Continuum Orbitals Research Articles

Continuum random-phase approximation for γ transitions between excited states in neutron-rich nuclei

A characteristic feature of collective and particle-hole excitations in neutron-rich nuclei is that many of them couple to unbound neutron in continuum single-particle orbits. The continuum random phase approximation (cRPA) is a powerful many-body method that describes such excitations, and it provides a scheme to evaluate transition strengths from the ground state. In an attempt to apply cRPA to the radiative neutron capture reaction, we formulate in the present study an extended scheme of cRPA that describes gamma-transitions from the excited states under consideration, which decay to low-lying excited states as well as the ground state. This is achieved by introducing a non-local one-body operator which causes transitions to a low-lying excited state, and describing a density-matrix response against this operator. As a demonstration of this new scheme, we perform numerical calculation for dipole, quadrupole, and octupole excitations in $^{140}$Sn, and discuss E1 and E2 transitions decaying to low-lying $2^{+}_{1,2}$ and $3^{-}_{1}$ states. The results point to cases where the branching ratio to the low-lying states is larger than or comparable with that to the ground state. We discuss key roles of collectivity and continuum orbits in both initial and final states.

Efficient Treatment of Correlation Energies at the Basis-Set Limit by Monte Carlo Summation of Continuum States.

The calculation of electron correlation is vital for the description of atomistic phenomena in physics, chemistry, and biology. However, accurate wavefunction-based methods exhibit steep scaling and often sluggish convergence with respect to the basis set at hand. Because of their delocalization and ease of extrapolation to the basis-set limit, plane waves would be ideally suited for the calculation of basis-set limit correlation energies. However, the routine use of correlated wavefunction approaches in a plane-wave basis set is hampered by prohibitive scaling due to a large number of virtual continuum states and has not been feasible for all but the smallest systems, even if substantial computational resources are available and methods with comparably beneficial scaling, such as the Møller–Plesset perturbation theory to second order (MP2), are used. Here, we introduce a stochastic sampling of the MP2 integrand based on Monte Carlo summation over continuum orbitals, which allows for speedups of up to a factor of 1000. Given a fixed number of sampling points, the resulting algorithm is dominated by a flat scaling of . Absolute correlation energies are accurate to <0.1 kcal/mol with respect to conventional calculations for several hundreds of electrons. This allows for the calculation of unbiased basis-set limit correlation energies for systems containing hundreds of electrons with unprecedented efficiency gains based on a straightforward treatment of continuum contributions.

Breit–Pauli R-Matrix approach for the time-dependent investigation of ultrafast processes

We have refactored the Breit–Pauli R-Matrix integral package within the RMatrxI package to employ a B-Spline basis to allow for level-resolved time-dependent R-Matrix calculations involving a laser pulse. The B-Spline approach independently verifies the accuracy of the current integral package pstg1r.f, but requires greater flexibility at the R-Matrix boundary when describing the continuum wavefunctions. This adaptation can be integrated with either the subsequent serial or parallel Breit–Pauli suite of codes. Program summaryProgram Title: BSplineStg1Program Files doi:http://dx.doi.org/10.17632/prk6fsn56y.1Licensing provisions: GPLv2Programming language: Fortran 2003Nature of problem: Previously, the time-dependent R-Matrix with Time-dependent (RMT) codes could only be used with atomic data from atomic physics codes which did not include spin–orbit effects. When we desire to include spin–orbit effects in a level resolved Time-dependent R-Matrix calculation, we must use a different atomic structure and integral package, such as the RMatrixI package. However, while the RMatrixI package is able to include spin–orbit effects, it was not previously compatible for use with RMT.Solution method: To enable compatibility between RMatrixI and RMT, we write an alternative to the Stg1 code within the RMatrixI package (titled BSplineStg1) that uses B-Spline techniques to create and describe continuum orbitals. The necessary one- and two-electron integrals for the remainder of the RMatrixI package are calculated using Gauss–Legendre integration. The remainder of the Breit–Pauli R-Matrix codes require no further modification.

