Continuum Part Research Articles

An integrated effective fragment—polarizable continuum approach to solvation: Theory and application to glycine

A new discrete/continuum solvation model has been developed by combining the effective fragment potential (EFP) for the discrete part and the polarizable continuum model (PCM) for the continuum part. The usefulness of this model is demonstrated by applying it to the calculation of the relative energies of the neutral and zwitterionic forms of glycine. These calculations were performed by treating glycine with ab initio wave functions. Water clusters were treated with both ab initio and EFP methods for comparison purposes, and the effect of the continuum was accounted for by the PCM model. The energy barrier connecting the zwitterionic and neutral three-water clusters was also examined. The computationally efficient EFP/PCM model gives results that are in close agreement with the much more expensive full ab initio/PCM calculation. The use of methods that account for electron correlation is necessary to obtain accurate relative energies for the isomers of glycine.

Discrete and continuum quantum states for the Kratzer oscillator

AbstractKratzer oscillator is a realistic zero‐order model for describing the anharmonic ro‐vibrational motion in diatomic molecules. Kratzer oscillator has an energy spectrum containing both discrete and continuum parts. Wavefunctions belonging to the continuum would be useful in the study of transitions to the continuum in molecular dissociation processes. In this article, bound and scattering wavefunctions of the Kratzer oscillator are reviewed and the bound–bound and the bound–free matrix elements are obtained. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Temperature dependent BCS-gap equations in the continuum

Pairing correlations for excited nuclei near the proton drip line are described by using finite-temperature BCS equations including the continuum part of the spectrum. The suitability of the proposed method to obtain stable solutions in the presence of single-particle resonances is shown.

Continuum discretization in a basis of transformed harmonic-oscillator states

The inclusion of the continuum in the study of weakly bound systems is discussed. A transformed harmonic-oscillator basis is introduced to provide an appropriate discrete and finite basis for treating the continuum part of the spectrum. As examples of application of the method the one-dimensional Poeschl-Teller and Morse potentials are worked out. The strength functions corresponding to different operators that couple the ground state to the continuum are investigated. It is found that the energy moments of those distributions are accurately reproduced with a small basis set.

Critical assessment of particle decay as a probe to study the continuum

Quantities that determine the structure of the continuum part of nuclear spectra are studied in detail. Measurable particle (proton) decay processes are discussed and formulas to evaluate the corresponding decay widths are analyzed. It is shown that the decaying resonances are so narrow that such processes can be treated as if the nucleus was bound and, therefore, they are not suitable to probe formalisms designed to treat the continuum. Other processes that are more sensitive to broad resonances are analyzed and suggestions for probable experiments are made.

A Possible New Effect of Self-Consistency in the Relativistic Hartree-Fock Approximation

A new effect of self-consistency in the relativistic Hartree-Fock (HF) approximation suggested previously is confirmed by a renormalized calculation with the on-shell renormalization conditions. Two self-consistency schemes, one requiring self-consistency in the HF potential (scheme P) and the other in the baryon propagator (scheme BP), are studied. It is pointed out that the on-shell renormalization conditions make the self-consistency requirement in scheme P automatically satisfied. Our calculated results show that scheme P is a good approximation to scheme BP for the calculation of the baryon propagator and the self-consistency in scheme BP diminishes the continuum part of the spectral representation for the baryon propagator, while scheme P yields a baryon propagator which is the same as the HF result contributed by the single particle part of the above spectral representation alone. Further, it is demonstrated that the region of validity of the quasi-particle approximation may depend on the renormalization conditions.

Kohn–Sham calculations using hybrid exchange-correlation functionals with asymptotically corrected potentials

The theory is presented for asymptotically correcting the potentials of hybrid exchange-correlation functionals, i.e., those that include a fraction of orbital exchange. The Kohn–Sham equations involve a multiplicative potential due to the continuum part of the hybrid functional and a nonmultiplicative term due to the orbital exchange. In asymptotic regions the multiplicative σ-spin potential is corrected to take the form (CX−1)/r+εHOMO,σ+Iσ, where CX is the fraction of orbital exchange; εHOMO,σ is the σ-spin self-consistent highest occupied Kohn–Sham eigenvalue; and Iσ is an approximate ionization energy. For the hydrogen atom, the asymptotic correction leads to a potential that closely resembles the exact potential; the eigenvalue spectrum is intermediate between the Schrödinger and Hartree–Fock eigenvalues, reflecting the presence of orbital exchange. Kohn–Sham orbitals and eigenvalues determined from this procedure have been used to calculate singlet vertical excitation energies for CO, N2, H2CO, C2H4, and C6H6. The correction significantly improves excitation energies to Rydberg states, with mean absolute errors below 0.2 eV. However, despite including orbital exchange, the results do not represent an improvement over the results obtained by asymptotically correcting a recently developed GGA functional. The asymptotic correction is also shown to reduce static isotropic polarizabilities.

