Ab initio anharmonic force field and equilibrium structure of carbonyl chlorofluoride

The quadratic, cubic, and semi-diagonal quartic force field of OCFCl has been calculated at the MP2 level of theory employing a basis set of triple-zeta quality. The spectroscopic constants derived from the force field are in excellent agreement with those from previous and new experiments. The equilibrium structure has been derived from experimental ground state rotational constants and ab initio rovibrational interaction parameters. This semi-experimental structure is in excellent agreement with the ab initio structure calculated at the CCSD(T) level of theory. This good agreement indicates that the derived structure is accurate. The equilibrium geometry is: r_e(CO)=1.173(1) Å; r_e(C-F)=1.323(1) Å; r_e(C-Cl)=1.721(1) Å; ∠_e(OCF)=124.0 (1)°; and ∠_e(OCCl)=126.4(1)°.     

Rovibronic Levels for the (1-3)Πg Manifold of B2: Exterior Complex Scaling Finite Element Calculations Based on an Adiabatic and a Strictly Diabatic Basis

A one-dimensional complex scaled finite element method was applied on an adiabatic basis of B₂ in order to find rovibronic term energy values for the (1)³Π_g; (v, N)=(0-8, 0-25) and (2)³Π_g; (v, N)=(0-10, 0-25) levels. The method was also applied to the (1)³Π_g; (v, N)=(0-8, 0-25) and (2)³Π_g; (v, N)=(0-1, 0-25) levels in a strictly diabatic framework. Adiabatic single-channel and diabatic coupled-channel total wavefunctions were obtained and used in order to identify the vibrational levels. Comparing levels for the interacting two-state (1-2)³Π_g and three-state (1-3)³Π_g system, a constant energy shift of about 1.7 cm⁻¹ is found. Comparisons between the adiabatic Born-Oppenheimer (BO) and the diabatic (1)³Π_g; (v, N)=(0-8, 0-25) levels show differences between −20 and 7 cm⁻¹, while the corresponding shifts for the (2)³Π_g; (v, N)=(0, 0-25) and (1, 0-25) levels are about 50 and 60 cm⁻¹, respectively. A comparison between our three-state approximation and experimental observations of the (2)³Π_g-A³Π_u electronic transition shows a difference in the line positions of about 665 cm⁻¹. The calculated widths for all but the (1)³Π_g; (v, N)=(7-8, 0-25), as well as the (2)³Π_g; (v, N)=(0, 0-25) BO and diabatic rovibronic levels, have small but with N increasing predissociation rates. The (1)³Π_g; (v>8, N=0-25) BO and the (2)³Π_g; (v, N)=(1, 0-25) diabatic levels are strongly predissociated with widths ≥16 cm⁻¹.     

N = 3 and N = 1 supersymmetry in a new class of solutions for d = 11 supergravity

We present a new class of compactifying solutions for d = 11 supergravity. The internal 7-spaces are described by coset manifolds N^pqr of the form SU(3) × U(1)/U(1) × U(1). The three integers p, q, r characterize the embedding of the stability subgroup U(1) × U(1) in SU(3) × U(1).Their supersymmetry content is quite remarkable. For a particular choice of p, q, r the isometry of N^pqr is SU(3) × SU(2): in this case we find that N = 3 supersymmetry survives. For all the other values of p, q, r, supersymmetry is broken to N = 1, and the isometry group is SU(3) × U(1).We also find a class of solutions with internal photon curl F_αβγδ ≠ 0, breaking all supersymmetries.     

Existence of Noncompact Static Spherically Symmetric Solutions of Einstein SU(2)-Yang–Mills Equations

We consider static spherically symmetric solutions of the Einstein equations with cosmological constant Λ coupled to the SU(2)-Yang–Mills equations that are smooth at the origin r=0. We prove that all such solutions have a radius r _c at which the solution in Schwarzschild coordinates becomes singular. However, for any positive integer N, there exists a small positive Λ _N such that whenever 0<Λ<Λ _N , there exist at least N distinct solutions for which the singularity is only a coordinate singularity and the solution can be extended to r≥r _c . Received: 5 June 2000 / Accepted: 13 March 2001     

The high-resolution infrared spectrum of pyrrole between 900 and 1500 cm revisited

