Relativistic corrections to order ${\alpha }^2$ to the Bethe sum rule have been obtained for a one-electron system employing the Foldy-Wouthuysen transformation. The results have wide applications in high-Z systems at large momentum transfers.

Results are presented for an electron-impact ionization study of Bi microclusters ranging in size from one to thirty-eight atoms. By comparing results from two different cluster source conditions, we identify a clear onset for fragmentation in the vicinity of 10.5 eV. This is below the threshold for double ionization, yet well above the first ionization thresholds. The latter thresholds have been found to be almost universally double and are quite sharp, suggesting unique structures for each cluster size. The first threshold energies vary with size in a manner consistent with the classical metal-sphere model.

An algebraic approach for the computation of resonance positions and widths by elimination of a noncompact generator by diagonalization is discussed. For an analytically solvable model potential, the method is shown to be completely equivalent to the method of complex rotation of coordinates.

We give a collective variable description of high-gain free-electron-laser amplifiers which reproduces the solution of microscopic equations quite satisfactorily both in the linear and in the nonlinear regime.

The annihilation rate of Ps is proportional to the square of the wave function at the origin. This can be affected by exposure of the atom to a resonant laser. The result depends upon whether the excited state is an S or a P state and whether it is singlet or triplet. An alteration of the decay rate by as much as a factor of 2 is possible.

We study a model for dynamic percolation relevant to the electrical conductivity of water in oil microemulsions. The charge carriers reside on percolation cluster sites and can propagate by hopping between nearest-neighbor sites. The cluster sites also undergo diffusion, so that the clusters continuously rearrange themselves. The conductivity below the percolation threshold $p_c$ is finite and, as p→$p_c$, increases as ‖$p_c$-p${\Vert }^{\mathrm{-}s̃}$ where s̃ differs from the static exponent s. In the neighborhood of $p_c$, the conductivity depends as a power law on the rate of cluster rearrangement.

The structure of periodic windows in one-dimensional maps is shown to become quantitatively universal approaching a period-doubling accumulation point from the opposite side. This implies, among other things, that the relative density of aperiodic solutions tends to a universal number ρ=0.892. . .. This universality of the window structure applies to the same class of maps as the period-doubling universality, and we have strong numerical evidence for it to apply to any dissipative multidimensional dynamics which goes through a complete period-doubling sequence.

A simple renormalization procedure is constructed directly from the mapping equation for the standard mapping. The spectrum is then derived. Some properties of the mapping are discussed.

The series of truncated entropies, approaching the Kolmogorov-Sinai entropy when correlations between increasingly distant signals are taken into account, is considered. A general expression for the related characteristic time is obtained for one-dimensional maps. Evaluating it analytically near the state of maximal entropy and at intermittency we find it reflecting faithfully the degree of stochasticity.

We demonstrate for the standard map how ‘‘Cantori’’ can be found by Newtonian methods. Of crucial importance is the optimal choice of initial conditions derivable from one of Aubry’s theorems.

Standard theory predicts that the familiar exponential decay law must be modified at long times, a result that was recently challenged by Mittleman and Tip, who claimed, in the context of a model problem, that the modification was strongly attenuated if the artificial assumption of an interaction that is suddenly switched on is replaced by a gradual onset. We have studied the same question, and find that while this reduction is present, it is of a different form than that predicted by Mittleman and Tip, and could be negligible in certain cases. We suggest that their model might actually be a promising candidate problem for an experimental investigation of departures from exponential decay.

The natural width of molecular vibrational spectra arises from anharmonic coupling with zero-point vibrations of other modes. The stochastic quantization of Nelson describes a quantum-mechanical state as a Markovian random process in which the position is a time-dependent random variable even in a pure stationary state. As a consequence, the zero-point fluctuation is shown to give rise to the quantum-mechanical motional narrowing of molecular vibrational spectra.

The laser-rf double-resonance technique has been used to make precise measurements of the hyperfine structure (hfs) of 10 excited 4$f^{11}$5d6$s^2$ levels in ${}_{}{}^{167}{}_{}{}^{}\mathrm{Er}$. The results are compared with multiconfiguration Dirac-Fock calculations using the known nuclear moment values. For the dipole hfs, agreement with experiment is reasonably good. For the quadrupole hfs, agreement is relatively good if allowance is made for different shielding in the 4f and 5d shells. The theory is less successful where the 4f and 5d contributions tend to cancel. Sternheimer factors $R_{\mathrm{nl}}$ are obtained. Precise hfs studies are also made for one highly excited 4$f^{12}$6s6p level. Less precise results are given for hfs of more highly excited levels and for isotope shifts of a number of optical lines.