Multireference quantum chemistry protocol for simulating autoionization spectra: Test of ionization continuum models for the neon atom

In this contribution we present a protocol to evaluate partial and total Auger decay rates combining the restricted active space self-consistent field electronic structure method for the bound part of the spectrum and numerically obtained continuum orbitals in the single-channel scattering theory framework. On top of that, the two-step picture is employed to evaluate the partial rates. The performance of the method is exemplified for the prototypical Auger decay of the neon $1s^{-1}3p$ resonance. Different approximations to obtain the continuum orbitals, the partial rate matrix elements, and the electronic structure of the bound part are tested against theoretical and experimental reference data. It is demonstrated that the partial and total rates are most sensitive to the accuracy of the continuum orbitals. For instance, it is necessary to account for the direct Coulomb potential of the ion for the determination of the continuum wave functions. The Auger energies can be reproduced quite well already with a rather small active space. Finally, perspectives of the application of the proposed protocol to molecular systems are discussed.

Self-Consistent Field Theory of Ionization Collisions

A self-consistent field (SCF) theory for treating scattering systems was formulated previously and is extended here to the ionization process, in which the continuum orbitals are made square integrable by an amputation procedure. The method is applied to the electron-hydrogen scattering system in the zero angular momentum coupling model, and the differential cross section is compared with the recent results obtained by several other approaches. It is shown that the amputated continuum functions provide an effective projection of the scattering equation. SCF continuum functions generated in the present formalism are used to analyze the effective charge approximation.

Interaction of molecular nitrogen with free-electron-laser radiation.

We compute molecular continuum orbitals in the single center expansion scheme. We then employ these orbitals to obtain molecular Auger rates and single-photon ionization cross sections to study the interaction of N2 with Free-Electron-Laser (FEL) pulses. The nuclei are kept fixed. We formulate rate equations for the energetically allowed molecular and atomic transitions and we account for dissociation through additional terms in the rate equations. Solving these equations for different parameters of the FEL pulse, allows us to identify the most efficient parameters of the FEL pulse for obtaining the highest contribution of double core hole states (DCH) in the final atomic ion fragments. Finally we identify the contribution of DCH states in the electron spectra and show that the DCH state contribution is more easily identified in the photo-ionization rather than the Auger transitions.

Merging quantum chemistry packages with B-splines for the multichannel scattering problem

SynopsisTo study the photoionization of polyelectronic systems by attosecond pulses, we take advantage of existing quantum chemistry packages for the description of correlated electronic states, and of hybrid Gaussian- B-splines basis for the representation of continuum orbitals. In our approach, a short-range region, which can host all the interacting electrons and is described by commercial packages, is matched to a long-range region which describes single-ionization states in terms of a close-coupling expansion. We validate this approach showing multichannel ionization results for He.

One of signatures of a memristor

In this paper, a model of memristor oscillator with a piecewise function is taken into account. The classification and stability of equilibria of the model are discussed in detail. One of main findings is that there exist at least three types of singular continuum (pinched hysteresis loop) whose loci always pass through the origin in the voltage–current plane, and whose loop areas will shrink to zero, which is one of signatures of a memristor which distinguishes it from non-memristive devices. By using qualitative analysis and bifurcation theory, we have obtained two Hopf bifurcation surfaces and a unique unstable periodic orbit, whose existence and uniqueness are proved in a strict mathematical way. By using numerical simulations, we have given different phase portraits of equilibria, singular continuum and periodic orbits, which verify all analytical results. Our results demonstrate that the three types of singular continuum and a unique unstable periodic orbit may become some of important signatures of a memristor distinguishing from non-memristive devices, and the associated memristor circuits with piecewise functions may reflect more real aspects of functional roles of a memristor in electronic circuits.

Self-consistent field theory of collisions: Orbital equations with asymptotic sources and self-averaged potentials

The self-consistent field theory of collisions is formulated, incorporating the unique dynamics generated by the self-averaged potentials. The bound state Hartree–Fock approach is extended for the first time to scattering states, by properly resolving the principal difficulties of non-integrable continuum orbitals and imposing complex asymptotic conditions. The recently developed asymptotic source theory provides the natural theoretical basis, as the asymptotic conditions are completely transferred to the source terms and the new scattering function is made fullyintegrable. The scattering solutions can then be directly expressed in terms of bound state HF configurations, establishing the relationship between the bound and scattering state solutions. Alternatively, the integrable spin orbitals are generated by constructing the individual orbital equations that contain asymptotic sources and self-averaged potentials. However, the orbital energies are not determined by the equations, and a special channel energy fixing procedure is developed to secure the solutions. It is also shown that the variational construction of the orbital equations has intrinsic ambiguities that are generally associated with the self-consistent approach. On the other hand, when a small subset of open channels is included in the source term, the solutions are only partiallyintegrable, but the individual open channels can then be treated more simply by properly selecting the orbital energies. The configuration mixing and channel coupling are then necessary to complete the solution. The new theory improves the earlier continuum HF model.

