Bound-state Region Research Articles

Analysis of Stabilization and Extrapolation Methods for Determining Energies and Lifetimes of Metastable Electronic States.

Two methods that make use of standard electronic structure tools, the stabilization and extrapolation methods, are discussed with an eye toward pointing out their relative strengths and weaknesses and for improving their applications. In the former, whether to utilize energy data from only one or from both branches of an avoided crossing between the quasi-bound and pseudo-continuum states is one issue that is focused on. Another is the decision of where along the stabilization plot's branches (i.e., far from or close to the avoided crossing) to create data points for optimal performance given a reasonable (10-5-10-7 eV) precision in the electronic energy. A third issue is how many parameters to use in fitting energy data to the (one or two) branches of the stabilization plot. In extrapolation methods, one uses energy data computed when the metastable state's energy has been rendered stable by the application of an external potential, which thus produces a one-branch function. The main issues in implementing this method are the functional form for how the energy E depends on the strength of the external potential especially as the energy evolves from the bound-state region toward the unbound region and how to choose data points so that energy values of a reasonable precision are capable of determining the parameters in the formula that produces the metastable state's energy E and half-width Γ/2 (inversely related to the state's lifetime). In addition to explaining, critiquing, and comparing these two methods, several suggestions are offered for their further testing and improvements.

Particle scattering and resonances involving avoided crossing

In molecules, the nonadiabatic couplings between two adiabatic potentials build the avoided crossing (AC) region. The rovibronic resonances in the AC region of two-coupled potentials are very special, since they are not in the bound state region of the adiabatic potentials, and they usually do not play important roles on the scatterings and are less discussed. Exemplified in particle scattering, resonances in the AC region are comprehensively investigated. The effects of resonances in the AC region on the scattering cross sections strongly depend on the nonadiabatic couplings of the system, it can be very significant as sharp peaks, or inconspicuous buried in the background. More importantly, it shows a simple quantity proposed by Zhu and Nakamura (1992 J. Chem. Phys. 97 8497) to classify the coupling strength of nonadiabatic interactions, can be well applied to quantitatively estimate the importance of resonances in the AC region. Example applications of the quantity for real molecules (MgH, CO and OH) can well explain the evolutions of cross sections in the AC region published in the literatures. This work provides a simple and practical way to determine the candidate molecules when studying the resonances in the AC region.

Search for η^{'} Bound Nuclei in the ^{12}C(γ,p) Reaction with Simultaneous Detection of Decay Products.

We measured missing mass spectrum of the ^{12}C(γ,p) reaction for the first time in coincidence with potential decay products from η^{'} bound nuclei. We tagged an (η+p) pair associated with the η^{'}N→ηN process in a nucleus. After applying kinematical selections to reduce backgrounds, no signal events were observed in the bound-state region. An upper limit of the signal cross section in the opening angle cosθ_{lab}^{ηp}<-0.9 was obtained to be 2.2 nb/sr at the 90%confidence level. It is compared with theoretical cross sections, whose normalization ambiguity is suppressed by measuring a quasifree η^{'} production rate. Our results indicate a small branching fraction of the η^{'}N→ηN process and/or a shallow η^{'}-nucleus potential.

Impurity-Induced Multibody Resonances in a Bose Gas.

We investigate the problem of N identical bosons that are coupled to an impurity particle with infinite mass. For noninteracting bosons, we show that a dynamical impurity-boson interaction, mediated by a closed-channel dimer, can induce an effective boson-boson repulsion which strongly modifies the bound states consisting of the impurity and N bosons. In particular, we demonstrate the existence of two universal "multibody" resonances, where all multibody bound states involving any N emerge and disappear. The first multibody resonance corresponds to infinite impurity-boson scattering length, a→+∞, while the second corresponds to the critical scattering length a^{*}>0 beyond which the trimer (N=2 bound state) ceases to exist. Crucially, we show that the existence of a^{*} ensures that the ground-state energy in the multibody bound-state region, ∞>a>a^{*}, is bounded from below, with a bound that is independent of N. Thus, even though the impurity can support multibody bound states, they become increasingly fragile beyond the dimer state. This has implications for the nature of the Bose polaron currently being studied in cold-atom experiments.

