Third-order Calculations Research Articles

Modelling one-dimensional unsteady flows in ducts: Symmetric finite difference schemes versus galerkin discontinuous finite element methods

The Euler equations for one-dimensional unsteady flows in ducts have been solved resorting to classical symmetric shock-capturing methods with second-order accuracy and to the recent discontinuous Galerkin finite-element method, with second- and third-order accuracy. In particular, the finite difference techniques adopted are the two-step Lax—Wendroff method and the MacCormack predictor—corrector method, with the addition of the flux corrected transport (FCT) or of the Davis nonupwind TVD scheme to suppress the spurious oscillations in the vicinity of discontinuous solutions. The finite-element method adopted is based on the weak formulation of the Euler equations, which are solved by introducing a discontinuous finite-element space discretization. A dissipative mechanism has been considered to supplement the FEM with a “discontinuity capturing” operator, adding a “viscous like” term to damp minor numerical overshoots arising in proximity of steep gradients of the solution. The numerical tests chosen to carry out a comparison between these schemes are the shock-tube problem and the shock—turbulence interaction problem. Both the test cases considered show the superiority of third-order FEM calculations, whereas the comparison between the computer run times points out the greater computational effort required.

A Lagrangian dynamical theory for the mass function of cosmic structures -- I. Dynamics

A new theory for determining the mass function of cosmic structures is presented. It relies on a realistic treatment of collapse dynamics. Gravitational collapse is analysed in the Lagrangian perturbative framework. Lagrangian perturbations provide an approximation of truncated type, i.e. small-scale structure is filtered out. The collapse time is suitably defined as the instant at which orbit crossing takes place. The convergence of the Lagrangian series in predicting the collapse time of a homogeneous ellipsoid is demonstrated; it is also shown that third-order calculations are necessary in predicting collapse. Then, the Lagrangian prediction, with a correction for quasi-spherical perturbations, can be used to determine the collapse time of a homogeneous ellipsoid in a fast and precise way. Furthermore, ellipsoidal collapse can be considered as a particular truncation of the Lagrangian series. Gaussian fields with scale-free power spectra are then considered. The Lagrangian series for the collapse time is found to converge when the collapse time is not large. In this case, ellipsoidal collapse gives a fast and accurate approximation of the collapse time; spherical collapse is found to poorly reproduce the collapse time, even in a statistical sense. Analytical fits of the distribution functions of the inverse collapse times, as predicted by the ellipsoid model and by third-order Lagrangian theory, are given. These will be necessary for a determination of the mass function, which will be given in Paper II.

Highly Accurate ab Initio π-Electron Hamiltonians for Small Protonated Schiff Bases

The effective valence shell Hamiltonian (ℋν) method is used to compute the ab initio π-electron Hamiltonians for the two small protonated schiff bases propeniminium (C3H4NH2+) and pentaniminium (C5H6NH2+), which are small analogs of the photoreceptor chromophore retinal. Because these small polyenes lack low lying, ionic excited states, the ℋν calculations perform remarkably well even though they employ constrained, semiempirical-like valence spaces. In contrast to studies on other polyenes, even the second-order ℋν performs extremely well, while the third-order ℋν calculations yield an effective π-electron Hamiltonian that resembles the familiar semiempirical Pariser−Parr−Pople (PPP) Hamiltonian but which also retains ab initio accuracy for the vertical excitation energies. The ab initio π-electron Hamiltonian need only retain the PPP-type αi, βi,j, and γi,j effective integrals to produce excitation energies that compare well with previous multireference single and double configuration interaction (MRSDCI) calculations. Since the ab initio π-electron effective integrals arise from first principles, they do not require any empirical corrections. However, they do lack the simple transferability generally assumed in semiempirical models for these short-chain polyenes.

Ab initio calculations of QED effects in Li-like, Na-like and Cu-like ions

We review ab initio calculations of the bound-state self-energy and vacuum polarization for resonant transitions of Li-like, Na-like and Cu-like ions with a degree of ionization of about ten or greater. The direct part of the Dirac-Fock potential is included to all orders in the rediative corrections, and exchange effects are added in first order. When combined with previous relativistic correlation calculations, the results agree in most cases with experiment, the most notable exception being the Cu isoelectronic sequence at low Z, where the third-order correlation calculations have probably not converged to experimental precision. For low and intermediate Z, the "screening" effect is found to scale closely as the normalization of the valence wavefunction at the origin, but at high Z deviations from this scaling principle can be significant.

