Hyperspherical Method

Using the hyperspherical adiabatic approach in a coupled-channel calculation, we present precise binding energies of excitons trapped by impurity donors in semiconductors within the effective-mass approximation. Energies for such three-body systems are presented as a function of the relative electron-hole mass in the range 1р1/р6, where the Born-Oppenheimer approach is not efficiently applicable. The hyperspherical approach leads to precise energies using the intuitive picture of potential curves and nonadiabatic couplings in an ab initio procedure. We also present an estimation for a critical value of (crit) for which no bound state can be found. Comparisons are given with results of prior work by other authors.

The nonadiabatic ground state for the helium atom is obtained with the hyperspherical adiabatic approach. Potential curves, nonadiabatic couplings, and channel functions are calculated by a numerically exact procedure based on the analytical expansion of the channel functions. The coupled radial equations are solved by standard techniques. The convergence of the procedure is investigated as nonadiabatic couplings are systematically introduced. The inclusion of 11 potential curves and channel functions gives a ground-state energy that difFers from the best variational calculation by 0.1 parts per million.

We present angular basis functions for the Schrödinger equation of two-electron systems in hyperspherical coordinates. By using the hyperspherical adiabatic approach, the wave functions of two-electron systems are expanded in analytical functions, which generalizes the Jacobi polynomials. We show that these functions, obtained by selecting the diagonal terms of the angular equation, allow efficient diagonalization of the Hamiltonian for all values of the hyperspherical radius. The method is applied to the determination of the 1 S e energy levels of the Li ϩ and we show that the precision can be improved in a systematic and controllable way.

We present within the hyperspherical adiabatic approach a nonadiabatic calculation of the low-lying doubly excited states for heliumlike atomic systems with Z ranging from 2 to 6. Potential curves, channel functions, and nonadiabatic couplings are accurately obtained using a well-established technique. By neglecting the lowest potential curve of each system, the doubly excited states are identified as bound states of the remaining coupled closed channels. We investigate the efficacy of this approximation by comparing the resonance positions with the values obtained when the coupling with the neglected open channel is included in the calculation.

A quantum three-body Coulomb system of two heavy and one light particles (rn 1 >> ma, m2 >> m3) is considered in the framework of the adiabatic hyperspherical approach. Two sets of approximate quantum numbers are found that specify the adiabatic hyperspherical potential curves at small and large hyperradii. A complete classification scheme is obtained for the potential curves of the mesic molecule dtlt.

The results of calculation of the properties of three-and four-particle systems within the framework of the recently suggested stochastic variational method (SVM) are presented. The 3N, 4N and 3~ model systems with two-particle S-wave central forces have been investigated. The results of all calculations are compared with similar results of other workers, and a higher efficiency of the method proposed as compared with other methods is demonstrated.

Using the hyperspherical adiabatic approach in a coupled-channel calculation, we present precise binding energies of excitons trapped by impurity donors in semiconductors within the effective-mass approximation. Energies for such three-body systems are presented as a function of the relative electron-hole mass in the range 1р1/р6, where the Born-Oppenheimer approach is not efficiently applicable. The hyperspherical approach leads to precise energies using the intuitive picture of potential curves and nonadiabatic couplings in an ab initio procedure. We also present an estimation for a critical value of (crit) for which no bound state can be found. Comparisons are given with results of prior work by other authors.

Rigorous definitions are presented for the kinematic angular momentum K of a system of classical particles ͑a concept dual to the conventional angular momentum J), the angular momentum L associated with the moments of inertia, and the contributions to the total kinetic energy of the system from various modes of the motion of the particles. Some key properties of these quantities are described-in particular, their invariance under any orthogonal coordinate transformation and the inequalities they are subject to. The main mathematical tool exploited is the singular value decomposition of rectangular matrices and its differentiation with respect to a parameter. The quantities introduced employ as ingredients particle coordinates and momenta, commonly available in classical trajectory studies of chemical reactions and in molecular dynamics simulations, and thus are of prospective use as sensitive and immediately calculated indicators of phase transitions, isomerizations, onsets of chaotic behavior, and other dynamical critical phenomena in classical microaggregates, such as nanoscale clusters.