Missing-mass spectroscopy with the Li6(π−,K+)X reaction to search for HΛ6

We searched for the bound state of the neutron-rich $\Lambda$-hypernucleus ${}^{6}_{\Lambda}$H, using the ${}^{6}$Li($\pi^{-}, K^{+}$)X double charge-exchange reaction at a $\pi^{-}$ beam momentum of 1.2 GeV/c at J-PARC. A total of $1.4 \times 10^{12}$ $\pi^{-}$ was driven onto a ${}^{6}$Li target of 3.5-g/$\rm cm^2$ thickness. No event was observed below the bound threshold, i.e., the mass of ${}^{4}_{\Lambda}$H + 2n, in the missing-mass spectrum of the ${}^{6}$Li($\pi^{-}, K^{+}$)X reaction in the $2^{\circ}$ < $\theta_{\pi K}$ < $20^{\circ}$ angular range. Furthermore, no event was found up to 2.8 MeV/$c^2$ above the bound threshold. We obtained the the double-differential cross section spectra of the ${}^{6}$Li($\pi^{-}, K^{+}$)X reaction in the angular range of $2^{\circ}$ < $\theta_{\pi K}$ < $14^{\circ}$. An upper limit of 0.56 nb/sr (90% C.L.) was obtained for the production cross section of the ${}^{6}_{\Lambda}$H hypernucleus bound state. In addition, not only the bound state region, but also the $\Lambda$ continuum region and part of the $\Sigma^{-}$ quasi-free production region of the ${}^{6}$Li($\pi^{-}, K^{+}$)X reaction, were obtained with high statistics. The present missing-mass spectrum will facilitate the investigation of the $\Sigma^{-}$-nucleus optical potential for $\Sigma^{-}$-${}^{5}$He through spectrum shape analysis.

Energy-dependent target widths in a coupled-channel scattering study

The question of how the scattering cross section changes when the spectra of the colliding nuclei have low-excitation particle-emitting resonances is explored using a multichannel algebraic scattering method. As a test case, the particle-unstable nucleus $^{8}\mathrm{Be}$ has been considered as a target. Nucleon-nucleus scattering cross sections, as well as the spectra of the compound nuclei formed, have been determined from calculations that consider, as well as those that disregard, particle-emission widths of the target nuclear states. The resonant character of the unstable excited states introduces a problem because the low-energy tails of these resonances can intrude into the subthreshold, bound-state region. This unphysical behavior needs to be corrected by modifying, in an energy-dependent way, the shape of the target resonances from the usual Lorentzian form. The resonance function must smoothly reach zero at the elastic threshold. We explore herein a way of achieving this condition.

Potential energy of ?H 2 - (2? u + in the bound-state region

An accurate calculation of the lowest negative electronic state of ∞H 2 - (fixed nuclei) is reported using the CCSD(T) method and doubly augmented cc-pv5z basis set. Comparison has been made with the reference data by Senekowitsch et al. [Chem. Phys. Lett. 111 (1984) 211]. Owing to larger size of the basisset and inclusion of triple excitations, no vertical shift in this work is necessary to reproduce the asymptotics of ∞H + ∞H -. In addition, the effect of basis-set truncation is estimated, based on the complete-basis-set extrapolation method. The contribution of correlated electron-proton motion to the electron-energy curve for H2 − dynamics is pointed out.

