A perturbation analysis with respect to the space dimension is used to construct singular solutions of the two-dimensional Schrödinger equation with cubic nonlinearity. These solutions blow up at a rate {ln ln[(${\mathit{t}}^{\mathrm{*}}$-t${)}^{\mathrm{-}1}$]/(${\mathit{t}}^{\mathrm{*}}$-t)${\mathrm{\}}}^{1/2}$, in contrast to the behavior in three dimensions where there is no logarithmic correction. The form of such solutions is supported by the results of high-resolution numerical simulations.

The interactions of elementary atoms, i.e., Coulomb bound states of elementary particles such as positronium or pionium, with hydrogen atoms are studied. The processes leading to excitation of elementary and/or the hydrogen atom are discussed. The numerical values of cross sections, in particular, the total ones, of seven elementary atoms are given.

The method that we have recently proposed to correct the local-density approximation of the density-functional theory for spurious self-interaction effects is used to calculate the interconfigurational energies and the ionization potentials of the transition-metal atoms. The results so obtained are compared with the corresponding quantities calculated by using the local-density approximation and the Perdew and Zunger self-interaction correction. In both cases we find very large improvements which are essentially due to a better description of the 3d electrons. The results of this paper show that the large local-density errors for these quantities are mainly due to the self-interaction terms. They also indicate that the latter are probably a major source of errors in the local-density calculations of the electronic properties of the transition-metal dimers and solids.

The method of Bonham, Peacher, and Cox for computing molecular multicenter integrals for 1s Slater-type orbitals is generalized to include all states. This was possible by using B functions as basis functions which have the simplest structure under Fourier transformation, compared with other commonly used exponential-type orbitals (ETO’s). Those ETO’s which differ from B functions, like Slater-type orbitals (STO’s), can be expressed by finite linear combinations of B functions. Therefore multicenter integrals occurring in molecular calculations with any of the commonly used ETO basis sets can be represented by integrals with B functions. In the present paper the three-center nuclear attraction integrals and the two-electron multicenter integrals with B functions are evaluated in a unified way via the Fourier-transform method and Feynman’s identity. The resulting expressions require a two- or three-dimensional numerical integration, respectively. The numerical and computational properties of the resulting formulas are discussed and various test values are given. Comparison is made with some values of integrals with STO’s which exist in the literature.

A general theoretical base and a general strategy for implementing semiclassical quantization using the adiabatic-switching method are presented for two-dimensional systems. The method proposed does not depend on specialized coordinates, trajectory, or surfaces-of-section studies and is generalizable to multidimensional systems. The choice of the initial tori for the switching procedure is accomplished by simple diagonalizations of small-dimensional matrix representations of invariant operators obtained from perturbation theory. The method gives quantum energies at a useful level of accuracy for the vast majority of states in many of the well-known nonresonant and resonant Hamiltonian cases. Many eigenvalues previously thought unobtainable when the adiabatic-switching method is used are obtained in a quite simple manner.

The formulation of a general strategy for implementing semiclassical quantization using the adiabatic-switching method, presented in the preceding paper for two-dimensional systems, is generalized to multidimensional systems. The method is shown to produce accurate semiclassical eigenvalues for the realistic polynomial potential describing ${\mathrm{SO}}_2$ as well as for model potentials for other multidimensional systems. The results are compared with semiclassical results obtained with use of other methods.

We present an accurate numerical solution of the relativistic Thomas-Fermi-Dirac-Weizsäcker variational equation for the case of neutral atoms and positive ions.