Absolute Cross Sections for Near-Threshold Electron-Impact Excitation of the 2s 2S → 2p 2P Transition of Li-Like C3+, N4+, and O5+ Ions

Excitation cross sections of 1s22s 2S1/2 → 1s22p 2P1/2,3/2 transition among the fine-structure levels in Li-like C3+, N4+, and O5+ ions are calculated for energies of the near-threshold by using the relativistic distorted-wave program REIE06. The target state wavefunctions are calculated by using the Grasp92 code. The continuum orbitals are studied in the distorted-wave approximation, in which the direct and exchange potentials among all the electrons are included. The results of the Li-like C3+ ion settle the discrepancy between several previous experiments by using the crossed-beams fluorescence method, in good agreement with the measurements of Savin et al. Moreover, the results in Li-like N4+, and O5+ ions are compared with the previous experiments, and a good agreement is obtained.

Construction of complex STO-NG basis sets by the method of least squares and their applications

Electronic resonance state energies and photoionization cross sections of atoms and molecules are calculated with the complex basis function method by using mixture of appropriate complex basis functions representing one-electron continuum orbitals and the usual real basis functions for the remaining bound state orbitals. The choice of complex basis functions has long been a central difficulty in such calculations. To address this challenge, we constructed complex Slater-type orbital represented by N-term Gaussian-type orbitals (cSTO-NG) basis sets using the method of least squares. Three expansion schemes are tested: (1) expansion in complex Gaussian-type orbitals, (2) expansion in real Gaussian-type orbitals, and (3) expansion in even-tempered real Gaussian-type orbitals. By extending the Shavitt–Karplus integral transform expression to cSTO functions, we have established a mathematical foundation for these expansions. To demonstrate the efficacy of this approach, we have applied these basis sets to the calculation of the lowest Feshbach resonance of H2 and the photoionization cross section of the He atom including autoionization features due to doubly excited states. These calculations produce acceptably accurate results compared with past calculations and experimental data in all cases examined here.

Dynamical photoionization observables of the CS molecule: the role of electron correlation.

Highly correlated calculations are performed on the primary ionic states and the prominent satellite present in the outer valence photoelectron spectrum of carbon monosulfide (CS). Dyson orbitals are coupled to accurate one particle continuum orbitals to provide a correlated description of energy dependent cross sections, asymmetry parameters, branching ratios, and molecular frame photoelectron angular distributions. The comparison with results obtained at the Hartree-Fock and Density Functional Theory level shows the strong sensitivity of these observables to details of the correlation in the bound states. The behaviour of the well characterized satellite state is analyzed in detail, and shows differences from the relevant primary states, revealing the limitations of a simple intensity borrowing mechanism. The results resolve the intensity disagreement with experiment obtained at the level of the sudden approximation.

Electron-impact ionization of W+ forming W2+

We have investigated electron-impact ionization (EII) for the ground level 5d46sD61/2 and the two metastable levels 5d5S65/2 and 5d36s2F43/2 of W+ forming W2+ by theoretical calculation using the flexible atomic code based on a distorted-wave (DW) approximation. It is found that the direct ionization (DI) of a 5d and 6s electron is largely sensitive to the choice of optimization configurations for the single, atomic local central potential used in bound and continuum orbitals descriptions of both the initial and the final states. The DI cross section is reduced by as much as 60% for the variation of the optimization configurations. Indirect excitation–autoionization (EA) cross sections for the 5d and 6s→nl (5≤n≤30) and 4f, 5s and 5p→nl (n=5 and 6) transitions have been calculated with the detailed branching ratios. Experiments are about 28% lower than our theoretical predictions for the total EII cross sections of the ground level 5d46sD61/2 and the metastable level 5d5S65/2 at the peak. This is due to the large contribution of the EA though the DI cross section is significantly reduced than previous theoretical works and experiments.