VariationalR-matrix calculations for singly and doubly excited singlet gerade channels inH2

Variational ab initio R-matrix theory is combined with generalized multichannel quantum defect theory, implemented in spheroidal coordinates, to calculate clamped-nuclei {sup 1}{sigma}{sub g}{sup +}, {sup 1} product {sub g}, and {sup 1}{delta}{sub g}{sup +} electron-ion scattering phase shift matrices for H{sub 2}. The calculations cover the bound state region below H{sub 2}{sup +} 1{sigma}{sub g}, the resonance region between H{sub 2}{sup +} 1{sigma}{sub g} and H{sub 2}{sup +} 1{sigma}{sub u}, and they extend beyond the H{sub 2}{sup +} 1{sigma}{sub u} threshold. They span the range of internuclear distances 1{<=}R{<=}5 a.u. The use of spheroidal instead of spherical coordinates allows a restricted partial wave expansion to be used, thus yielding a compact set of interaction parameters pertaining to the electron-ion scattering dynamics in H{sub 2}. The accuracy of our fixed-nuclei quantum defects is generally of the order of about 0.02. At the same time the quantum defect matrices obtained here exhibit a smooth behavior across the ionization thresholds and their elements also vary rather smoothly with internuclear distance. These results represent a step toward the goal of constructing a unfied theoretical description of ionization and dissociation fragmentation dynamics of H{sub 2}.

Model Study of the Vibrational Level Pattern of Bent Symmetric Triatomic Molecules: Application to CH2 and CD2

A simple but flexible model was devised to examine the effects of barrier to linearity and mode coupling on spectral pattern of bent symmetric triatomic molecules. Despite simplicity, the model was found to represent the potential energy surface reasonably well in the bound state region. While the results from the model showed the usual behavior of bending spectral pattern of bent molecules for the cases of low and moderate barrier height to linearity, we also observed the increase of successive bend level spacings with energy even before the bend energy reaches the top of the barrier for a molecule with a significant barrier height to linearity and near-linear equilibrium geometry. We applied our model to simulate the potential energy surface for the ground (X̃3B1) and first excited (ã1A1) electronic states of CH2 and CD2. The vibrational levels of CH2 and CD2 obtained from the model show a reasonable agreement for the X̃3B1 electronic state and excellent agreement for ã1A1 state with the experimental and other theoretical results. From our high-quality ab initio electronic study of X̃3B1 state for CH2, we determine the equilibrium bond length of 1.0776 Å, bond angle of 133.69°, and barrier height to linearity of 1983 cm-1 which are in close agreement with experimental results by Bunker et al. (J. Chem. Phys. 1986, 85, 3724). It is suggested that the discrepancy in the fundamental bending frequency for X̃3B1 state of CH2 between the experiment and previous theoretical results could be due to the unusual floppiness of potential energy surface near equilibrium as well as lower barrier height to linearity for this molecule.

Two-electron excitations and one-electron multiple-scattering resonances in the x-ray absorption of solid neon.

The local and partial density of states of the conduction band of solid neon in a large energy interval has been extracted from the analysis of the K-edge x-ray-absorption near-edge structure (XANES) spectrum by using a full multiple-scattering method. The absorption cross section for one-electron transitions from the Ne 1s level to the conduction-band final states in the solid Ne is predicted by first principles. The absorption cross section is described as due to an atomic factor and to a structure factor determined by the local and partial density of states for a large cluster formed by about 80 atoms. The experimental spectrum is obtained by describing the final state in the fully relaxed potential in the presence of the core hole. The exciton peak at threshold is shown to be due to the lowest unoccupied valence state pushed down in the bound-state region by the core hole. In the 100-eV XANES energy range the single-scattering approximation fails to predict the XANES of solid Ne indicating that the full multiple-scattering approach is required. The present analysis allows us to identify strong two-electron excitations 1${\mathit{s}}^{2}$2${\mathit{s}}^{2}$2${\mathit{p}}^{6}$\ensuremath{\rightarrow}1${\mathit{s}}^{1}$2${\mathit{s}}^{2}$2${\mathit{p}}^{5}$${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{i}}$${\mathit{p}}^{2}$ in the energy range of 30 eV above the K edge, involving monopole 2p\ensuremath{\rightarrow}\ensuremath{\varepsilon}p transitions to the lowest valence states at energy \ensuremath{\varepsilon}. The intensity of the two-electron excitations is found to be so strong that it should be considered in standard extended x-ray-absorption fine-structure--XANES data analysis for local structure investigation.