A surface dipole layer due to adsorbates on a fine mesh can produce localized electric fields capable of ionizing highly excited atoms. This effect is observed when an atomic beam containing high-Rydberg states of Li is incident on a degassed Au mesh having square openings 6 μm across. During the formation of an adlayer on the surface, the adsorbate field is analyzed through measurements of the excited-state atomic-beam flux transmitted through the mesh. The beam-transmission experiments confirm that the excited-state survival probability is surface-layer dependent. Two distinct dips are observed in the transmission as a function of time, during submonolayer adsorption of ${\mathrm{H}}_2$O and Li on the mesh surface. A cusp between the dips is interpreted as evidence for a change in the sign of the net surface dipole-moment density. These results are compatible with a Langmuir-type model for the survival probability of excited atoms subject to electric-field ionization as they pass through the mesh. Demonstrating that high-Rydberg atoms may be useful as probes for localized electric fields, the experiments serve as feasibility studies for a method of analysis which can readily determine surface dipole orientations and which may have the ability to detect changes in adsorbate coverage at the level of ${10}^{\mathrm{-}4}$–${10}^{\mathrm{-}5}$ monolayer.

The shifted large-N expansion method, which was developed to obtain accurate energy eigenvalues for nonrelativistic-potential problems, has been extended to deal with a relativistic particle (with or without spin) bound in a spherically symmetric potential. The calculations are carried out for any arbitrary quantum state using expansion in terms of a parameter 1/k¯, where k¯ contains the dimension of the space N and the so-called shift parameter. Similar to the work of T. Imbo, A. Pagnamenta, and U. Sukhatme [Phys. Rev. D 29, 1669 (1984)] we suggest determination of the shift parameter in such a way that the exact analytic result for the nonrelativistic Coulomb binding energy is restored. As a consequence of this choice, we obtain also a highly convergent expansion for the relativistic part of the energy eigenvalue. Although the formalism is developed for spin-zero and spin-(1/2 particles in any arbitrary spherically symmetric potential, it is illustrated for the Coulomb potential as a special case. Our results are consistently better than those previously obtained by using the unshifted 1/N expansion technique. The shifted 1/N expansion is seen to be applicable to a much wider class of relativistic potentials which may have applications in atomic processes. A few interesting aspects of our approach are briefly discussed.

The discrete energy levels $E_{\mathrm{nl}}$ of the Schrödinger Hamiltonian H=-(1/2Δ+V(r), where V(r)=-(v/r)[1-rλ(1-1/Z)/(1+rλ)] have been analyzed using the dynamical group formulation of the scaling variational method. A simple analytical formula has been obtained which gives the energy value of different states for various values of Z. We have also found out the exact numerical eigenvalues for the above potential for the 1S, 2S, 2P, 3S, 3P, and 3D cases, and the energy values given by the dynamical group method are in excellent agreement with the exact numerical ones.

A matrix-continued-fraction algorithm which basically relies on an expansion of the eigenfunctions into complete sets is presented for calculating the eigenvalues of the one-dimensional Schrödinger equation with rational potentials. The method also applies to the corresponding three-dimensional central-force potentials. In particular, we investigate the one-dimensional nonpolynomial-oscillator potential ${\mathit{V}}_{\mathrm{NPO}}$(x)=${\mathit{x}}^2$+λ${\mathit{x}}^2$/(1+${\mathit{gx}}^2$), the eigenvalues of which are obtained for a large range of parameters λ and g. In the three-dimensional case we check the shifted 1/N expansion which has been applied to the nonpolynomial potential $V_{\mathrm{NPO}(\mathrm{r}}$) very recently. In addition to the numerical results, analytically exact solutions are given for a restricted class of such potentials. Moreover, we point out that the continued-fraction algorithm applies to a wider class of polynomial and nonpolynomial (rational) potentials. In summary, the matrix-continued-fraction algorithm is shown to produce the eigenvalues of essentially one-dimensional Schrödinger equations for a wide class of rational potentials with very high accuracy, and the proposed technique seems to offer several computational advantages compared to other numerical approaches applied in the field. Finally, an application of the continued-fraction algorithm to the two-variable Schrödinger equation is suggested.

