systematic fashion an orthonormal set of symmetry-projected functions, with good total angular momentum, and carrying the irreducible representations of the molecular symmetry group.     

The rovibrational levels of ammonia

In this paper we report variational rovibrational studies on ammonia and its isotopomers. We use six internal coordinates (one of which describes the umbrella motion). The expansion functions are products of one-dimensional functions of these internal coordinates, multiplied by appropriate functions of the Euler angles to describe the rotational motion. We use a previously published high accuracy six-dimensional potential energy surface [LEONARD, C., HANDY, N. C., CARTER, S., and BOWMAN, J. M., 2002, Spectrachim. Acta, 58, 825]. We derive the full kinetic energy operator for the rovibrational motion in these coordinates. This operator has been completely checked to give a hermitian secular matrix. All matrix elements are evaluated numerically by quadrature. The symmetry of the expansion functions is fully described in D_3h, C_2v and C_s. It is not possible to perform the calculations in D_3h, but complete degeneracy in the appropriate levels is obtained with the C_2v program. The algebraic complexity of this program has been far greater than for any other variational study we have undertaken for a tetra-atomic molecule.

We present J = 0, 1, 2 energy levels for the experimentally observed band origins of NH₃, and J = 0, 1 energy levels for the ground state and fundamentals of NH₂D, ND₂H and ND₃. For the asymmetric isotopomers, identical results are obtained for both C_2v and C_s, thus confirming the validity of the method. The levels we obtain are completely converged. Agreement with observation is of the order of 0.5% (of course being dependent upon the accuracy of the potential energy surface); therefore the ordering of the rotational levels and their splitting is completely predicted and understood.     

Calculation of Metric for Gaussian Weave State

Using the recoupling theorem and graph calculation in loop quantum gravity, it is demonstrated that the action of metric matrix operator on Gaussian weave state is an eigenaction, the representation matrix elements of the metric operator and their expectation values are calculated. The values of length of tangent vectors of edges adjacent to the vertex of Gaussian weave state, as well as the angles between them are also obtained in the cases of k=0 and k=2.     

Central configurations of nested rotated regular tetrahedra

In this paper we prove that there are only two different classes of central configurations with convenient masses located at the vertices of two nested regular tetrahedra: either when one of the tetrahedra is a homothecy of the other one, or when one of the tetrahedra is a homothecy followed by a rotation of Euler angles α=γ=0 and β=π of the other one.We also analyze the central configurations with convenient masses located at the vertices of three nested regular tetrahedra when one them is a homothecy of the other one, and the third one is a homothecy followed by a rotation of Euler angles α=γ=0 and β=π of the other two.In all of these cases we have assumed that the masses on each tetrahedron are equal but masses on different tetrahedra could be different.     

度量算符对Gauss编织态的本征作用及自旋几何

对圈量子引力中标架度量矩阵算符对Gauss编织态的作用为本征作用，提供了完整的证明.求得了全部标架度量矩阵算符的表示矩阵，及其期望值.利用自旋几何定理，在内腿颜色k=0和k=2两种情况下，算得了Gauss编织态顶角毗邻的4条腿(P=1)的相位位形切方向间的全部夹角，以及切矢量的长度. 关键词： 度量算符的表示矩阵 度量期望值 切方向间夹角 切矢量长度     

Effective field theory approach to structure functions at small $x_{\mathrm{Bj}}$

We relate the structure functions of deep inelastic lepton-nucleon scattering to current-current correlation functions in a Euclidean field theory depending on a parameter r. The r-dependent Hamiltonian of the theory is P ⁰ -(1-r)P ³ , with P⁰ the usual Hamiltonian and P³ the third component of the momentum operator. We show that a small in the structure functions corresponds to the small r limit of the effective theory. We argue that for there is a critical regime of the theory where simple scaling relations should hold. We show that in this framework Regge behaviour of the structure functions obtained with the hard pomeron ansatz corresponds to a scaling behaviour of the matrix elements in the effective theory where the intercept of the hard pomeron appears as a critical index. Explicit expressions for various analytic continuations of the structure functions and matrix elements are given as well as path integral representations for the matrix elements in the effective theory. Our aim is to provide a framework for truly non-perturbative calculations of the structure functions at small for arbitrary Q². Received: 16 July 2002 / Published online: 9 December 2002 RID="a" ID="a" e-mail: O.Nachtmann@thphys.uni-heidelberg.de     

An Extension of the Beale-Kato-Majda Criterion for the Euler Equations

 The Beale-Kato-Majda criterion asserts that smooth solutions to the Euler equations remain bounded past T as long as is finite, ohgr; being the vorticity. We show how to replace this by a weaker statement, on , where Δ^j is a frequency localization around . Received: 27 February 2002 / Accepted: 29 July 2002 Published online: 14 November 2002     