The Fourier transform gas-phase infrared spectrum of pyrrole, C₄H₅N, has been recorded with a resolution of ca. 0.003 cm⁻¹ in the 900-1500 cm⁻¹ spectral region. Four fundamental bands, ν₈(A₁; 1016.9 cm⁻¹), ν₂₃(B₂; 1049.1 cm⁻¹), ν₇(A₁; 1074.6 cm⁻¹), ν₂₀(B₂; 1424.4 cm⁻¹) and the overtone band 2ν₁₆(A₁; 962.7 cm⁻¹) have been analysed using the Watson model. The ν₈ and 2ν₁₆ bands are unperturbed; the ν₇ and ν₂₃ bands are locally perturbed, while the ν₂₀ band is globally perturbed by weak c-Coriolis resonance. Upper state vibrational term values, and rotational and centrifugal distortion constants, have been obtained from fits using S-reduction and I^r-representation as well as A-reduction and III^r-representation. A set of ground state rotational and centrifugal distortion constants using A-reduction was obtained from a simultaneous fit of ground state combination differences from all five bands and previous microwave and millimetre-wave data.     

The elastic behavior of dense C3N4 under high pressure: First-principles calculations

We have carried a detailed theoretical study on the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c- and p-C₃N₄ compounds under a maximum of pressure up to 100 GPa by using first principles calculations. The optimized lattice constants under zero pressure and zero temperature agreed well with the previous experimental and theoretical results. The band gaps of the four types of dense C₃N₄ were widened gradually with the increase of pressure. The calculated Poisson’s ratio γ and B/G values suggest α-, c- and p-C₃N₄ are brittle materials under 0–100 GPa, whereas β-C₃N₄ will become a ductile material as external pressure reaches 57 GPa. We found that the Debye temperature of the four dense C₃N₄ gradually reduces in the order of c-C₃N₄>p-C₃N₄>α-C₃N₄>β-C₃N₄ at 0 GPa and 0 K. However, the Debye temperature of c-C₃N₄ was lower than p-C₃N₄ when external pressure exceeds 6.3 GPa. It may hint that the results could be served as a valuable prediction for further experiments.     

N-Vortex Equilibria for Ideal Fluids in Bounded Planar Domains and New Nodal Solutions of the sinh-Poisson and the Lane-Emden-Fowler Equations

We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain ${\Omega\subset\mathbb{R}^2}We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain W ì \mathbbR²{\Omega\subset\mathbb{R}^2} , which is not necessarily simply connected. On an arbitrary bounded domain we obtain new equilibria for N = 3 or N = 4. If Ω has an axial symmetry we obtain a symmetric equilibrium for each N ? \mathbbN{N\in\mathbb{N}} . We also obtain new stream functions solving the sinh-Poisson equation -Dy = rsinhy{-\Delta\psi=\rho\sinh\psi} in Ω with Dirichlet boundary conditions for ρ > 0 small. The stream function y_r{\psi_\rho} induces a stationary velocity field v_r{v_\rho} solving the Euler equation in Ω. On an arbitrary bounded domain we obtain velocitiy fields having three or four counter-rotating vortices. If Ω has an axial symmetry we obtain for each N a velocity field v_r{v_\rho} that has a chain of N counter-rotating vortices, analogous to the Mallier-Maslowe row of counter-rotating vortices in the plane. Our methods also yield new nodal solutions for other semilinear Dirichlet problems, in particular for the Lane-Emden-Fowler equation -Du=|u|^p-1u{-\Delta u=|u|^{p-1}u} in Ω with p large.     

The complex Lorenz equations

We have undertaken a study of the complex Lorenz equations

$\text{x}\text{?}\text{= ?σx + σy}$

.

$\text{y}\text{?}\text{= (r ? z)x ? ay}$

.

$\text{z}\text{?}\text{= ?bz +}\text{1}\text{2}\text{(x}{}^1\text{y + xy}{}^1\text{)}$

. where x and y are complex and z is real. The complex parameters r and a are defined by r = r₁ + ir₂; a = 1 ? ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r₂ = 0. We have been able to determine analytically two critical values of r₁, namely r1_c and r1_c . The origin is a stable fixed point for 0 < r₁ < r_1c, but for r₁ >r_1c, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ <b + 1. If σ > + 1 then this limit is only stable in the region r_1c < r₁ < r_lc. When r₁ >r_lc, a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.     

