A shell-model calculation of the mass 18 nuclei and the effective interaction in the sd-shell

A shell-model calculation of the energy levels of ¹⁸O and ¹⁸F is performed in the 2 particle and 3 particle-1 hole configuration space using the bare G-matrix elements of Kuo and Brown. The objective is to test the convergence of perturbative calculations of the effective interaction. The results are compared with those of first order perturbation theory and are found to be generally similar.     

Theoretical determination of the vibrational levels of NH+ 3 and its isotopomers

The vibrational levels associated with the electronic ground state X²A₂″ of NH⁺ ₃ have been determined up to 5000 cm^?1 by perturbation and variational calculations with full dimensionality of the molecule. For the variational part a new version of MULTIMODE was used which uses the ab initio electronic energy and its first derivative to define the potential energy function. These quantities were generated by the B97-1 density functional and RCCSD(T) approaches. For ND⁺ ₃, ND₂H⁺ and NDH⁺ ₂ the vibrational levels were calculated only by perturbation theory. The rotational constants for all the isotopomers were determined and the first transition dipole moments for NH⁺ ₃ and ND⁺ ₃ were plotted. A critical comparison of the perturbation and variational techniques suggests a possible further modification to the MULTIMODE algorithm for large systems.     

Radiative corrections to positronium energy levels

A systematic perturbation theory for the computation of positronium energy levels is presented. The proposed method is based on the Bethe-Salpeter equation and a zeroth-order kernel for which exact solutions are known explicitly. The cancellation of ultraviolet divergences arising in the perturbation series is discussed, and new contributions to the ground-state hyperfine splitting of positronium are evaluated up to order α⁶.     

Finite-size corrections to the energy levels of light muonic atoms

An analytic expression is developed in perturbation theory for the nuclear finite-size correction to the s-state energy levels of light muonic atoms. Using first-, second- and third-order perturbation theory, the finite-size corrections of order (Zα)⁴, (Zα)⁵, and (Zα)⁶ are calculated analytically for an arbitrary charge distribution. Application is made to the case of the μ? ⁴He atom, where the error in our finite-size expression is shown to be less than 10ppm.     

Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory

We introduce a generalization of Wick-ordering which maps the anharmonic oscillator (AO) Hamiltonian for mass m and coupling λ exactly into a “Wick-ordered” Hamiltonian with an effective mass M which is a simple analytic function of λ and m. The effective coupling $\text{Λ =}\text{λ}\text{M}{}^3$ is bounded. We transform the AO perturbation series in λ into one in Λ. This series may then be summed using Borel summation methods. We also introduce a new summation method for the AO series (which is a practical necessity to obtain accurate energy levels of the excited states). We obtain a numerical accuracy for $\text{(E}{}_{\mathrm{PT}}\text{? E}{}_{\mathrm{exact}}\text{)}\text{E}{}_{\mathrm{exact}}$ of at least 10^?7 (using 20 orders of perturbation theory) and 10^?3 (using only 2 orders of perturbation theory) for all couplings and all energy levels of the anharmonic oscillator. The methods are applicable also to the double-well potential (DWP, the AO with a negative mass-squared). The only change is that now the effective coupling is unbounded as λ → 0. The series in Λ is, however, still summable. The relative accuracy in the energy levels for 20 orders of perturbation theory varies from 10^?7 for large coupling to 1% at λ = 0.1 and to 10% at λ = .05. We also present results for the sextic oscillator.     

Application of Huang-Yang pseudopotential to a few-body system

Huang and Yang replace a hard core of radius c by their pseudopotential (HYPP), which can be used in perturbation theory. I show that HYPP and boundary value perturbation theory give the same energy shift, to first order in c. I model a two-body potential as an oscillator potential (angular frequency ω) and a hard core. For an A-body system, the ratio of the first order energy correction to the unperturbed energy is 0.448 A^3/4 c a, where a = (M ω/2h)^1/2.     