Remarks on completeness of many-electron Sturmians

A question of completeness of a discrete many-electron Sturmian set proposed in a series of recent publications is considered. It is shown that already in the simplest case of a two-electron system the proposed Sturmians do not form a complete set since the spectrum of a generating eigenproblem is mixed: apart from discrete eigenvalues, with which the discrete Sturmians are associated as corresponding eigenfunctions, the spectrum also contains a continuum part. A peculiar feature of the spectrum found is that infinitely many discrete eigenvalues are embedded in the continuum.

Relativistic Hartree-Fock schemes and effect of self-consistency

A new effect of self-consistency in the relativistic Hartree-Fock (HF) approximation is studied by a simple model and a renormalized calculation. A comparison is made between two different HF schemes: one requiring self-consistency in the HF potential (scheme P) and the other in the baryon propagator (scheme BP). Our results show that scheme P is a good aproximation to scheme BP for the calculation of the baryon propagator and the self-consistency requirements make the results obtained by the two schemes closer to each other, because the self-consistency in scheme BP diminishes the continuum part of the spectral representation for the baryon propagator, while the self-consistency in scheme P yields a baryon propagator which approximates closely to the HF result contributed by the converged single particle part of the above spectral representation alone.

Vibronic state specific predissociation rates from excited electronic states

It is shown that predissociation can be perceived as a primary process due to the continuum part of a Morse oscillator potential. In the model proposed here internal conversion to the ground state is thus not necessarily the primary process of a consecutive dissociation but may be a simultaneous decay. As a consequence, dissociation rates should show strong variations from specific (ro-) vibrational states of the first excited electronic states that are similar to those known from “pure” internal conversion rates. This behaviour is demonstrated by calculating predissociation rates for the \(\) process. Especially the out-of-plane modes seem to play an extraordinary role in the excess energy behaviour of the predissociation rate. At lower excess energies, rates from single vibronic levels with out-of-plane mode characteristics may show an increase by several orders of magnitude.

Analysis of Impurity-Induced Circular Currents for the Chiral Superconductor Sr2RuO4

The microscopic mechanism of circular currents induced in the vicinity of a non-magnetic impurity is analyzed for the (time-reversal symmetry breaking) chiral superconducting state ${\bf d}({\bf k}) = \hat{{\bf z}} (k_x \pm i k_y)$ which is very likely realized in Sr_2RuO_4. From the analytic, not self-consistent solution of the Bogolyubov-de Gennes equations we find two types of quasiparticle states, a bound state and the continuum state. Through impurity scattering the condensate transfers angular momentum to the quasiparticle states which generate a circular current. At low temperature the continuum part of the quasiparticle spectrum gives the main contribution. The non-selfconsistent solution yields a state of finite angular momentum. The comparison with the corresponding Ginzburg-Landau theory reveals the existence of a compensating counterflow created by the superconducting condensate. The currents and magnetic fields appearing around the impurity have possibly been observed by muon spin relaxation measurements.

Energy Levels of Interacting Fields in a Box

We study the influence of boundary conditions on energy levels of interacting fields in a box and discuss some consequences when we hange the size of the box. In order to do this we calculate the energy levels of bound states of a scalar massive field χ nteracting with another scalar field φ through the Lagrangian \(\mathcal{L}_{\operatorname{int} } \) = \(\frac{3}{2}g\phi ^2 \mathcal{X}^2 \)> in a one-dimensional box on which we impose Dirichlet boundary conditions. We find that the gap between the bound states changes with the size of the box in a nontrivial way. For the case where the masses of the two fields are equal and for large box the energy levels of Dashen-Hasslacher-Neveu (DHN model) are recovered and we have a kind of boson condensate for the ground state. Below a critical box size \(L \sim 2.93\left( {2\sqrt 2 /M} \right)\) the ground-state level splits, which we interpret as particle-antiparticle production under small perturbations of box size. Below other critical sizes, \(L \sim \left( {6/10} \right)\left( {2\sqrt 2 /M} \right)\) and \(L \sim 1.17\left( {2\sqrt 2 /M} \right)\), of the box, the ground state and firstexcited state merge in the continuum part of the spectrum.