Effects of bound pairs on the low-density transport properties of gases

In some derivations of the Boltzmann equation, the equation which describes the time evolution of the distribution function in the phase space of the atoms of a low-density gas, it has been assumed that all of the trajectories in the phase space of the pairs of atoms are such that transforming back sufficiently far in time leads to separated atoms. In a recent development, it has been shown that if this is not true the ‘‘bound-state’’ trajectories lead to an addition to the collision integral which, in turn, leads to corrections to the values of the low-density transport coefficients. In the present alternate development, it is shown that the ‘‘bound-state’’ trajectories may be joined to trajectories in the ‘‘unbound region’’ through the introduction of a complex time and analytic continuation. With this generalization of the concept of trajectories, one may introduce the ‘‘molecular chaos assumption’’ in the usual manner, and the ‘‘collision integral’’ may be generalized to include the bound-state region of the phase space. This leads to ‘‘corrections’’ to the low-density limiting values of the transport coefficients. These effects of the bound pairs of atoms on the low-density transport coefficients of a gas of atoms which interact according to a Lennard-Jones potential are investigated numerically. It is found that the effects become important at temperatures, scaled in the usual manner by ε/k, below about unity. Specifically, it is found that at a scaled temperature of unity, corresponding to a temperature in the range of about 100–200 K, the effect is to lower the self-diffusion coefficient by about 3.7% and the viscosity and thermal conductivity by about 0.7%. The effects become significantly greater at lower temperatures.

A critical interpretation of the near-edge X-ray absorption fine structure of chemisorbed benzene

Near-edge X-ray absorption fine structure (NEXAFS) measurements for submonolayer benzene coverages on Ag(110) are reported and interpreted in light of recent electronic structure calculations for free benzene. Quantitative analysis of the continuum resonances indicates that benzene is bound parallel to Ag(110) while analysis of the C(1s) → π ∗ (e 2u) transition intensities yields a maximum tilt of ~ 10–15°. The apparent tilting suggested by the π ∗ resonances is most likely not a real and global tilt, but probably arises from the presence of tilted minority species. A comparison of the NEXAFS spectra for benzene on Ag(110) with those previously reported for benzene on Pt(111) and Mo(110) and phenyl thiolate on Mo(110) indicates that strong adsorbate-substrate interactions may result in significant perturbations in the bound-state region of the near-edge structure, as well as shifting of the continuum resonances due to changes in intramolecular bond lengths.

Spurious bound states in relativistic three-body equations

We point out that the most commonly used version of the relativistic Faddeev equations has spurious bound-state solutions and therefore it is unsuitable for calculations in the bound-state region. We propose a way to modify these equations which would correct the problem, but which leads to the loss of the clustering property. We present numerical examples for the case of the ${\ensuremath{\pi}}^{\ensuremath{-}}$ nn system.NUCLEAR REACTIONS ${\ensuremath{\pi}}^{\ensuremath{-}}$ nn bound states; calculated Fredholm determinant $0\ensuremath{\le}\sqrt{S}\ensuremath{\le}2018$ MeV.

Variable phase method for the calculation of the scattering phase shift: The bound-state region

Variable phase methods for the calculation of the scattering phase shift in non-relativistic, quantum mechanical potential scattering are investigated with emphasis on their use for negative energies (bound-state region) and general (not necessarily integral) angular momenta. Of the various phase methods known to be valid for positive energies (scatering region), some remain valid and some become invalid for negative energies. One of the phase methods demonstrated to be valid for both positive and negative energies is fast for numerical computation of the scattering phase shift and so is useful for phenomenological investigations.