Adjacency networks

We consider a finite set S={x₁,…,x_r} and associate to each element x_i a probability p_i. We then form sequences (N-strings) by drawing at random N elements from S with respect to the probabilities assigned to them. Each N-string generates a network where the elements of S are represented as vertices and edges are drawn between adjacent vertices. These structures are multigraphs having multiple edges and loops. We show that the degree distributions of these networks are invariant under permutations of the generating N-strings. We describe then a constructive method to generate scale-free networks and we show how scale-free topologies naturally emerge when the probabilities are Zipf distributed.     

Extracting d-orbital occupancy from magnetic Compton scattering in bilayer manganites

We consider the shape of the magnetic Compton profile (MCP), J_mag(p_z), in La_1.2Sr_1.8Mn₂O₇ for momentum transfer p_z along the [110] direction and the associated reciprocal form factor B(r) defined by taking the one-dimensional Fourier transform of J_mag(p_z). B(r) is shown to contain a prominent dip at r≈1 Å, where the minimum value B_min of B(r) can be related to the occupancies of the e_g orbitals of dx²−y² and d3z²−r² symmetry in the system. We illustrate our procedure in detail by analyzing the measured MCP at 5 K and the MCP computed within the framework of the local spin density approximation (LSDA) and comment on the differences between the measured and computed e_g occupancies as a reflection of the limitations of the LSDA in treating electron correlation effects.     

A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows

An immersed boundary method (IBM) with second-order spatial accuracy is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method is based on the computationally efficient direct-forcing method of Uhlmann [M. Uhlmann, An immersed boundary method with direct forcing for simulation of particulate flows, J. Comput. Phys. 209 (2005) 448–476] that is embedded in a finite-volume/pressure-correction method. The IBM consists of two grids: a fixed uniform Eulerian grid for the fluid phase and a uniform Lagrangian grid attached to and moving with the particles. A regularized delta function is used to communicate between the two grids and proved to be effective in suppressing grid locking. Without significant loss of efficiency, the original method is improved by: (1) a better approximation of the no-slip/no-penetration (ns/np) condition on the surface of the particles by a multidirect forcing scheme, (2) a correction for the excess in the effective particle diameter by a slight retraction of the Lagrangian grid from the surface towards the interior of the particles with a fraction of the Eulerian grid spacing, and (3) an enhancement of the numerical stability for particle–fluid mass density ratios near unity by a direct account of the inertia of the fluid contained within the particles. The new IBM contains two new parameters: the number of iterations N_s of the multidirect forcing scheme and the retraction distance r_d. The effect of N_s and r_d on the accuracy is investigated for five different flows. The results show that r_d has a strong influence on the effective particle diameter and little influence on the error in the ns/np condition, while exactly the opposite holds for N_s. A novel finding of this study is the demonstration that r_d has a strong influence on the order of grid convergence. It is found that for spheres the choice of r_d = 0.3Δx yields second-order accuracy compared to first-order accuracy of the original method that corresponds to r_d = 0. Finally, N_s = 2 appears optimal for reducing the error in the ns/np condition and maintaining the computational efficiency of the method.     

The model of the optical switching center and the polarization anomalies in absorption spectra of cupric oxide after fast particles irradiation

We consider the model of the optical switching center—a system with the following properties: it has two (or more) metastable states |1〉, |2〉, separated by a potential barrier U; it can switch from one state to another by absorbing the photons with energy ?ω∼U; the transition |1〉→|2〉 is allowed only for a certain light wave polarization p₁ and the transition |2〉→|1〉—for other polarization p₂; these polarizations p₁, p₂ are orthogonal. The optical properties of this system are studied and are found to exhibit unconventional polarization dependence. In particular, the absorption spectrum observed in natural (unpolarized) light can display new features, that are absent in the spectra, obtained in two independent polarizations. We discuss these results in connection with the (yet unexplained) experimental findings [N.N. Loshkareva, Yu.P. Sukhorukov, B.A. Gizhevskii, A.S. Moskvin, T.A. Belykh, S.V. Naumov, A.A. Samokhvalov, Phys. Solid State 40 (1998) 383], where the similar anomalous polarization dependence of the absorption spectra of cupric monoxide CuO after the fast particle bombardment is reported.     