Excited States of11Li

The excited states of the ``halo'' nucleus, ${}^{11}\mathrm{Li}$, have been investigated by means of the pion capture reaction, $^{14}C({\ensuremath{\pi}}^{\ensuremath{-}},pd{)}^{11}\mathrm{Li}$. Excited states have been identified at $1.02\ifmmode\pm\else\textpm\fi{}0.07$, $2.07\ifmmode\pm\else\textpm\fi{}0.12$, and $3.63\ifmmode\pm\else\textpm\fi{}0.13\mathrm{MeV}$. The continuum part of the ${}^{11}\mathrm{Li}$ missing mass spectrum is found to contain a major component which is consistent with ${}^{11}\mathrm{Li}$ breakup into one of the ${}^{9}\mathrm{Li}$ excited states and two strongly correlated neutrons.

Constraints on hadronic spectral functions from continuous families of finite energy sum rules

Hadronic τ decay data is used to study the reliability of various finite energy sum rules (FESR's). For intermediate scales ( s 0∼2–3 GeV 2), those FESR's with weights s k are found to have significant errors, whereas those with weights having a zero at the juncture ( s= s 0) of the cut and circular part of the contour work very well. It is also shown that a combination of two such sum rules allows rather strong constraints to be placed on the hadronic spectral function without sacrificing the excellent agreement between OPE and hadronic representations. The method then is applied to the strangeness S=−1 and pseudoscalar isovector channels, where it is shown to provide novel constraints on those ansatze for the unmeasured continuum parts of the spectral functions employed in existing extractions of m u + m d , m s . The continuum ansätz in the latter channel, in particular, is shown to produce a rather poor match to the corresponding OPE representation, hence re-opening the question of the value of the light quark mass combination m u + m d .

A, C, and D electronic states of the Ar–NO van der Waals molecule revisited: Experiment and theory

The A–X transition of ArNO has been reinvestigated by laser induced fluorescence (LIF) both in the bound–free and bound–bound region. The discrete part of the spectrum is at least two orders of magnitude weaker than the continuum part, indicative of a large change in geometry from the ground state. This very different configuration, both from the ground state and from the C and D states, can only be explained by strong interactions, induced by the perturbing argon atom, between the excited states of the van der Waals complex converging to the 3sσ,A, 3pπ,C, and 3pσ,D Rydberg states of NO. In order to quantitatively understand the observed structure of the A–X, C–X, and D–X excitation spectra, a global theoretical approach is proposed, based on ab initio calculations of the potential energy surfaces in the planar A′ and A″ symmetries, including a configuration interaction between the states of same symmetry. Small adjustments of the diabatic energy surfaces lead to a satisfactory agreement between the observed and calculated spectra. In contrast to the ground state, the Renner–Teller splitting of the 3pπ,C state into two A′ and A″ components is very large, of the order of 4000 cm−1. This effect is complicated by further mixing between the states of A′ symmetry induced by the argon atom. The A state is anisotropic and weakly bound with a small potential well at the linear configuration (the argon atom being on the side of the oxygen). The C(A″) and the bound electronic component of the strongly mixed C+D(A′) states exhibit a vibrational structure close to that of the ion and, consequently, present some Rydberg character even if the Coulomb field central symmetry (s-p) is broken by the perturbing argon atom.

Pairing Correlation and Quadrupole Deformation Effects on the14CDecay

A microscopic description of heavy-cluster decay using a single particle basis suited to describe the continuum part of the spectrum is presented. The role of pairing correlations and quadrupole deformations is analyzed. The stability of the results as a function of the number of major shells included in the calculation is attested.