Crystal field parametrizations for low symmetry systems

For lanthanide ions in sites with no axis of symmetry (C₁ or C_i symmetry) there is no obvious choice for the quantization axis. In these cases, the orientation of the lanthanide site may be rotated by three arbitrary Euler angles ?₁ , θ, and ?₂ , where ?₁ and θ determine the orientation of the z axis, and ?₂ determines the orientation of the crystal about the z axis. We show that appropriate selection of these three Euler angles allows the number of independent crystal-field parameters to be reduced from 27 to 24. This paper explores the possible ways of performing this parameter reduction, and presents the relationship between the different sets of resulting crystal-field parameters.     

General multipole expansion of polarization observables in deuteron electrodisintegration

Formal expressions are derived for the multipole expansion of the structure functions of a general polarization observable of exclusive electrodisintegration of the deuteron using a longitudinally polarized beam and/or an oriented target. This allows one to exhibit explicitly the angular dependence of the structure functions by expanding them in terms of the small rotation matrices d ^j _m'm(θ), whose coefficients are given in terms of the electromagnetic multipole matrix elements. Furthermore, explicit expressions for the coefficients of the angular distributions of the differential cross-section including multipoles up to L _max = 3 are listed in tabular form. Received: 19 November 2002 / Accepted: 7 May 2002     

Calculation of the geometry of the complex of spatially hindered quinones and phenols from E.S.R. spectra of radical pairs

Geometry of the binary complex of 3,6-di-t-butyl-o-quinone and 2,4,6-tri-t-butyl phenol is investigated. Photolysis of this system produces two types of radical pairs with unequal zero-field splitting tensors. The values of the components, D and E, for each type of radical pair were calculated by diagonalizing the dipole-dipole interaction matrix; the elements of this matrix were calculated by using spin density distributions, either known from the experiment or obtained theoretically. The four parameters (three coordinates of the centre of the benzene ring in the phenoxy radical, and one Euler angle) were found by variational search techniques, through comparing the experimental and calculated values of the zero-field splittings. It is shown that benzene rings of the two radicals are displaced with respect to one another by approximately 1·6 Å and rotated relative to one another by 86·5°. This configuration is interpreted in terms of interaction between the t-butyl groups, hydroxyl and carbonyl groups.     

The measurement of frequency separations in high-resolution nuclear magnetic resonance spectra using a double-lock spectrometer

The rotation-ring-puckering problem of four-membered ring cyclobutane-type molecules with D _4h -D _2d conformations is formulated as a dynamical system in the eulerian angles and the polar coordinate ?. Classical and quantum-mechanical hamiltonians and their symmetry group O ⁺⁺(3) × D _∞h are discussed. The kinetic energy matrix coefficients g_mm are found to be irrational functions of the trigonometric polynomial 3 + cos 2? in the internal coordinate ?. It is shown that the energy matrix coefficients may all be expressed as Laurent coefficients by means of the hypergeometric function ₂ F ₁, the convergence being determined by the roots of the kinetic determinant. Selection rules and matrix elements for rotation-ring-puckering Raman transitions are given.     

Vibration-rotation wavefunctions and energies for any molecule obtained by a variational method

This paper presents an alternative method to the usual approach via perturbation theory for the determination of vibrational-rotational energy levels of a molecule in a given electronic state. It is assumed that the electronic Born-Oppenheimer equation has been solved, by an ab initio method, to give a potential function which is used in the nuclear Born-Oppenheimer equation. But the method can also be used with any potential obtained by any method. An approximate solution to the nuclear equation is derived in the form of a linear combination of expansion functions, the coefficients being determined by the standard linear variational method. Angular momentum theory is used to show that the nuclear wavefunction for m = 0 can be represented by a linear combination of functions of the form

where qi are variables which are closely related to the vibrational normal coordinates, and β, γ are two of the Euler angles. m is the eigenvalue of the Z-component of angular momentum operator in space-fixed axes OXYZ. The H_vi (qi) denote Hermite polynomials while Y^J8 (β, γ) are spherical harmonics. It is explicitly shown how all the matrix elements can be evaluated using a (3N–6) dimensional numerical integration technique. The theory in its present form is not suitable for molecules which are linear in the equilibrium configuration. In the following paper the method is used in a calculation on the water molecule.     