A relationship between the total wave function describing electron-impact ionization of hydrogen and the one representing scattering of two electrons and a proton in the continuum is revealed. On the basis of this relationship, forms of the scattered wave for the ionization process valid in all asymptotic domains are obtained. When all interparticle distances become large, the new wave functions reduce to the well-known Peterkop asymptotic wave function obtained in the hyperspherical approach. In particular, the Peterkop wave function is obtained by direct application of the present approach. This allows one to resolve the long-standing amplitude-phase ambiguity problem, which is an artifact of the hyperspherical approach to the ionization process. The Peterkop wave function is invalid when the two electrons are close to each other. This causes problems in practical calculations even in the domain where all particles are far apart. Our formulation provides a solution to this problem.

Ground-state energies and wavefunctions for helium atom (He) in exponential-cosine-screened Coulomb potentials (ECSCP) with screening parameter λ, V (r) = − 1 r e −λr cos (λr) (in au), have been obtained for various values of λ within the framework of Ritz's variational principle. Highly correlated and extensive wavefunctions have been used to obtain energy eigenvalues. The ground-state energies of He for various values of λ have been shown to converge with the increase of the terms in the wavefunction. The ground-state energies of He + along with the ionization potentials of He have also been reported. Our present work represents a first calculation, for the first time in the literature, of the ground-state energies of He interacting with ECSCP.

The hyperspherical adiabatic approach is used to obtain the highly excited series 1sns 1 S e and 1s(n + 1)p 1 P o of the helium atom. The introduction of appropriate asymptotic conditions at large values of the hyperspherical radius results in a stable algorithm that allows the calculation of the full atomic spectrum with precision of a few parts per million. Comparison with the variational calculations available in the literature shows that the accuracy of the results improves with increasing principal quantum number. We present the energies up to n = 31 which is the typical value used in multiphoton excitation experiments.

We present a nonadiabatic calculation, within the hyperspherical adiabatic approach, for the ground state energy of the alkali-metal negative ions. An application to the sodium negative ion (Na Ϫ ) is considered. This system is treated as a two-electron problem in which a model potential is used for the interaction between the Na ϩ core and the valence electrons. Potential curves and nonadiabatic couplings are obtained by a direct numerical calculation, as well as the channel functions. An analysis of convergence is made and comparisons of the electron affinity with results of prior work of other authors are given.

We present within the hyperspherical adiabatic approach a nonadiabatic calculation of the low-lying doubly excited states for heliumlike atomic systems with Z ranging from 2 to 6. Potential curves, channel functions, and nonadiabatic couplings are accurately obtained using a well-established technique. By neglecting the lowest potential curve of each system, the doubly excited states are identified as bound states of the remaining coupled closed channels. We investigate the efficacy of this approximation by comparing the resonance positions with the values obtained when the coupling with the neglected open channel is included in the calculation. 250 J. J. De Groote and M. Masili Low-Lying Doubly Excited States of the Helium Isoelectronic Series 253

Molecular dynamics simulations and both normal mode and hyperspherical mode analyses of NO-doped Kr solid are carried out in order to get insights into the structural relaxation of the medium upon electronic excitation of the NO molecule. A combined study is reported on the time evolution of the cage radius and on the density of vibrational states, according to the hyperspherical and normal mode analyses. For the hyperspherical modes, hyper-radial and grand angular contributions are considered. For the normal modes, radial and tangential contributions are examined. Results show that the first shell radius dynamics is driven by modes with frequencies at ϳ47 and ϳ15 cm −1 . The first one is related to the ultrafast regime where a large part of the energy is transmitted to the lattice and the second one to relaxation and slow redistribution of the energy. The density of vibrational states ␥͑͒ is characterized by a broad distribution of bands peaking around the frequencies of ϳ13, ϳ19, ϳ25, ϳ31, ϳ37, ϳ47, and ϳ103 cm −1 ͑very small band͒. The dominant modes in the relaxation process were at 14.89, 23.49, and 53.78 cm −1 ; they present the largest amplitudes and the greatest energy contributions. The mode at 14.89 cm −1 is present in both the fit of the first shell radius and in the hyper-radial kinetic energy spectrum and resulted the one with the largest amplitude, although could not be revealed by the total kinetic energy power spectrum.