A new algorithm for one-dimensional problems

A new computation scheme in perturbation theory is developed for one-dimensional nonrelativistic problems with static potentials. If λ is a measure of the strength of the perturbation, after n steps in this scheme, the energy and wave function corrections are known to order λ^2n?1, whereas the same are known only to order λⁿ in the ordinary perturbation expansion.     

Perturbative approach to the hydrogen atom in a strong magnetic field

We consider states of the hydrogen atom with the principal quantum number n≤3 and zero magnetic quantum number in a constant homogeneous magnetic field ?. The perturbation theory series is summed using the Borel transformation and conformal mapping of the Borel variable. Convergence of the approximate energy eigenvalues and their agreement with the corresponding existing results are observed for external fields up to n³?/?₀～5, where ?₀ is the atomic magnetic field. The possibility of restoring the asymptotic behavior of energy levels using perturbation theory coefficients is also discussed.     

The Breit equation as a zeroth-order approximation for calculation of the bound-state energy

An equation whose kernel is determined by the one-photon exchange between interacting particles is analyzed in the framework of the Bethe-Salpeter formalism. This equation is considered as a zeroth-order approximation in the calculation of the corrections to the energy levels. This is the Breit equation that takes into account additional small terms (with respect to the parameter (Zα)²). A procedure that employs the Brillouin-Wigner perturbation theory and makes it possible to sequentially take into account the corrections to the energy levels is considered.     

Perturbation theory in terms of electron density

When an inhomogeneous electron gas is subjected to a perturbation, its energy and density both change by small amounts. We calculate the changes in the energy explicitly in terms of the density changes within the density-functional theory of many-electron systems. We also derive the equations for the induced densities, and using these show that a density correct up to order n in terms of the perturbation parameter is sufficient to give energy changes up to order (2n + 1). As a corollary, the even-order energy changes E⁽²ⁿ⁾ are variational with respect to the density changes ?⁽ⁿ⁾. The equations for the induced densities also follow from this corollary. The even-order corollary also gives a variational method of calculating the induced densities. The theory is demonstrated by applying it to calculate the polarizability and hyperpolarizability of the hydrogen, helium and neon atoms.     

Canonical transformation approach to the intermediate-coupling T-t Jahn-Teller effect

A canonical transformation is proposed to handle the T-t dynamical Jahn-Teller problem. The coupling term of the transformed Hamiltonian is treated as a small perturbation. The vibronic energy lowering for the ground state of the system is evaluated and compared with the exact numerical calculation of Cancer and Englman.¹ It is shown that the second-order perturbation theory gives accurate results in the weak- and intermediate-coupling regions.     

Energy levels and probabilities of radiative transitions in the Kr IX ion

The energy levels of the 3d ⁹4l (l = 1–3) and 3p ⁵3d ¹⁰4l (l = 0, 1) configurations in the Kr IX ion and the probabilities of radiative transitions to the ground state from these excited states have been calculated within the relativistic perturbation theory using a zero-approximation model potential. The results are shown to be stable within this approximation. The well-known problem of anomalously low accuracy of the calculations of some higher lying singlet levels is considered.     

Classical Hamiltonian perturbation theory without secular terms or small denominators

We consider perturbation theory in ? for the classical Hamiltonian H = H₀ + ?H₁, where H₀ gives rise to a known motion and ? is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ? and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator (s ? L)^?1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ?, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ?ⁿ⁺¹ when the accuracy of the theory is order ?ⁿ.     

Calculation of the energy levels of excited vibrational states of the HD16O molecule by summing divergent series of the Rayleigh-Schrödinger perturbation theory. The shift of zero-order levels

The Rayleigh-Schrödinger perturbation theory is applied to calculation of the energy levels of excited vibrational states of the HD¹⁶O molecule. The model of coupled anharmonic oscillators is considered, with the anharmonic part of potential energy being taken into account as the perturbation. The calculations are carried out for the vibrational states that correspond to three-to seven-fold vibrational excitations. Since the perturbation series diverge in the case of strong resonance interactions and their approximations by the Padé and Padé-Hermite methods do not yield sufficiently correct results, a calculation technique is applied that allows the zero-order approximation to be modified. The zero-order Hamiltonian is modified by shifting the vibrational frequencies, which decreases the mixing of states. The new Rayleigh-Schrödinger series can be summed using the quadratic Padé-Hermite approximation method.     

