Vibration-rotation matrix elements for diatomic molecules; vibration-rotation interaction functions Fv′v (m) for CO

General expressions for the fourth-order vibrational matrix elements are obtained for the transitions v → v′ (v ? v′ v + 4) using a sixth-power internuclear potential and a quartic dipole moment function. The results for the dipole moment coefficients M₀ to M₄ of CO and for some transition moments R^v′_v deduced from a calculation including successively second, third and fourth order perturbation theory are compared.Using the rotational potential function expanded through the cubic term as a perturbation, we have also obtained general expressions for the vibration-rotation matrix elements. The vibration-rotation interaction functions F^v′_v (m) are calculated for the transitions v → v′ (v = 0, 10, 20 and v ? v′ ? v + 4) of CO and the coefficients C, D, E and G of the quartic polynomials fitting these functions are deduced. Taking account of uncertainties in the matrix elements R⁰₀ to R⁴₀, an error estimate for the coefficients of F²₀(m) and F³₀(m) is given.     

Fourth-order contributions to the rotational-vibrational matrix elements for diatomic molecules; application to the Fς′v(m) functions for CO

Using the rotational potential function expanded through the quartic term as a perturbation, and considering successively a quartic and a quintic dipole moment function, general expressions are obtained for the vibration-rotation matrix elements corresponding to the transitions vJ→?′J′ with ?′?v+5.The vibration-rotation interaction functions F^?′_v(m) are deduced for the transitions v→?′ (v=0, 10, 20 and?′?v+4) of CO from calculations including third then fourth-order contributions to the matrix elements. The results obtained for the coefficients of the quartic polynomials fitting these functions are compared and their validity is discussed.     

Theoretical infrared transition matrix elements for the water molecule

Using the Hartree-Fock-Roothaan method, symmetric potential energy and dipole moment functions have been obtained for the ground state of the water molecule. These functions were transformed and analyzed in terms of the normal symmetric stretching (v₁) and bending (v₂) coordinates for the calculation of i.r. transition matrix elements. There is satisfactory agreement between this work and the experimental values for the transition matrix element of the v₂ fundamental. However, a discrepancy of a factor of 2.5 is observed between calculated and experimental matrix elements for the v₁ fundamental.     

Perturbation theory for the intensity of Stark lines of a hydrogen atom

Perturbation theory for the wave function of a hydrogen-like atom in a homogeneous electric field of strength F makes it possible to obtain the Rayleigh-Schrödinger series with the coefficients of F ^N (N=0, 1, 2,...) being linear combinations of the Sturm function, which represents the unperturbed state, with 8N ² functions of the corresponding complete set with indices adjacent to the parabolic quantum number of the initial level. A method for recursive analytic calculation of the coefficients of the linear combination for any order N is developed. General expressions for corrections to the matrix elements and intensities of the radiation transitions between Stark sublevels are obtained. Analytic formulas and numerical values of the corrections up to the fourth order for the Lyman and Balmer series are presented. A comparison with the available data for transitions between the Stark components of Rydberg states is given.     

A generalized internal axis method for high barrier tunneling problems,as applied to the water dimer

When more than one large-amplitude vibrational motion is present in a molecule, it is often not possible to define a global internal-axis-method (IAM) coordinate system and set of basis functions. In the present work, a method is presented for extending the IAM treatment to tunneling problems in such cases, using as an illustration a model for the water dimer with three large-amplitude vibrational coordinates. The method involves the construction of two different sets of local IAM-like coordinate systems. The first of these contains n coordinate systems, one for the small neighborhood surrounding each of the n equilibrium frameworks. The second contains on the order of $\text{n}{}^2\text{2}$ coordinate systems, one for each feasible tunneling path between each pair of frameworks. Basis functions written in the second set of local IAM-like coordinates are used to determine the complex phase factors associated in this method with tunneling matrix elements of the phenomenological rotational Hamiltonian in the high barrier limit. These phase factors govern the way in which the various real tunneling frequencies in the molecule constructively and/or destructively interfere in the Hamiltonian matrix elements and final energy expressions. Various mathematical approximations are involved in using the local IAM-like basis sets to obtain matrix elements; the full extent of the adverse effects of these approximations will not be known until an attempt to fit experimental data is carried out.     

Application of symmetry coupling coefficients in zero-field splitting of d5 ion complexes

Symmetry coupling coefficients and symmetry reduced matrix elements are used in calculating the minimum required matrix elements required to determine the analytical expressions for the three ⁶ A ₁ ground-state energy levels for a d⁵ ion in a weak crystal field of tetragonal symmetry. Expressions for the spin hamiltonian parameters a, D, F and G are derived, from which the effect of spin-orbit coupling and the tetragonal component of the crystal field are examined. The parameters describe the form of the hamiltonian and consequently the spin eigenfunctions and eigenvalues of the ground state where a is a measure of the isotropic component whereas D, F and G measure the tetragonal component of the interaction giving rise to the zero-field splitting. Some experimental results are examined.     