New measurements of direct single and double ionization of helium by hydrogen, helium, carbon, nitrogen, and oxygen ions are presented for projectile charge states ranging from 0 to 3+. These data are combined with previously published values to provide a detailed description of direct double ionization of helium for a wide range of experimental parameters, i.e., projectile ions include protons through oxygen, impact energies from 25 to 5000 keV/u, and projectile charge states ranging from neutral to fully stripped. The charge-state and impact-velocity dependencies of the measured cross sections are compared with the dependencies predicted by an independent electron multiple-ionization model. It is found that a multistep mechanism (i.e., the independent ionization of both target electrons) is the dominant direct double-ionization mechanism for charged-particle impact energies above 100 keV/u. This mechanism dominates until some maximum impact velocity, which depends on the projectile charge, is reached. For higher velocities a single-step ionization mechanism becomes important. The limited neutral-particle impact data indicate that the single-step mechanism becomes dominant at much lower impact velocities and there is no evidence of the multistep mechanism in the neutral-particle impact energy range investigated.

The authors outline an asymptotic calculation of adiabatic potential curves for diatomic quasimolecular systems that happens to be quite accurate in the domain of long-range forces, even for atoms with complex electronic configurations. A formula for the molecular oscillator strength expressed via the oscillator strengths of the constituent atoms and the eigenfunctions of the effective Hamiltonian is presented. Quite general theoretical results on the electrostatic interaction within these quasimolecular systems are then applied to a calculation of the dipole-dipole interaction for the quasimolecule Eu-Sr, leading to the adiabatic potentials in the region of interatomic distances larger than approximately 1.0 nm. These results are checked by comparing the quasistatic absorption coefficient for the blue quasistatic wing of the Eu resonance line broadened by the influence of Sr atoms to the experimentally derived profile [K. Niemax, Phys. Rev. Lett. 55. 56 (1985)]. Excellent agreement is obtained. A great influence of the molecular oscillator strengths on the shape of the quasistatic profile is found, so that, in general, the character of the interaction potential could not be judged from log-log plots of the absorption coefficient alone.

Depolarized interaction-induced light-scattering (DILS) spectra of dense gaseous krypton have been measured at room temperature (297 K) in the pressure range 10–270 bars. The absolute values of the integrated intensity and the second spectral moment have been evaluated, at each density, from the measured spectra, by means of a calibration procedure which makes use of a mixture containing a small fraction of molecular hydrogen, whose rotational Raman lines are used as an intensity standard. Monte Carlo computer simulation has been used to evaluate the first two even spectral moments at various densities and room temperature. The simulation was performed by use of the pairwise-additive potential and polarizability. The pair potential was the one given by R. A. Aziz [Mol. Phys. 38, 177 (1979)] while models for the pair-polarizability anisotropy were derived from the low-density collisional spectrum. The comparison between theory and experiment suggests that the second spectral moment is affected by irreducible many-body terms. It is suggested that, in addition to neutron scattering, DILS can be used as an experimental tool to probe irreducible many-body properties.

Electric quadrupole and magnetic dipole transition probabilities between the 3$p^6$4s ${}_{}{}^2{}_{}{}^{}S_{1/2}^{}{}_{}{}^{}$ and the 3${\mathit{p}}^6$3d ${}_{}{}^2{}_{}{}^{}$${\mathit{D}}_{3/2}$, ${}_{}{}^2{}_{}{}^{}$${\mathit{D}}_{5/2}$ levels for K through ${\mathrm{Mo}}^{23+}$ have been calculated in the (relativistic) Dirac-Fock single-configuration approximation. The positions of the (excited) ${4\mathrm{s}}^2$${\mathit{S}}_{1/2}$ level for ${\mathrm{Mn}}^{7+}$-${\mathrm{Mo}}^{23+}$ are predicted along with the fine-structure splitting in the ground configuration 3$p^6$3d.

We present the principles, applications, and performance of new techniques developed to measure the frequency spectrum of freely precessing polarized noble-gas atoms. The noble-gas polarization technique of spin exchange with laser optically pumped Rb enables us to simultaneously polarize several species ${(}^3$He, ${}_{}{}^{21}{}_{}{}^{}\mathrm{Ne}$, and ${}_{}{}^{129}{}_{}{}^{}\mathrm{Xe}$). With signal-to-noise ratios of ${10}^4$, the free precession frequencies are determined with a precision of less than ${10}^{\mathrm{-}6}$ Hz out of ${10}^3$ Hz in 1024 s. The technique will be applied to determine frequency shifts due to dipole couplings to applied fields and quadrupole splittings due to local Lorentz-invariance-violating spatial anisotropy. In all cases, the frequencies of two or more species are measured simultaneously in order to provide magnetometry for increased precision and to provide measurement and cancellation of systematic effects.