Self-Consistent Solutions for the Scattering State with Two Free Electrons

Wave functions for the scattering states with two free electrons in the field of an ion core are explicitly calculated by the self-consistent, continuum Hartree–Fock (CHF) theory. Typically, such states are associated with the three-body recombination, collisional ionization and photo-double ionization, but have never been directly studied previously. The calculated continuum orbitals are found to be predominantly of the plane-wave forms, as though the system is translation invariant, in the context of many-body HF theory. The symmetry is mildly broken by the presence of the core ion, at about fifteen-percents level, indicating that the orbitals are largely delocalized and the effect of the core potential is an important but minor perturbation. The properties of channel orthogonality and completeness are preserved by the nearly plane wave forms. To test the validity of this finding and the CHF, the continuum orbitals are used to evaluate the amplitudes for the electron impact ionization, and the amputation procedure, that is crucial in the theory, is also critically re-examined.

The Ratip program for relativistic calculations of atomic transition, ionization and recombination properties

The Ratip program for relativistic calculations of atomic transition, ionization and recombination properties

O1s photoionization dynamics in oriented NO2

We have performed extensive density functional theory (DFT) calculations, partial cross sections, dipole prepared continuum orbitals, dipole amplitudes and phase shifts, asymmetry parameters β, and molecular frame photoelectron angular distributions, to elucidate the O1s photoionization dynamics of NO(2) molecule with emphasis on the shape resonances in the O1s ionization continuum. In the shape resonance region, the β parameters and photoelectron angular distributions have been compared with our experimental results. Fairly good agreement between the theory and experiment has confirmed that the DFT level calculations can well describe the photoionization dynamics of the simple molecule such as NO(2). Interference due to equivalent atom photoionization is theoretically considered, and the possibility of detection of the effect in the two degenerate channels with different combinations of light polarization and photoemission direction is discussed.

Differential and Integral Cross Sections for Electron Impact Excitation of Lithium

The differential and integral cross sections for electron impact excitation of lithium from the ground state 1s22s to excited states 1s22p, 1s23l (l = s,p,d) and 1s24l (l = s,p,d,f) at incident energies ranging from 5 eV to 25 eV are calculated by using a full relativistic distorted wave method. The target state wavefunctions are calculated by using the Grasp92 code. The continuum orbitals are computed in the distorted-wave approximation, in which the direct and exchange potentials among all the electrons are included. A part of the cross sections are compared with the available experimental data and with the previous theoretical values. It is found that, for the integral cross sections, the present calculations are in good agreement with the time-independent distorted wave method calculation, for differential cross sections, our results agree with the experimental data very well.

Methods for the calculation of photoionization cross sections using the extended Koopmans' theorem

A new approach to the calculation of photoionization cross sections for atoms and molecules is described. The procedure requires for input only a wave function for the parent initial state, obviating the necessity of a separate calculation on the final bound state by making use of the recently developed Extended Koopmans' Theorem. Explicit formulas for the photoionization cross sections, in terms of the one- and two-particle reduced density matrices of the initial state wave function, have been obtained at several levels of sophistication, including the sudden approximation, an orbital model for the photoelectron, and a close-coupling model. In the latter two cases, equations have been derived for the optimization of the continuum orbitals required for the description of the outgoing photoelectron. Approximations which make the optimization equations more tractable are also discussed. 26 references.

Relativistic Distorted-Wave Collision Strengths of Ni-, Cu- and Zn-like Au Ions

Excitation energies and electron impact excitation strengths from the ground states of Ni-, Cu- and Zn-like Au ions are calculated. The collision strengths are computed by a 213-levels expansion for the Ni-like Au ion, 405-levels expansion for the Cu-like Au ion and 229-levels expansion for the Zn-like Au ion. Configuration interactions are taken into account for all levels included. The target state wavefunctions are calculated by using the Grasp92 code. The continuum orbits are computed in the distorted-wave approximation, in which the direct and exchange potentials among all the electrons are included. Excellent agreement is found when the results are compared with previous calculations and recent measurements.

Cross sections and asymmetry parameters for photoionization of [formula omitted] in the VUV region

Calculated cross sections and asymmetry parameters for photoionization of the three outermost valence orbitals of CH 3 F for photon energies from near threshold to 50 eV are reported. The photoelectron continuum orbitals are obtained using the Schwinger variational method at the level of static-exchange plus correlation potential. Our results are compared with other theoretical and experimental results showing a good quantitative agreement. Correlation effects do not significantly affect the computed total cross sections and photoelectron asymmetry parameters. The broader structure that appears in the experimental cross section are well described. However, the narrow resonant features near threshold are not found in the present one-electron treatment of the photoionization process.