Representation of the Coulomb Matrix Elements by Means of Appell Hypergeometric Function <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

Exact analytical representation for the Coulomb matrix elements by means of Appell’s double series F₂ is derived. The finite sum obtained for the Appell function F₂ allows us to evaluate explicitly the matrix elements of the two-body Coulomb interaction in the lowest Landau level. An application requiring the matrix elements of Coulomb potential in quantum Hall effect regime is presented.     

F-theorem, duality and SUSY breaking in one-adjoint Chern-Simons-Matter theories

We extend previous work on N=2 Chern-Simons theories coupled to a single adjoint chiral superfield using localization techniques and the F-maximization principle. We provide tests of a series of proposed 3D Seiberg dualities and a new class of tests of the conjectured F-theorem. In addition, a proposal is made for a modification of the F-maximization principle that takes into account the effects of decoupling fields. Finally, we formulate and provide evidence for a new general non-perturbative constraint on spontaneous supersymmetry breaking in three dimensions based on Q-deformed S³ partition functions. An explicit illustration based on the known analytic solution of the Chern-Simons matrix model is presented.     

Effective interactions and charges in58Ni

The structure of the low-lying states of⁵⁸Ni has been calculated in shell model by assuming an inert⁵⁶Ni core plus two valence nucleons in the p_3/2, f_5/2 and p_1/2 orbitals. The two-body matrix elements are first expressed in terms of seven radial matrix elements and these are then parametrized to give best fit between the computed and the observed energies of the levels below 4 MeV. The wave-functions obtained using these two-body matrix elements are used to study the concept of effective charges. It is found that a single effective charge is not sufficient to predict theB(E2) rates equally well for the thirteen known transitions for which experimental values are available. Assumption of state-dependent effective charges gives a far better agreement. An analysis using wavefunctions obtained with Kuo’s two-body matrix elements also gives a similar result.     

Stereo vision sensor calibration based on random spatial points given by CMM

This paper presents a new, high-precision calibration method for stereo vision sensor based on virtual template. Given 3D spatial points coordinates by coordinates measurement machine (CMM) and corresponding image points coordinates, the technique is realized by taking the projection matrix elements as unknown and using singular value decomposition to solve the least-squares solutions. The method avoids solving intrinsic and extrinsic parameters of each camera and introducing the calibration error which caused by template machining and measurement error. We measure the 3D coordinates of a group of random spatial points by the calibrated stereo vision sensor then compare the results with CMM measurement values, the errors are less than 0.05 mm in the Z axis direction, and less than 0.01 mm in the X and Y axis direction, the experiment results show that the technique is feasible and effective.     

Su(4) approach to fourfold degenerate electronic states of some hexafluoride molecules

We show that two appropriate realizations of the su(4) algebra allow the construction of all electronic operators needed for the study of vibronic and rovibronic interactions in a G^′ electronic state. In each case a full bosonic realization is made and all matrix elements are calculated. Illustrations of our formalism and comparisons with previous approaches are made in the case of ν₅(F_2g) and ν₃(F_1u) modes.     

Spectroscopy of XY2Z2 (C2v) Molecules: A Tensorial Formalism Adapted to the O(3)⊃Td⊃C2v Chain. Application to the Ground State of SO2F2

A tensorial formalism adapted to the case of quasi-spherical XY₂Z₂ asymmetric tops such as SO₂F₂ has been developed as an extension of the usual one for the tetrahedral molecules. We use the O(3)⊃T_d⊃C_2v group chain. All the coupling coefficients and formulas for the computation of matrix elements are given for this chain. Such relations are then deduced in the C_2v group itself. We also present a development of the Hamiltonian, dipole moment, and polarizability operators for the molecules under consideration using this formalism. These operators are involved in the calculation of the energies and intensities of rovibrational transitions and are essential for spectrum simulations. Expressions for the matrix elements are derived for these operators. A first application to the ground state of SO₂F₂ is presented. Programs for spectrum simulation and fit using these methods are freely available at the URL http://www.u-bourgogne.fr/LPUB/c2vTDS.html.     