Quantum dynamics calculations for HBr ] Cl ] Br ] HCl are carried out using the hyperspherical elliptic coordinates. The concepts of potential ridge and nonadiabatic transitions at avoided crossings introduced previously for isoergic and symmetric three-dimensional heavyÈlightÈheavy reactions are conÐrmed to be useful to clarify the mechanisms of vibrationally nonadiabatic reactions in this exo-(or endo-)ergic system. The role of important avoided crossings which dominate the reaction dynamics is illustrated.

In the past, several efficient methods have been developed to solve the Schrödinger equation for fournucleon bound states accurately. These are the Faddeev-Yakubovsky, the coupled-rearrangement-channel Gaussian-basis variational, the stochastic variational, the hyperspherical variational, the Green's function Monte Carlo, the no-core shell model, and the effective interaction hyperspherical harmonic methods. In this article we compare the energy eigenvalue results and some wave function properties using the realistic AV8Ј NN interaction. The results of all schemes agree very well showing the high accuracy of our present ability to calculate the four-nucleon bound state.

We have made an investigation to study the plasma screening effect of dense quantum plasmas on the photodetachment cross section of hydrogen negative ion within the framework of dipole approximation. Plasma screening effect has been taken care of by the exponential cosine-screened Coulomb potential ͑EC-SCP͒. The asymptotic forms of highly correlated wave functions for the initial bound states of H − and the plane wave form for the final e − − H states are used to evaluate the transition matrix elements. Results for photodetachment cross section in dense quantum plasmas are reported for the plasma screening parameter in the range ͓0.0,0.6͔ ͑in a 0 −1 ͒. In respect of the photodetachment process of H − , we have also compared the plasma screening effect of a dense quantum plasma with that of a weakly coupled plasma for which plasma screening effect has been represented by the Debye model. Our results for the unscreened case agree nicely with some of the most accurate results available in the literature.

In the past, several efficient methods have been developed to solve the Schrödinger equation for fournucleon bound states accurately. These are the Faddeev-Yakubovsky, the coupled-rearrangement-channel Gaussian-basis variational, the stochastic variational, the hyperspherical variational, the Green's function Monte Carlo, the no-core shell model, and the effective interaction hyperspherical harmonic methods. In this article we compare the energy eigenvalue results and some wave function properties using the realistic AV8Ј NN interaction. The results of all schemes agree very well showing the high accuracy of our present ability to calculate the four-nucleon bound state.

Electron affinities of the alkali atoms sodium to eka-francium are calculated by the intermediate Hamiltonian Fock-space coupled cluster approach, which allows very large P spaces. Large basis sets are used (37s32p23d18f 10g7h for most atoms͒, and many electrons are correlated ͑from 10 for Na Ϫ to 52 for E119 Ϫ ͒ to account for core polarization. While the usual Fock-space method gives errors of 5%-9% for K, Rb, and Cs, the intermediate Hamiltonian results agree with all known values to 5 meV or 1%. The EA of Fr, not known experimentally, is predicted at 491Ϯ5 meV. While EAs decrease from Li to Cs, the Fr value is 20 meV higher than that of Cs, with E119 EA being much higher at 662 meV. This trend reversal is due to relativistic stabilization of s orbitals, which has been shown ͓Eliav et al., Phys. Rev. Lett. 74, 1079 ͑1995͔͒ to give the rare gas E118 positive electron affinity.

For the first time the electric-dipole matrix elements and oscillator strengths for the discrete ISC_1P0 transitions in helium are calculated within the HSA-approach taking into account the nonadiabatic channel coupling. The dependence ofthe energyvalues for the ground and single excited states and oscillator strengths on the number ofchannels are investigated.

For the first time the hyperspherical adiabatic representation is used to carry out the multichannel calculation of the groundstate energies of the He atom and H, Li~and Be 2ĩons. Convergence ofthe hyperspherical adiabatic basis is studied numerically.

For the first time the electric-dipole matrix elements and oscillator strengths for the discrete ISC_1P0 transitions in helium are calculated within the HSA-approach taking into account the nonadiabatic channel coupling. The dependence ofthe energyvalues for the ground and single excited states and oscillator strengths on the number ofchannels are investigated.