On the real-space renormalization-group study of some 2D quantum spin systems

Abstract Within a real-space renormalization-group framework, the critical behaviour of the spin- 1 2 anisotropic quantum Heisenberg model on the three planar lattices has been studied. To examine the stability of the results with respect to changes in the approximation scheme, various divisions of the lattices and various truncation procedures have been used. It has been shown that there is a non-trivial fixed point and corresponding critical temperature for the nearest-neighbour XY model for each two-dimensional lattice irrespective of the approximation used. Concerning the Heisenberg model, to first order in the cumulant expansion there is no non-trivial fixed point, however, it appears in the second- and third-order calculations. This suggests that the two-dimensional Heisenberg model also exhibits some kind of a critical behaviour. In addition, the phase transitions to the many-sublattice structure have been discussed.

Valence ionization of HCl. An investigation of many-body effects

Many-body Green’s-function calculations which employ large configurational expansions and which are accurate to third and fourth order of perturbation theory are reported for the valence ionic states of HCl. Large polarized basis sets including Rydberg functions s-, p-, and d-type symmetries have been used. Third-order calculations are sufficient in the outer valence region, but in the inner valence region where the complete breakdown of the one-particle picture of ionization is observed a fourth-order theory is necessary in conjunction with large basis sets. The synchrotron and x-ray excited spectra can be assigned nearly up to the double-ionization threshold in a very satisfactory way. No indication of strong outer valence satellite lines is found.

Ab initio g-Hartree calculations on the atoms Be and Ne that employ fully numerical and basis-set techniques.

We report fully numerical calculations of correlation energies in second-order perturbation theory based on Hartree-Fock and g-Hartree mean fields for Be and Ne atoms and show that basis-set techniques give essentially the same results. An extension to higher-order perturbation theory is relatively straightforward in the basis-set approach. We report third-order calculations that yield in the case of Be significantly improved values for the g-Hartree correlation energy which now agrees within 4% with the total correlation energy known from complete configuration-interaction estimates, while third-order Hartree-Fock correlation energies calculated in the same basis set still disagree by 10% due to difficulties in describing the near degeneracy of the 2s and 2p orbitals. In the case of the Ne atom, we find that for the g-Hartree mean field still higher orders of perturbation theory are required to obtain agreement with experiment.

Is charmonium a perturbative system?

The results of a numerical study of the convergence of the pertubation for a quarkonium potential are presented. When the Coulomb interaction is treated as the perturbation, the third-order calculations of the energy leve l differences and the leptonic decay rates for the S-states are in good quantitative agreement with exact computer solutions of Eichten et al. and other authors. Comparisons with first- and second-order calculations indicate reasonable convergence in third order. The rate of convergence of the perturbation theory expressions for the energy levels and leptonic decay rates obtained here may indicate to what extent spin-orbit and spin-spin terms in the Hamiltonian can be treated in low orders perturbatios theory.

Effective electrostatic operators calculated with second quantisation

A general method is given for the calculation of products of electrostatic operators, using the formalism of coupled creation and annihilation operators. Such products appear both in inner products (on which the concept of operator orthogonality is based) and in perturbation theory (where the summations can be evaluated explicitly in the case of so-called 'weak' perturbations). A second-order calculation for the lN configuration is given in detail. In addition, third-order calculations are discussed with stress on the simplifying cancellations that occur between various contributions; an example of a class of perturbations of interest is worked out. The present method has, when compared with previous calculations, the advantage of leading in a natural and a relatively quick way to the final equations. Moreover, the operators are directly classified according to their pure n-particle character (customarily, n is restricted to 1, 2, 3). Such a classification is indispensable in ab initio calculations of parameters.