Allowing for Coulomb effects in the effective range expansion for two coupled channels

A Coulomb-modified matrix of scattering amplitudes (an [(F)\tilde]\tilde F matrix) is considered for the case of two coupled channels of elastic scattering of charged particles with different orbital angular momenta (l ₁ and l ₂ = l ₁ + 2). Matrix elements of the [(F)\tilde]\tilde F matrix are expressed in terms of the matrix elements of a [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix inverse to a modified reaction K matrix. The elements of the [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix are written as expansions that are generalizations of single-channel effective range expansion with allowance for the Coulomb interaction. If a system of colliding particles involves a bound state, the analytic continuation of these expansions into the region of negative energies makes it possible to obtain both the position of the pole corresponding to the bound state and the scattering amplitude residues in this pole, in terms of which the corresponding vertex constants and asymptotic normalization coefficients are expressed.     

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.     

Twist-4 gluon recombination corrections for deep inelastic structure functions

We calculate twist-4 coefficient functions for the deep inelastic structure function F₂(x,Q²) associated to 4-gluon operator matrix elements for general values of the Bjorken variable x and study the numerical effect on the slope ∂F₂(x,Q²)/∂logQ². It is shown that these contributions diminish the strongly rising twist-2 terms towards small values of x.     

An analytical model of the heat of chemisorption: A perturbation LCAO MO approach

An analytical LCAO MO perturbation model has been developed for treating the heat of chemisorption Q of an adsorbate A monolayer on a transition metal M film. The model combines parameters of the metal band (the Fermi level E_F, band width W = W^occ + W^vac, the d occupancy N_d, density of states n(?), etc.) with those of the local A-M interactions (the adorbital energy ?_A, off-diagonal matrix elements β_AM, etc.). The major cases of A's having lone pair, singly occupied, and vacant adorbitals have been considered, and the analytical expressions for Q as well as some numerical estimations are presented. The relative values of Q seem to be crucially dependent on the ratio β/(?_A ? E_F). The Q vs. N_d plots for the donor and radical A's are rather flat, typically Q decreasing monotonically as N_d increases, but for the acceptor A's the plots are very parameter dependent and show a variety of trends. The results obtained agree with straightforward computations and (scarce) experimental data.     

Hopping conductivity of carbynes modified under high pressures and temperatures: Galvanomagnetic and thermoelectric properties

The conductivity, thermopower, and magnetoresistance of carbynes structurally modified by heating under a high pressure are investigated in the temperature range 1.8–300 K in a magnetic field up to 70 kOe. It is shown that an increase in the synthesis temperature under pressure leads to a transition from 1D hopping conductivity to 2D and then to 3D hopping conductivity. An analysis of transport data at T ≤ 40 K makes it possible to determine the localization radius a ～ (56?140) Å of the wave function and to estimate the density of localized states _g(E _F) for various dimensions d of space: _g(E _F) ≈ 5.8 × 10⁷ eV^?1 cm^?1 (d=1), _g(E _F) ≈5×10¹⁴ eV^?1 cm ^?2 (d=2), and _g(E _F)≈1.1×10²¹ eV^?1 cm_?3 (d=3). A model for hopping conductivity and structure of carbynes is proposed on the basis of clusterization of sp ² bonds in the carbyne matrix on the nanometer scale.     

Out-of-plane C m H vibrations and electronic structure of Zn complexes of porphyrins and their π anions

Based on the calculation of frequencies of the normal vibrations of Zn complexes of octaalkylporphyrins (Zn-OAPs), Zn porphin (Zn-P), and their meso-deuterated derivatives by the method of density functional theory (B3LYP/6–31G(d, B3LYP/6–311+G(d)), a new interpretation of C _m H(D) out-of-plane deformation vibrations of porphyrins is proposed. It is shown that the vibration patterns of these modes are characterized by high contributions (>90%) of the coordinates of displacement out of the macrocycle plane of the H(D) atoms in meso positions. At the same time, a small change in the vibration pattern of such a mode in the d ₄ isotopomer (～5%) is accompanied by a large change in its intensity. The geometry of free π anionic forms of Zn-OAPs and Zn-P and their complexes with cations Na⁺ and K⁺ is optimized, and the normal vibrations of these compounds are calculated. A comparison with experimental IR spectra of solid samples shows that anion-cation complexes form under these conditions. The interaction with cations lowers the symmetry of the nuclear skeleton of reduced forms of Zn-OAPs, considerably decreases their charge, and markedly affects the structure of this skeleton and its vibrations.     

On the temperature dependence of the g shift in Nd3+:YES

A temperature-dependent contribution to the g factor of Nd³⁺:YES has been observed. A semi-empirical theory of the phenomena is presented in which the matrix elements of the dynamic crystal field potential are evaluated starting from spin-lattice relaxation rates. The evaluated contribution is in good agreement with the observed g shift.