E.S.R. evidence for the planarity of the t-butyl radical

Monte Carlo calculations are reported for the radial distribution function g ₂(r; λ) of a fluid in which the intermolecular pair potential is [u ^ref(r) + λu ^p(r)], u ^ref(r) being the Weeks-Chandler-Andersen (WCA) reference fluid, and [u ^ref(r) + u ^p(r)] being the Lennard-Jones (6, 12) fluid. The calculations are performed for λ values in the range 0 to 1, at the state condition ρσ³ = 0·80, kT/ε = 0·719. It is shown that at high densities the perturbation expansion of g ₂(r; λ = 1) about g ₂(r; λ = 0) is rapidly convergent, but that the corresponding expansion for y ₂(r; λ) = exp [βu(r; λ)] × g ₂(r; λ) is not. In addition Monte Carlo estimates of the individual terms that contribute to the first-order perturbation term, (?g ₂/?λ)_λ=0, are presented. It is shown that these terms are individually large, but that (?g ₂/?λ)_λ=0 is small because there is strong cancellation between the various terms. Consequently, the calculation of (?g ₂/?λ)_λ=0 is highly sensitive to the approximation used to evaluate the individual terms.     

Hadron diffractive processes: the structure of soft pomeron and colour screening

On the basis of the experimental data on diffractive processes in πp, pp and pˉp collisions at intermediate, moderately high and high energies, we restore the scattering amplitude related to the t-channel exchange by vacuum quantum numbers by taking account of the diffractive s-channel rescatterings. At intermediate and moderately high energies, the t-channel exchange amplitude turns, with a good accuracy, into an effective pomeron which renders the results of the additive quark model. At superhigh energies the scattering amplitude provides a Froissart-type behaviour, with an asymptotic universality of cross sections such as σ^tot _πp/σ^tot _pp→ 1 at s→∞. The quark structure of hadrons being taken into account at the level of constituent quarks, the cross sections of pion and proton (antiproton) in the impact parameter space of quarks, σ_π(r _1⊥, r _2⊥; s) and σ_p(r _1⊥, r _2⊥, r _3⊥; s), are found as functions of s. These cross sections implicate the phenomenon of colour screening: they tend to zero at |r _i⊥−r _k⊥|→ 0. The effective colour screening radius for pion (proton) is found for different s. The predictions for the diffractive cross sections at superhigh energies are presented. Received: 15 December 1998     

High-resolution infrared and theoretical study of the fundamental bands ν6, ν7, ν9 and ν13 of 1,2,3-thiadiazole

The Fourier transform gas-phase IR spectrum of 1,2,3-thiadiazole, C₂H₂N₂S, has been recorded with a resolution of ca. 0.003 cm⁻¹ in the 700-1100 cm⁻¹ spectral region. Four fundamental bands ν₆(A^/; 1101.8 cm⁻¹), ν₇(A^/; 1038.8 cm⁻¹), ν₉(A^/, 858.9 cm⁻¹), and ν₁₃(A^//; 746.2 cm⁻¹) have been analyzed using the Watson model in A-reduction. Two additional bands, ν₈ (A^/; 894.6 cm⁻¹) and ν₁₂(A^//; 881.2 cm⁻¹) were assigned by their weak Q-branches. Ground state rotational and quartic centrifugal distortion constants as well as upper state spectroscopic constants have been obtained from fits. A number of weak global and local interactions are present in the bands. The resonances identified were qualitatively explained by Coriolis type perturbations with neighboring levels. Ground state rotational and quartic centrifugal distortion constants, anharmonic frequencies, and vibration-rotational α-constants predicted by quantum chemical calculations using a cc-pVTZ basis and B3LYP methodology, have been compared with the present experimental data, where there is generally good agreement.     