Heavy ion e+e− pairs to all orders in Zα

The heavy ion cross section for continuum e⁺e^? pair production has been calculated to all orders in Zα. Comparison is made with available CERN SPS and RHIC STAR data. Computed cross sections are found to be reduced from perturbation theory with increasing charge of the colliding heavy ions and for all energy and momentum regions investigated. Au or Pb total cross sections are reduced by 28% (SPS), 17% (RHIC) and 11% (LHC). For very high energy (E _e ⁺, E _e ^?>3 GeV) forward pairs at LHC the reduction from perturbation theory is a bit larger (17%). Use of zero degree calorimeter triggering (and thus small impact parameter weighting) makes impact parameter representation of exact pair production useful. Preliminary exact calculations in the zero impact parameter limit show a much larger reduction from perturbation theory (about 40%) at both RHIC and LHC.     

Absence of diffusion in the Anderson tight binding model for large disorder or low energy

We prove that the Green's function of the Anderson tight binding Hamiltonian decays exponentially fast at long distances on ? ^v , with probability 1. We must assume that either the disorder is large or the energy is sufficiently low. Our proof is based on perturbation theory about an infinite sequence of block Hamiltonians and is related to KAM methods.     

The study of the radiative lifetimes of 3pndF3 and 3pndD3 of Si I

The energy levels and lifetimes of 3pnd¹F₃ (n = 3-15) and 3pnd³D₃ (n = 3-18) of neutral silicon are calculated and predicted by means of the multichannel quantum defect theory (MQDT). In addition, this paper corrects the assignment mistakes of 3pnd¹F₃, 3pnd³F₃ for n > 8, and predicts many positions and lifetime values of the energy state which are not obtained in theory and experiment up to now, and analyzes the perturbation images between the channels.     

Das Konsistenzproblem der Greenfunktion eines A-Hyperons in Kernmaterie

An iteration procedure is derived for determining a self-consistent approximation for the one-particle Green's function in any order, starting from perturbation theory. This method is used to calculate the second Born approximation of the self energy of a Λ-particle in nuclear matter. After one iteration the approximation is nearly self-consistent. The value of the binding energy of the Λ-paiticle is only slightly decreased from the value of perturbation theory. Because of the vanishing particle density of the Λ-hyperon the quasiparticle energy depends on the square of the momentumk, the width of the spectral function is proportional tok ⁴ and the imaginary part of the self energy operator is proportional to (Ω-μ)^5/2 for Ω approaching the chemical potential μ.     

A many-electron perturbation theory study of the hexagonal boron nitride bilayer system*

In this article we explore methods to reduce the computational cost in many-electron wave function expansions including explicit correlation and compact one-electron basis sets for the virtual orbitals. These methods are applied to the calculation of the interlayer binding energy of the h-BN bilayer system. We summarize the optimized interlayer distances as well as their binding energies for various stacking faults on different levels of theory including second-order Møller-Plesset perturbation theory and the random phase approximation. Furthermore, we investigate the asymptotic behavior of the binding energy at large interlayer separation and find that it decays as D^-4 in agreement with theoretical predictions, where D is the interlayer distance.     

The 41Σ+ electronic state of LiCs molecule

The 4¹Σ⁺ state of LiCs molecule is observed experimentally for the first time. The inverted perturbation approach (IPA) method is used to derive the potential energy curve of the state from the measured spectra. The experiment is accompanied by theoretical calculations of adiabatic potentials for excited states in LiCs including 4¹Σ⁺, performed with the MOLPRO program package. The irregular shape of the 4¹Σ⁺ state potential predicted by theory is confirmed in the experiment.