Electronic density of states of a one-dimensional superlattice

The electronic density of states in a long-period superlattice is investigated. Using the transfer matrix formalism, exact solutions are obtained for a one-dimensional model consisting of two square well potentials of finite depth and width. New energy bands and minigaps are found to develop not only in the bound state region but also in the ionisation region.

A study of the Galitskii-Feynman T matrix for liquid 3He

The Galitskii-Feynman T matrix, which sums the infinite ladder series in a many-fermion system for both particle-particle and hole-hole scattering, is studied in detail for a family of realistic He-He interactions. The structure of the S-wave bound-state singularity, reported previously, and its dependence on the bare interaction are documented at length. Special attention is devoted to the T matrix in the scattering region, where the c.m. energy of the interacting pair is positive. In particular, the on-energy-shell T matrix in this region is parametrized in terms of real “effective” phase shifts incorporating many-body effects. The critical behavior discussed previously in the bound-state region manifests itself clearly in the zero-energy limit of these phase shifts for the S wave. Below (above) a certain critical density, which is a function of both temperature and c.m. momentum, this limit approaches the value 0(−π) radians. A generalized Levinson's theorem relates this behavior to the existence of fermion-fermion pairing. An especially striking feature of these many-body phase shifts is the cusp behavior exhibited at the Fermi surface in the lowtemperature limit, which turns out to arise essentially from the structure of the particle and hole occupation probabilities. Throughout this study the temperature dependence of the T matrix is particularly emphasized.

Bound states of 4He with local interactions

The Yakubovsky four-body equations for the 0 +-state are solved in the boundstate region for the case of local central forces of the Yukawa type, using the Hilbert-Schmidt expansion. As a result the groundstate energy of 4He is found to be −29.6 MeV.

Investigation of thePb208(p, p′)Reaction atE=54MeV

Collective excitations of $^{208}\mathrm{Pb}$ were investigated by inelastic proton scattering at 54 MeV. The spectra covered the entire bound-state region for scattering angles between 11 and 58\ifmmode^\circ\else\textdegree\fi{}, and scattered particles were detected in a broad-range spectrograph with energy resolution \ensuremath{\approx} 35-40 keV full width at half maximum. The experimental angular distributions were compared to those predicted by distorted-wave Born-approximation calculations, utilizing a collective-model form factor. The location of the low-spin members of a sequence (${2}^{+}$-${8}^{+}$) of positive-parity states was confirmed. Likely candidates for $J&gt;8$ states were found, as well as several collective fragments of ${3}^{\ensuremath{-}}$ and ${4}^{+}$ strengths. No appreciable fragmentation of the ${2}^{+}$ strength was found. Energy-weighted sum-rule strengths were estimated for all multipoles studied and were compared with the measured values.NUCLEAR REACTIONS $^{208}\mathrm{Pb}(p, {p}^{\ensuremath{'}})$, $E=54$ MeV; measured absolute $\ensuremath{\sigma}(E, \ensuremath{\theta})$. $^{208}\mathrm{Pb}$ levels, deduced $L$, ${{\ensuremath{\beta}}_{L}}^{2}$. Spectrograph.

Calculations of Three-Particle Scattering Amplitudes Using Local Yukawa Potentials

Calculations of three-particle phase shifts for a system interacting via local Yukawa potentials are presented. The method involves numerical analytic continuation from the bound-state region to the scattering region. The method is also applied to separable potentials to verify that it in fact gives correct results.

Static Model Using Crossing-Symmetric Off-Shell Equations Satisfying Two-Particle Unitarity

Off-shell equations for the ${\ensuremath{\pi}}^{\ensuremath{-}}p$ and ${\ensuremath{\pi}}^{+}p$ elastic scattering amplitudes are derived for a static model, based on charged scalar theory. These amplitudes satisfy elastic unitarity below the three-particle cut and are related by crossing symmetry. The equations are solved numerically in the bound-state region for the entire range of coupling, and the results compared with the known results of charged scalar theory. In particular, calculations of the bound-state ${N}^{++}$ are reasonable. Possible extensions to more realistic models are discussed.