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

 A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in L ^p and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields naturally arise in the study of the behavior of entropy solutions of the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics. Received: 7 May 2002 / Accepted: 2 December 2002 Published online: 2 April 2003 Communicated by P. Constantin     

Elastic scattering of 81 keVγ rays by aluminium,nickel, tantalum,gold and lead

A recently reported study [Phys. Rev. A49, 3664 (1994)] of elastic scattering of 81 keVγ rays in the angular range from 60° to 133° has been extended to smaller and larger angles. Previously reportedS matrix calculations of atomic Rayleigh scattering have been shown to require a subtraction of contributions from spurious resonances. Most of the experimental data are in agreement with the calculations. Calculations (MF + ASF) based on a combination of relativistic modified form factors (MF’s) and angle independent anomalous scattering factors (ASF’s) are found to be inadequate for an explanation of experimental cross-sections in the case of highZ elements at angles larger than about 120°.     

Relativistic corrections for the vibration-rotation levels of the ground electronic state of the hydrogen molecular cation H+ 2

Previous work [Moss, R. E., and Valenzano, L., 2002, Molec. Phys., 100, 649 and 1527] on the non-adiabatic properties of the vibration-rotation levels of the ground electronic state of the hydrogen molecular cation is extended to the calculation of relativistic corrections. Unlike the earlier calculations, in which all matrix elements were evaluated analytically, numerical methods are needed for some of the integrals.     

The β angle as the CP violating phase in the CKM matrix

The CKM matrix describing quark mixing with three generations can be parameterized by three Euler mixing angles and one CP violating phase. In most of the parameterizations, the CP violating phase chosen is not a directly measurable quantity and is parameterization dependent. In this work, we propose to use the most accurately measured CP violating angle β in the unitarity triangle as the phase in the CKM matrix, and construct an explicit β parameterization. We also derive an approximate Wolfenstein-like expression for this parameterization.     

Spontaneous decay controlled inversionless laser in a V-type four level system

By using a semiclassical approach, we obtain 16 coupled optical Bloch equations involving density matrix elements for a V-type four level system with three closely spaced upper levels irradiated by a single mode of the electromagnetic field. The off-diagonal elements of the density matrix ρ_ij are in general complex and arise due to the interference between the probability amplitudes of the levels i and j. Through the complex off-diagonal density matrix elements, we introduce the phase angles between the levels participating in dipole allowed/forbidden transitions. Therefore, the phase angles may be regarded as the outcome of the quantum coherence. Under rotating wave approximation, we use a perturbative approach to solve these Bloch equations analytically. These solutions for density matrix elements are used to obtain a closed form analytical expression for the imaginary part of the polarization and hence the absorptive lineshape. The proper choice of the decay constants induces the necessary coherence responsible for inversionless laser.     

Attempt at a gauge-theory-based explanation of the mass spectrum and number of light neutrinos

The symplectic group Sp(n/2) of invariance of flavors of n Majorana states (n is even) does not admit the existence of invariant Majorana masses. Only a specific mass matrix involving diagonal and off-diagonal elements is possible. A mass matrix as a result of spontaneous flavor- and chiral-symmetry breaking may appear here only in the case where the number of flavors is n = 6 and only together with spontaneous R- and L-symmetry violation—that is, parity violation. As a result, three light and three heavy Dirac particles (neutrinos) are present if the seesaw mechanism is operative. Special features of the observed spectrum of light neutrinos—in particular, the fact that two states are far off the third one—can be explained by simple properties of the mass matrices arising in Sp(3). The arrangement of states corresponds to an ordinary mass hierarchy. The mixing angles for physical neutrinos cannot be determined without understanding the mechanisms responsible for the formation of the charged-lepton spectrum and the weak current of Majorana states.     

Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy

Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated under the localized induction approximation for the Euler equations. Using the method of matched asymptotic expansions in a small parameter ε, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(ε³), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. It is found that there is an integrable line in the core whose evolution obeys a summation of the first and third terms of the localized induction hierarchy. Received 2 October 2001 / Received in final form 10 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: yasuhide@math.kyushu-u.ac.jp     

Effective-range expansion for two coupled channels and properties of bound states

The S matrix and the scattering-amplitude matrix (F matrix) are considered for the case of two coupled elastic-scattering channels differing by the values of the orbital angular momentum (l ₁ and l ₂ = l ₁ + 2). The matrix elements of the S and F matrices in the absence of Coulomb interaction are expressed in terms of the matrix elements of the matrix K ⁻¹ inverse to the reaction K matrix. The elements of the K ⁻¹ matrix are written in the form of expansions that are generalizations of the single-channel effective-range expansion. If there is a bound state in the system of colliding particles, then an analytic continuation of these expansions to the region of negative energies makes it possible to obtain both the position of the pole corresponding to this bound state and the residues of scattering amplitudes at this pole, the respective vertex constants and asymptotic normalization coefficients being expressed in terms of these residues. By way of example, the developed formalism is applied to describing triplet neutron-proton scattering.