Electron temperature measurements with high time resolution in RFX

Core electron temperature measurements have been performed in the large, high-current RFX reversed field pinch experiment (a = 0.46 m, R = 2 m, target plasma current = 2 MA) with the double filter technique. With this technique the soft x-ray emission is measured by two Si detectors, covered by different Be-filters, and viewing the same region of the plasma. By choosing the filters in such a way that only the emission from the continuum part of the spectrum is measured, the ratio of the two signals is a function of the highest electron temperature along the line of sight of the detectors. With this diagnostic, the electron temperature has been measured in RFX with a bandwidth of approximately 2 kHz. This allows the direct measurement, for the first time in a RFP on a single shot basis, of coherent oscillations, associated with relaxation events and showing under some conditions a sawtooth-like character. The measurements of electron temperature obtained with this method are in good agreement with those performed with other diagnostics.

Theory of photoinduced charge transfer in a molecularly doped conjugated polymer.

In a molecularly doped conjugated polymer, it is possible for an exciton, photoexcited on the polymer, to decay into a free charge in the polymer's valence or conduction band and a charge of opposite sign in an available level on an adjacent dopant molecule. Such pair processes, which conserve energy, can occur if the energy ${\mathrm{\ensuremath{\omega}}}_{0}$ of the polymer exciton lies within the continuum part of the charge-transfer electron-hole excitation spectrum of the polymer/molecule composite. We calculate the corresponding exciton decay rate 1/\ensuremath{\tau}(\ensuremath{\Delta}) for two types of matrix element that describe the microscopic charge-transfer event. This rate is a maximum when the energy \ensuremath{\Delta} of the molecular acceptor level lies below the conduction-band edge by an amount that is a little more than the exciton's binding energy, a similar result occurring for the case of hole transfer to the molecule. The reduction of 1/\ensuremath{\tau} by the Coulomb interaction between the electron and hole in the final state is important. \textcopyright{} 1996 The American Physical Society.

Feeding of hole states by proton decay of Gamow-Teller and isobaric analog state resonances.

The study of giant resonances in nuclei is attracting much attention both experimentally and theoretically @1,2#. The importance of these studies is that they may provide information about the continuum part of nuclear spectra as well as an understanding of the nuclear dynamics in the continuum. However, there is not much experimental evidence regarding partial decay widths from giant resonances—even the theoretical results are scarce. The reason for this is that the processes of formation and decay of giant resonances are time dependent and a proper treatment of such processes is a very difficult task. Experimentally, it is not easy to subtract the interesting signals from the background. Moreover, if the resonance is not isolated what one measures is not related to the resonance itself but rather to the result of the contribution of all the overlapping resonances, including their interference. The relativistic random-phase approximation ~RRPA !@ 3 # is a natural extension of the random-phase approximation ~RPA! which allows for the treatment of particle-hole states in the continuum. The eigenvalues of the RRPA can be complex due to the admixture of bound and resonant states in the unperturbed particle-hole basis ~Berggren’s representation !. The imaginary part of a complex eigenvalue is associated with the escape width of the state @4#. The formalism has been presented in detail previously and several applications of it have been given, particularly for the description of charge-conserving multipole excitations in 208 Pb @7#. As has been discussed in @4# the residues of the S matrix at a complex energy v n are the partial decay widths of the resonance n if the resonance is isolated ~nonoverlapping!. For such a resonance there exists a direct correspondence between the total escape width and the imaginary part of the energy, i.e., G n522Im(v n). This result has been confirmed for the case of the decay of the giant monopole resonance in 208 Pb @7# .I n the present case we shall show that similar results are also valid for charge-exchange modes, i.e., the isobaric analog ~IAS! and Gamow-Teller giant resonances ~GTGR’s! in 208 Bi.

Coupling effects of resonant and discretized non-resonant continuum states in 4He + 6Li scattering at 10 MeV/A

Alpha-particle scattering from the resonant (3 1 + ) and non-resonant continuum states of 6Li is studied at incident energy 10 MeV/A. The α + d breakup continuum part within the excitation energy E ex = 1.475–2.475 MeV is discretized in two energy bins. Unlike the results at higher incident energies, here the coupled-channel calculations show significant breakup continuum coupling effects on the elastic and inelastic scattering. It is shown that even when the continuum-continuum coupling effects are strong, the experimental data of the ground state and the resonant as well as discretized non-resonant continuum states impose stringent constraint on the coupling strengths of the non-resonant continuum states.