Direct Perturbation Theory Research Articles

Configuration interaction calculations with terms linear in the interelectronic coordinate for the ground state of H+3. A benchmark study

The CISD-R12 method is applied to the ground states of H2 and H+3 at their equilibrium configurations, with both bare nuclear Hamiltonian (BNH) and self-consistent field (SCF) references with eight basis sets that range from 10 s (i.e., ten contracted s-type Gaussians) to 30s20p12d9f. With the largest basis set, the energy of H2 is obtained with an error of 2μEh, while for H+3, the error is probably smaller and the best calculated energy of −1.343 835Eh is supposed to be accurate to all indicated figures. The BNH reference is always superior to the SCF reference. The relativistic corrections are evaluated at the SCF level by means of ‘‘direct perturbation theory’’ both for H2 and H+3. The Hylleraas-SCF methods and related concepts for the construction of an optimum reference function are discussed in the Appendix.

Unitarity and ????? in chiral perturbation theory

The direct chiral perturbation theory (ChPT) one-loop prediction for the production of two pions from two photons is compared with the recent Crystal Ball experimental data. The one-loop computation does not fit data even close to threshold because unitarity effects are important even at very low energies. However, when the constraints coming from unitarity and analyticity are included in a suitable way, the range of application of one-loop ChPT can be largely extended in agreement with the experiment.

Particle in a random magnetic field on a plane

We study the properties of a two-dimensional spinless particle moving in a random magnetic field. This problem arises in the context of a modern theory of strongly correlated systems as well as in the theory of vortex-lines dynamics in high-${\mathit{T}}_{\mathit{c}}$ materials. The problem is investigated with a variety of methods including direct perturbation theory, quasiclassical approximation, the method of an optimal fluctuation, and Monte Carlo simulations. We obtain a shape of the density of states near the unrenormalized lower boundary of the spectrum, a particle mobility, and its diamagnetic orbital susceptibility.

Perturbative relativistic calculations for one-electron systems in a Gaussian basis

The “direct perturbation theory” of relativistic corrections (in which singularities of the traditional relativistic perturbation theory are avoided) is applied in a basis of atom-centered Gaussian orbitals to several states of the H atom and the ions H + 2 and HeH 2+. Almost exponential convergence with the basis size is achieved for the non-relativistic energy as well as for the relativistic corrections E 2, E 4, E 6, respectively. An overall 9-figure accuracy of the relativistic energies of the 1σ g and 1σ u states of H + 2 is achieved. The Pauli—Hamiltonian, that only furnishes E 2, is much inferior to direct perturbation theory even for E 2 (at least for states with non-vanishing electron density of the nuclei).

Geometry of the steady-state approximation: Perturbation and accelerated convergence methods

The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold ℳ, before it reaches the point equilibrium of the system. Locating ℳ describes the slow time evolution precisely, and allows all rate constants to be obtained from steady-state data. The line set ℳ is found by solution of a functional equation derived from the flow differential equations. For planar systems, the steady-state (SSA) and equilibrium (EA) approximations bound a trapping region containing ℳ, and direct iteration and perturbation theory are formally equivalent solutions of the functional equation. The iteration’s convergence is examined by eigenvalue methods. In many dimensions, the nullcline surfaces of the flow in Γ form a prism-shaped region containing ℳ, but this prism is not a simple trap for the flow. Two of its edges are EA and SSA. Perturbation expansion and direct iteration are now no longer equivalent procedures; they are compared in a three-dimensional example. Convergence of the iterative scheme can be accelerated by a generalization of Aitken’s δ2 extrapolation, greatly reducing the global error. These operations can be carried out using an algebraic manipulative language. Formally, all these techniques can be carried out in many dimensions.

Effects of higher-order dispersion on envelope solitons

The soliton modifications resulting from the addition of a small third derivative term, due to higher-order dispersion, to the nonlinear Schrödinger equation are investigated. Based on direct perturbation theory, it is shown that through first order, the soliton phase and velocity are modified, but the shape, amplitude, and width are unchanged. The radiation stimulated by the third derivative term, which is not predicted by the perturbation theory, is also derived. The expression obtained confirms the result of Wai (Ph.D. thesis, University of Maryland, 1988), which was obtained by a different method. The rate of change of the soliton amplitude due to this radiation, which emerges in front of the soliton, is calculated. The nonlinear term that drives the radiation is shown to grow to a value that is much larger than the unperturbed value.