We present a study of the influence of the constant uniform electric field on the doubly excited resonances of the positronium negative ion associated with the Nϭ3 positronium threshold. The calculation is performed in the framework of the complex coordinate rotation method which obtains results both for the real part of the energy ͑position of the resonance͒ and for the imaginary part of the energy ͑related to the width of the resonance͒. Results are given for the M ϭ0

We have examined in detail the construction of state-dependent dynamic potentials (SDPs) based on an exact separation scheme of the wavefunction as a product of coupled conditional and marginal probability amplitudes. For particles of equal mass and for rather general forms of the kinetic energy operator in curvilinear coordinates, it is proven that the SDP becomes the expectation value of the total hamiltonian with respect to the conditional amplitude obtained from an assumed real wavefunction. The use of these SDPs for the description of localized wavefunctions is also discussed. As an example we have undertaken the analysis of SDPs arising from various separation schemes of Hylleraas coordinates for two simple wavefunctions approximating the ground state of helium. The separation of s = r, + rz, t = r, -r, and rtz leads to SDPs exhibiting minima in the regions where localization of the wavefunction takes place. The possbility of an adiabatic breakup, Born-Oppenheimer type, of the problem is also considered. bur results show that strong "non-adiabaticity" is a distinct feature of all the separations we considered. The separation of r, is treated too as an example of a non-collective variable and the corresponding SDP resembles an effective Hartree-like one-electron potential. We finally sketch some possible applications of the present approach for the analysis of localization and non-adiabaticity in more complex situations in atomic and molecular cases.

Methods for applying the variationally stable procedure for Nth-order perturbative transition matrix elements of Gao and Starace [Phys. Rev. Lett. 61, 404 (1988); Phys. Rev. A 39, 4550 (1989)] to multiphoton processes involving systems other than atomic H are presented. Three specific cases are discussed: one-electron ions or atoms in which the electron-ion interaction is described by a central potential; two-electron ions or atoms in which the electronic states are described by the adiabatic hyperspherical representation; and closed-shell ions or atoms in which the electronic states are described by the multiconfiguration Hartree-Fock representation. Applications are made to the dynamic polarizability of He and the two-photon ionization cross section of Ar.

Bound and resonance states of a two-electron quantum dot are studied using a variational expansion with real basis-set functions. The two-electron entanglement ͑von Neumann entropy͒ is calculated as a function of the quantum-dot size at both sides of the critical size, where the ground ͑bound͒ state becomes a resonance ͑unbound͒ state. The use of von Neumann entropy is proposed as a method for the determination of the energy of a resonance.

Using a rapidly convergent composite basis of Frankowski-Pekeris and Frankowski functions, we have accurately calculated the nodal surfaces of low-lying excited states of the helium atom to investigate Bressanini and Reynolds' conjecture ͓D. Bressanini and P. J. Reynolds, Phys. Rev. Lett. 95, 110201 ͑2005͔͒ that these nodal surfaces are rigorously independent of the interelectronic angle 12 . We find that in fact there is a slight dependence of the nodal surfaces on 12 , but it is so small that the assumption of strict independence may well yield extremely useful approximations in fixed-node quantum Monte Carlo calculations. We explain how Kato's cusp conditions determine the qualitative features of these nodal surfaces, which can accurately be modeled using the familiar ansatz of a symmetric or antisymmetric linear combination of products of hydrogenic orbitals, with some adjustments of the parameters. We explain why a similar near independence of the nodal surfaces on the angular variables can be expected for the ground and singly excited states of the lithium atom, but generally not for larger atoms.

Energy levels of highly excited bound Rydberg states, the position and widths of autoionizing states, and oscillator strengths are calculated for He 3 S, 3 P e , 3 P o , 3 D e and 3 D o symmetries up to the N = 5 He + excitation threshold. The calculations are performed with the Kmatrix B-spline method with maximum orbital angular momentum ' max = 8. Reliable doubly excited-state parameters up to the n = 20 multiplet below each ionization threshold are presented. One thousand and six hundred newly identified bound and metastable states, seven times those available in literature, fill many gaps, reveal a dozen intruder states, and allow new speculations on propensity rules and radiative decays of triplet Rydberg states.