Franck-condon factors for diatomic molecules and the factorization method

By means of the factorization method, recurrence relations for the matrix elementS$\text{<ψ}{}_{\mathrm{<mtext>v\text{'}</mtext>}}\text{¦r}{}^{\mathrm{N}}\text{¦ψ}{}_{\mathrm{<mtext>v</mtext>}}\text{<}$ for an arbitrary positive integer N are obtained in the framework of α-averaging for the Morse potential. The method of calculating the matrix element$\text{<ψ}{}_{\mathrm{<mtext>v\text{'}</mtext>}}\text{¦R}{}_{\mathrm{<mtext>e</mtext>}}\text{¦ψ}{}_{\mathrm{<mtext>v</mtext>}}\text{<}$ is outlined, provided a dependence of the electronic transition moment R_e on the internuclear distance r can be approximated by the polynomial R_e(r)=a₀+a₁r+a₂r²+…+ainNr^N. Recurrence relations are also obtained for the case of the exponential dependence R_e(r).     

Anharmonic force field and equilibrium structure of nitric acid

The quadratic, cubic and semi-diagonal quartic force field of nitric acid has been calculated at the CCSD(T) level of theory employing a basis set of triple-ζ quality. A semi-experimental equilibrium structure has been derived from experimental ground state rotational constants and rovibrational interaction parameters calculated from the ab initio force field. It is found that the A and B semi-experimental equilibrium rotational constants of the ¹⁸O isotopologues (for which the rotation of principal axes is large) cannot be accurately reproduced. This problem is discussed and a remedy is proposed. Finally, the semi-experimental structure is in agreement with the ab initio structure calculated at the CCSD(T) level of theory using a basis set of at least quadruple-ζ quality and a core correlation correction, except for the long NO single bond for which the CCSD(T) value is too short due to inadequate treatment of electron correlation. The empirical structures are also determined and their accuracy is discussed. The best equilibrium structure is: r_e(NO_syn) = 1.209(1) Å, r_e(NO_anti) = 1.194(1) Å, r_e(NO) = 1.397(1) Å, r_e(OH) = 0.968(1) Å, ∠(ONO_syn) = 115.8(1)°, ∠(ONO_anti) = 114.2(1)° and ∠(NOH) = 102.2(1)°.     

A positive energy dynamics and scattering theory for directly interacting relativistic particles

Starting from the tensor product of N irreducible positive energy representations of the Poincaré group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ($\text{(|k|}{}^2\text{+ 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, k) and a free “reduced Hamiltonian” H_r⁰, which is a positive operator acting only on internal variables, and then replace H_r⁰ by an interacting reduced Hamiltonian H_r = H_r⁰ + V, where V commutes with the Lorentz group and is such that H_r is a positive operator. The resulting product form is shown to imply that the wave operators interwine the free and interacting representations so that the S-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrödinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the $\text{spin}\text{-}\text{1}\text{2}$ case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the $\text{spin}\text{-}\text{1}\text{2}$ case, coupling of a quantized radiation field. In view of eventual applications to “completely integrable” one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.     

Local existence of the borel transform in the Φ24 quantum field theory: A fixed point method

A fixed point argument is applied to the system of equations B = $\text{M}$(B) to be satisfied by the set of Borel transformed euclidean Green functions B = {B_2N(p₁,…,p_2N;μ)} of Φ₂⁴ in a neighbourhood of μ = 0, μ being the Borel variable associated with the coupling λ. We show (for small μ) the existence in an appropriate Banach space, of a unique set B = {B_2N} of functions B_2N satisfying these equations, to which the Borel-transformed perturbative series converge for each N (the convergence of the latter is obtained here as a by-product). An analogous application of this method to the set of one-particle irreducible Green functions of Φ₂⁴, which yields improved results, is also presented.     

The Inviscid Burgers Equation with Initial Value of Poissonian Type

We study the discontinuities (shocks) of the solution to the Burgers equation in the limit of vanishing viscosity (the inviscid limit) when the initial value is the opposite of the standard Poisson process p. We show that this solution is only defined for t ε (0, 1). Let T ₀ = 0 and T _n , n≧1, be the successive jumps of p. We prove that for all M > 0 the inviscid limit is characterized on the region x ε (-∞, M], t ε (0, 1) by the increasing process $N(t) = \sup \{ n \in \mathbb{N} {\text{| }}M + nt > T_n \} $ and the random set I(x) = {n ε {0,..., N(t)}‖T _n -nt≦x<T _n+1 - nt}. The positions of shocks are given in a precise manner. We give the distribution of N(t) and also the distribution of its first jump. We also prove similar results when the initial value is u _μ(y, 0) = -μp(y/μ²) + μ^-1 max(y, 0), μ ε (0, 1).