Perturbation theory of relativistic corrections

After a discussion of the problems associated with the non-relativistic limit of the Dirac equation and of the expansion of the exact eigenvalues and eigenfunctions of the H atom in powers ofc−2 the traditional approaches for a perturbation theory of relativistic effects are critically reviewed. Then a direct perturbation theory is presented, that is characterized by a change of the metric in 4-component spinor space such that the Levy-Leblond equation appears as the straightforward non-relativistic limit of the Dirac equation. The various orders in perturbation theory of the energy and the wave function are derived first in a direct way, then in a resolvent formalism. The formulas are very compact and easily generalizeable to arbitrary order. All integrals that arise to any order exist, and no controlled cancellation of divergent terms (as in other approaches) is necessary. In the same philosophy an iterative approach towards the solution of the Dirac equation is derived, in which the solution of the Schrodinger equation is the first iteration step.

VISCOELASTIC MODULATION OF SOLITARY PRESSURE PULSES IN NONLINEAR FLUID-FILLED DISTENSIBLE TUBES

The propagation of small-but-finite-amplitude weakly-dispersive pressure pulses in a viscoelastic fluid-filled tube is shown to be governed by a Korteweg-de Vries-Burgers (KdVB) equation. We present a direct singular perturbation theory to describe the viscoelastic modulation of solitary pressure pulses for small retardation times. The main pulse evolves so as to satisfy a leading-order energy balance resulting in an algebraic decay in the amplitude and translation speed of the solution. Higher-order energy balances imply a monotonically increasing positive phase shift. A complete description of the first-order perturbation pressure is given. A pressure shelf of finite extent is continuously excited in the lee of the propagating pulse. The shelf extends from the current location of the solitary pressure pulse to the Korteweg-Moens phase position. The transition of the pressure shelf back to a zero background state at the Korteweg-Moens phase position is accomplished through a series of viscoelastically modified spatially-decaying high-wavenumber oscillations. Ahead of the main pulse a uniformly valid perturbation pressure field is obtained via a combination WKB-power-series and similarity-solution procedure.

Is there a microscopic foundation for the theory of superconductivity in heavy fermion systems? (abstract)

An attempt is made of collecting reliable information from microscopic models, which may be relevant to the problem of heavy fermion superconductivity. For this purpose direct perturbation theory with respect to hybridization between band states and local states is used to cope with the influence of strong local correlations in lattice periodic systems.1,2 Results taken into account concern the one-particle excitation spectra and the collective behavior, resulting from quasiparticle interactions, of such lattice systems. We discuss the prospects of electronic and phononic mechanisms for superconductivity and the possible resulting symmetry of the order parameter. The particular difficulties found in a microscopic treatment of superconductivity in heavy Fermion systems are pointed out and compared to standard BCS theory.

Unidirectionalization of hamiltonian waves

We show how the restriction of certain bidirectional hamiltonian systems modelling nonlinear, one-dimensional wave propagation to waves moving in a single direction preserves the hamiltonian structure, even though the perturbation expansion of the bidirectional hamiltonian is not correct. A combination of the two approaches of direct hamiltonian perturbation theory and the method of multiple scales helps explain the apperance of integrable bihamiltonian wave models.

N-point functions and low-energy scattering by N centres

The scattering of a low-energy particle by a potential of a small range is known to be described satisfactorily by the s-wave alone. In the present paper we give a method of describing low-energy scattering by N potentials with the aid of N waves. For this purpose, a special system of Laplacian eigenfunctions is suggested. The scattering amplitude depends on only N parameters, irrespective of overlapping of potentials. The physical significance of these parameters δ λ, λ=1,2,… N, is shown by exp (2 iδ λ )= S λ where S λ is the eigenvalue of the S matrix. The parameters δ λ may be obtained by direct methods and perturbation theory. The low-energy scattering by an arbitrary configuration of N centres is discussed. The differential cross-section is averaged over all orientations of the configuration and radially about the direct beam, giving it as a function of the scattering angle. This formula may be used for the phase shift analysis.

On the damping of an elementary excitation of the quantum system interacting with a gaussian random field

Abstract A direct stochastic perturbation theory is developed to investigate the damping of elementary excitations in the presence of a gaussian random field. The method is applied to several quantum statistical models, the ideal Fermi or Bose gas and the isotropic Heisenberg model.

Electronic Spin Polarization around a Magnetic Impurity Using Perturbation Theory

The electronic spin polarization $p(\mathcal{r})$ arising from the $s\ensuremath{-}d$ exchange interaction is computed using iterated solutions of Nagaoka's truncated equations and direct perturbation theory. Most of our detailed calculations are only to order ${J}^{2}$ (where $J$ is the exchange constant). However, arguments are given to suggest that certain qualitative features [such as the oscillatory behavior of $p(\mathcal{r})$] will remain even if we work to all orders in $J$. We critically discuss the work of Falk and Fullenbaum as well as Suhl, whose results agree with ours apart from the important difference that our Ruderman-Kittel-Kasuya-Yosida term is proportional to the average of an effective spin rather than the bare impurity spin. This effective spin also enters the static susceptibility $\ensuremath{\chi}$. In one of the Appendices, we briefly consider the effect of potential scattering on the spin polarization.