Perturbative pseudopotential calculations of vacancy formation energies in simple metals

The perturbative pseudopotential method for the calculation of vacancy formation energies for simple metals is examined. The vacancy formation energies at constant volume for Na, Mg, and Al are calculated within the second- and third-order perturbation theories. Two first-principle model pseudopotentials and the linear screening theory are used. The convergence of the perturbative series is examined. The calculated vacancy formation energies are then compared with the experimental values. A detailed analysis of the results shows that the second- and third-order perturbative calculations of the vacancy formation energies of alkali metals and possibly of the alkali-earth metals are applicable. For higher-valency metals, such perturbative calculations are definitely questionable.

Scaling in second-order electron correlation calculations using systematic sequences of even-tempered basis sets

Improved results can often be obtained from second-order Rayleigh-Schrodinger perturbation calculations of electron correlation energies using large basis sets by introducing a scaling factor in the zero-order Hamiltonian. The scaling parameter may be determined from full third-order calculations using a smaller basis set. This scaling procedure can be applied in a systematic fashion by employing a sequence of even-tempered basis sets. Calculations illustrating this approach for the beryllium atom and the neon atom are presented. The scaling procedure is also employed in conjunction with a universal systematic sequence of basis functions. Calculations illustrating this Correlation energy — Mang-body perturbation theory.

Pade approximants and many-body perturbation theory: an application to the CH+ion

Pade approximants to the diagrammatic many-body perturbation expansion for electron correlation energies in molecules are discussed. The numerical convergence properties of the perturbation expansion and the Pade approximants which can be formed from it are compared. Calculations of the ground-state potential energy curve for the CH+ ion are reported. Systems, such as CH+, which contain low-lying excited states, can be handled by forming (2/1) Pade approximants from third-order calculations using 'shifted' denominators.

Third-order perturbation calculations in nuclear matter for realistic potentials

The particle-ladder diagrams up to third order in the basic interaction are calculated for nuclear matter with some recent local soft-core potentials. A detailed comparison to the G-matrix results is carried out. The prescription of zero potential energy for states above the Fermi sea is used. For the de Tourreil-Sprung (dTS) interaction the total third order contribution is 17% of the second order. The de Tourreil-Rouben-Sprung (TRS) potential shows poor convergence in the triplet-odd subspace. A new set of triplet-odd parameters is defined, and the overall convergence for the modified TRS is only slightly less rapid than that for dTS.

Higher order contributions to the transport coefficients of binary gas mixtures

The rate of decrease of higher order corrections to various transport coefficients of binary gas mixtures has been established for a temperature range (200K<or=T<or=2000K) in the case of the LJ (12-7) potential. The authors study shows that, depending upon the experimental accuracy, at least second- order calculations for viscosity and third-order calculations for other coefficients of binary mixtures should be performed to check the validity of an intermolecular potential. In obtaining these results, general expressions for transport coefficients have been written in terms of the inverse of matrices. The utility of these expressions for computer programming has been indicated.

Third-order corrections to the Hartree–Fock approximation for the binding energy of 4He

We present third-order calculations of the binding energy of 4He with an effective (finite) nucleon–nucleon interaction of the Hamada–Johnston type. The intermediate (unoccupied) states are approximated by plane waves in a first calculation and in a second by oscillator functions. In both approximations third-order graphs contribute only a few percent of the second-order terms to the binding energy.

Hamilton's Principle and the Maximum-Emission Coincidence

Self-locking of frequencies in laser oscillators is considered. The equations of motion for the cavity fields are restated as an extremum principle, which is a modified form of Hamilton's principle. This extremum condition is then compared with the ``maximum-emission'' condition hypothesized by Tang and Statz. It is found that the latter is not derivable from the former, unless one assumes infinitely fast-population relaxation of the laser medium and central tuning of laser modes with respect to the atomic resonance, and performs calculations by means of third-order perturbation theory. If the requirement of third-order perturbation calculations is relaxed, the maximum-emission hypothesis leads to a statement which is different from that of Hamilton's principle, but which coincidentally has the same solutions. If either of the other two assumptions is relaxed, the maximum-emission hypothesis not only gives a statement different from that of Hamilton's principle but leads to incorrect results.