We consider an exactly soluble model consisting of two electrons attracted to a common center by harmonic-oscillator forces with mutual Coulomb repulsion between the electrons. The exact wave function and energy for the singlet ground state is obtained by separating the Schrödinger equation in center-of-mass and relative coordinates and numerically integrating the relative-coordinate eigenvalue equation. The exact Kohn-Sham (KS) orbital for the two equivalent electrons with opposite spins is constructed from the exact density, and the Kohn-Sham equation is inverted, yielding the exact exchange-correlation potential as a function of position.

The exact correlation potential, as a function of position, is then obtained by subtracting the exact exchange potential from the exact exchange-correlation potential. The exact ground-state energy, exchange potential, correlation potential, KS single-particle energy eigenvalues ${\epsilon }_{\mathrm{KS}}$, exchange energy, and correlation energy are compared to the results given by density-functional theory employing the local-density approximation (LDA) plus correction terms and the exact KS orbitals for values of the harmonic-oscillator spring constant that vary over 4 orders of magnitude. We find that although total energies are accurately given by LDA plus correction terms, the values of ${\epsilon }_{\mathrm{KS}}$ are significantly in error. We show that this is mainly due to errors in the exchange potential.

In addition we show analytically that in a singlet ground state of a two-electron system, the exact expectation value of the exchange potential equals the exact exchange energy, whereas the LDA expectation value of the exchange potential is only (2/3) the LDA exchange energy. Furthermore, we show that although the gradient expansion approximations for the exchange energy are significant improvements over the LDA, neither of these approximations significantly decreases the error in the LDA for the expectation value of the exchange potential. Both the correlation energy and the expectation value of the correlation potential in the LDA plus corrections are significantly in error when compared with the results of the exact calculation, the most accurate results being given by the LDA with self-interaction correction.

Strong configuration-interaction effects between autoionizing configurations of the form (ns${)}^1$(np${)}^{m+1\mathrm{}}$ and bound configurations of the form (ns${)}^2$(np${)}^{m\mathrm{-}1}$(n’l’) have been treated with truncated-discrete-basis calculations, either neglecting or approximating the continuum. It is shown that Fano’s treatment of a level interacting with the continuum can be extended to the threshold region by including the Rydberg series of discrete levels. Inclusion of the infinite Rydberg series leads to a logarithmic singularity of matrix elements and the eigenvalue equation near the threshold region, which cancels the singularity arising from the integral over the continuum.

The cancellation leads to a relatively simple eigenvalue equation for the strongly perturbed eigenvalues. Numerical evaluation of the eigenvalue equation leads to eigenvalues comparable to those found with multiconfiguration Hartree-Fock calculations. Concentration on the threshold regions leads to a unique prescription for passive continua normalization independent of the number of continua used. Numerical evaluation of the eigenvalue equation predicts a (4s${)}^1$(4p${)}^5$ ${}_{}{}^3{}_{}{}^{}P$ level in Se 10 000 ${\mathrm{cm}}^{\mathrm{-}1}$ below the first ionization threshold, though no evidence for such a level is observed spectroscopically. By including the discrete levels in the Fano formalism, it is shown that the infinite discrete level sum plus integral over the continuum can be rewritten as an integral of the interaction potential and a modified one-electron Green’s function, leading to a ‘‘dressed’’ one-electron orbital, which is readily computed and which contains the interesting effects of the strong interaction. Such dressed orbitals are used to calculate oscillator strengths to the ground state for the neutral halogens, ${\mathrm{Ar}}^{1+}$, and the neutral elements of column VI in the periodic table. For Cl, I, and Se, the calculated oscillator strengths from the ground state to the (ns${)}^1$(np${)}^{m+1\mathrm{}}$ configuration are less than ${10}^{\mathrm{-}4}$, offering a plausible explanation for the nonidentification (Se) or misidentification (I) of these levels.

The composite-particle representation method is utilized to understand why spin-polarized atomic hydrogen (H↓) behaves as a bosonic system. We define a quantity called the bosonic probability ($P_B$) as a measurement of the accuracy of the bosonic description for a fermion system. The results show that when the distance between two hydrogen atoms (R) is greater than 1 bohr, $P_B$ increases rapidly with R. Above 6 bohr the bosonic probability becomes essentially 1. This strongly supports the assertion that H↓ be a Bose gas. The bosonic probability for the ground state of the ${\mathrm{H}}_2$ molecule is also discussed.