We present a nonadiabatic calculation, within the hyperspherical adiabatic approach, for the ground state energy of the alkali-metal negative ions. An application to the sodium negative ion (Na Ϫ ) is considered. This system is treated as a two-electron problem in which a model potential is used for the interaction between the Na ϩ core and the valence electrons. Potential curves and nonadiabatic couplings are obtained by a direct numerical calculation, as well as the channel functions. An analysis of convergence is made and comparisons of the electron affinity with results of prior work of other authors are given.

 We present within the hyperspherical adiabatic approach a nonadiabatic calculation of the low-lying doubly excited states for heliumlike atomic systems with Z ranging from 2 to 6. Potential curves, channel functions, and nonadiabatic couplings are accurately obtained using a well-established technique. By neglecting the lowest potential curve of each system, the doubly excited states are identified as bound states of the remaining coupled closed channels. We investigate the efficacy of this approximation by comparing the resonance positions with the values obtained when the coupling with the neglected open channel is included in the calculation.

We present a generalization of the variationally stable method of Gao and Starace ͓B. Gao and A. F. Starace, Phys. Rev. Lett. 61, 404 ͑1988͒; Phys. Rev. A 39, 4550 ͑1989͔͒ for two-electron atoms and ions that incorporates a coupled-channel adiabatic hyperspherical approach. Using this approach, we report results for twophoton detachment of H Ϫ , in which we have coupled one, two, three, and four adiabatic hyperspherical channels within each term level of the initial, intermediate, and final states. We present results also for the dynamic polarizability of H Ϫ as well as for the one-photon detachment cross section. Comparisons are given with results of prior work.

Using a generalization(M. Masili and A. F. Starace, Phys. Rev. A 62), 033403 (2000). of the variationally stable method of Gao and Starace(B. Gao and A. F. Starace, Phys. Rev. Lett. 61), 404 (1988); Phys. Rev. A 39, 4550 (1989). for two-electron atoms and ions, which incorporates a coupled-channel adiabatic hyperspherical approach, we report results for the dynamic polarizability of the helium atom for frequencies below the single photon ionization threshold. Comparison of results of coupling one, two, three, and four adiabatic hyperspherical channels within each term level of the initial and intermediate states show good convergence. Comparisons are also given with results of prior work by others. ^*Supported by FAPESP (Brazilian agency) under Process No. 99/11363-8 and by the U.S. D.O.E, B.E.S., Div. Chem. Sci. under Grant No. DE-FG03-96ER14646.

We present angular basis functions for the Schrödinger equation of two-electron systems in hyperspherical coordinates. By using the hyperspherical adiabatic approach, the wave functions of two-electron systems are expanded in analytical functions, which generalizes the Jacobi polynomials. We show that these functions, obtained by selecting the diagonal terms of the angular equation, allow efficient diagonalization of the Hamiltonian for all values of the hyperspherical radius. The method is applied to the determination of the 1 S e energy levels of the Li ϩ and we show that the precision can be improved in a systematic and controllable way.

The hyperspherical adiabatic approach is used to obtain the highly excited series 1sns 1 S e and 1s(n + 1)p 1 P o of the helium atom. The introduction of appropriate asymptotic conditions at large values of the hyperspherical radius results in a stable algorithm that allows the calculation of the full atomic spectrum with precision of a few parts per million. Comparison with the variational calculations available in the literature shows that the accuracy of the results improves with increasing principal quantum number. We present the energies up to n = 31 which is the typical value used in multiphoton excitation experiments.

Energies and wavefunctions are calculated for the bound states of the helium atom in the hyperspherical adiabatic approach by the full inclusion of nonadiabatic couplings. We show that the use of appropriate asymptotic radial boundary conditions not only allows the efficient calculation of energies accurate up to a few ppm for the ground state but also gives increasingly precise results for high-lying excited states with a unique set of equations. The accuracy of the wavefunctions is demonstrated by the calculation of oscillator strengths in the length form for transitions between states n 1 S e and (n + 1) 1 P o up to n = 29, in agreement with variational calculations.

Binding energies for excitons trapped by impurities in ZnSe and CdS are calculated within the hyperspherical adiabatic approach. Nonadiabatic couplings are included in the radial equations, leading to energies lower than the variational values available in the literature. Resonant states similar to autoionizing lines in atoms are predicted to lie above the first electron-impurity ionization threshold.