A variational method has been employed for the calculation of the rates of radiative and nonradiative decay of excited states of HeH and NeH. The calculated predissociation rates for the A ${}_{}{}^2{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ state of HeH are larger than the corresponding radiative dissociation rates by about 5 orders of magnitude. For the other excited states of HeH, the predissociation rates are larger by 1–2 orders of magnitude. The theoretical lifetimes of the v=0 level of the A ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$, ${\mathit{B}}^2$Π, C ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$, D ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$, and E ${}_{}{}^2{}_{}{}^{}\mathrm{\Pi }$ states of HeH are 1.2 ps, 17.8 ns, 1 ns, 0.65 ns, and 31 ns, respectively. In NeH, the calculated rate of predissociation of the A ${}_{}{}^2{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ state is comparable to that of radiative dissociation, while for C ${}_{}{}^2{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ radiative dissociation has a higher rate than predissociation. The theoretical lifetimes of the A ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$, B ${}_{}{}^2{}_{}{}^{}\mathrm{\Pi }$, and C ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$ states of NeH (for v=0) are 8.8, 6.0, and 6.9 ns, respectively.

Employing the novel synchrotron photoionization method, the Coster-Kronig decay yields $f_{12}$, $f_{13}$, and $f_{23}$ as well as the ratios of radiative yields ${\omega }_1$/${\omega }_3$ and ${\omega }_2$/${\omega }_3$ have been measured for a series of neighboring heavy elements with a typical uncertainty of ±0.01 and ±0.03, respectively. The data for $f_{23}$ and ${\omega }_2$/${\omega }_3$ are compared to published experimental data obtained with the Kα-Lα coincidence method that have small uncertainty; reasonable agreement is found. In contrast, our $f_{12}$ values are considerably smaller (factor of 2–3) than published experimental values obtained by various techniques; this result indicates systematic problems of these techniques. Using the well-known fluorescence yield ${\omega }_3$ and calculated absolute transition rates of the radiative decay, we have derived absolute transition rates of the nonradiative decay from our experimental data. Comparison of the observed values with theoretical transition rates shows that the theory substantially overestimates the rates of such nonradiative transition channels in which the excess energy carried away by the emitted electron is substantially smaller than the energy of the initial vacancy state.

The predictions of a quantum-mechanical theory of forward four-wave mixing in a homogeneously broadened system of two-level atoms are presented. In the limit of a very short interaction region, the predictions of this theory reproduce those of well-known theories for the spontaneous-emission spectrum of an atom in the presence of an intense laser field. More generally, the theory predicts how the emission spectrum is modified due to propagation effects for a medium of arbitrary length. For long propagation path lengths, the emitted radiation can be quite intense and has a spectrum that is strongly peaked at the Rabi sidebands of the incident laser frequency. The theory shows that Rabi sideband generation in the forward direction can be understood as parametric amplification of weak radiation emitted spontaneously at the Rabi sidebands. The quantum noise that initiates the four-wave-mixing process has contributions both from fluctuations in the incident vacuum radiation field and from fluctuations in the polarization of the atomic dipoles. Both contributions are important for the case of a radiatively broadened medium, although the material fluctuations make the dominant contribution for the case of a medium in which the broadening is largely collisional. Under certain conditions large amounts of squeezing in the radiated field are predicted.

An extended source located near the axis of a nearly circular, sufficiently nonlinear lens can produce extended-arc-like images. Examples of these may have been found near three distant clusters of galaxies which seem to be acting as gravitational lenses. We propose a simple geometrical method for locating images of objects using the Einstein ring and the corresponding caustic. We also give an approximate theory of lenses for which deviations from circularity are small perturbations. We discuss noncircular perturbations of practical interest for gravitational lensing and properties of the arcs that are thus formed.