The existence of ${\mathrm{Be}}^{-}$ is established from electron correlation calculations which include one-, two-, three-, and four-orbital excitations. The discrete spectrum of ${\mathrm{Be}}^{-}$ consists of three states: $1s^{2}2s2p^2 {}_{}{}^4{}_{}{}^{}P$, $1s^{2}2p^3 {}_{}{}^4{}_{}{}^{}S_{}^o{}_{}{}^{}$, and $1s2s2p^3 {}_{}{}^6{}_{}{}^{}S_{}^o{}_{}{}^{}$. The ${}_{}{}^4{}_{}{}^{}S_{}^o{}_{}{}^{}-{}_{}{}^4{}_{}{}^{}P$ transition offers the possibility of experimental detection of ${\mathrm{Be}}^{-}$, e.g., by beam-foil methods. The theoretical wavelength and probability for the transition are $\lambda =2671\mathrm{}$ Å and $A=0.67\mathrm{}\times {10}^9$ ${\sec }^{-1\mathrm{}}$, respectively.

A single table is presented which can be used with a scaling relationship to predict the intensity of synchrotron radiation for any electron energy, wavelength, and observation angle. The accuracy of the scaling relationship and the table has been tested to a level of 1 part in ${10}^5$ using current values for the fundamental constants.

Integral elastic-scattering cross sections were measured for ${\mathrm{C}}^{+}$ ions with kinetic energies in the range 0.5-3.0 keV scattered by stationary He atoms through effective laboratory angles greater than 5 × ${10}^{-3\mathrm{}}$ rad. The cross sections are compared with two kinds of the weighted means of the cross sections for the ${}_{}{}^2{}_{}{}^{}\Sigma$ and ${}_{}{}^2{}_{}{}^{}\Pi$ potentials of He${\mathrm{C}}^{+}$ obtained by ab initio calculation. Estimation of the population distribution of ${\mathrm{C}}^{+}$ ions of ${}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ involved in the projectile ion beams may be achieved by the comparison.

We have observed a new component in the signals produced by degenerate four-wave mixing of 5-nsec optical pulses by the $D$-line resonances of sodium vapor. This component is comparable in intensity to that previously observed from a grating pattern of excited sodium atoms, but arises rather from a grating of coherently aligned (but not optically pumped) ground-state atoms. This component exists when there are no spatial intensity beats between input waves.

We consider the Navier-Stokes (Euler) equation ($0\leftarrow \frac{1}{L}<k<\mathrm{}\infty$) with an arbitrary pumping acting in the region $k\simeq \frac{1}{L}\to 0$. It is shown that after being projected onto a smaller space $\frac{1}{L}<<k_0<k<\mathrm{}\infty$ this model is equivalent to the one with random stirring [P. C. Martin and C. DeDominicis, Suppl. Prog. Theor. Phys. 64, 108 (1978) C. DeDominicis and P. C. Martin, Phys. Rev. B 11, 4077 (1975) and V. Yakhot, Phys. Rev. A 23, 1486 (1981)], and yields the energy spectrum $E\left(k\right)=\frac{a{k}^{-\frac{5}{3}}}{\ln \mathrm{kL}}$.

The blue phases (BP) in a system of chiral biphenyl mixtures have been studied in detail. It is found that when the cholesteric pitch is below a critical value, two BPII lattice parameters coexist, with the shorter one being dominant and the longer one transforming continuously from the BPI lattice. The relative contraction of the dominant lattice at the BPI-BPII transition is apparently universal. Above the critical pitch, the shorter BPII lattice vanishes.

Results are presented here of a three-dimensional numerical analysis of the radiation fields produced in a free-electron laser. The method used here to obtain the spatial and temporal behavior of the radiated fields is based on the coherent superposition of the exact Lienard-Wiechert fields produced by each electron in the beam. Interference effects are responsible for the narrow angular radiation patterns obtained and for the high degree of monochromaticity of the radiated field.

Reflection of a circularly polarized electromagnetic wave by a stationary ripple is considered in a uniformly expanding plasma in a magnetic field. The effect of the external magnetic field on the total plasma reflectivity is investigated.

The $n$-dimensional Thomas-Fermi equation is derived and some of its properties are studied. An analytic solution is obtained in two dimensions for both neutral atoms and positive ions. It is shown that for $n\ge 4$ the density is too singular at the origin to be normalizable.

The orthogonally spin- and orbital-symmetry-adapted ($\mathrm{LS}$-coupled) coupled-pair many-electron theory and its extended version involving monoexcited and biexcited cluster coefficients is derived in a form suitable for application to closed-shell atomic systems. A diagrammatic approach, based on second quantization, the time-independent Wick theorem, and the graphical methods of spin algebras, is employed.

The analytic forms of both the $\mathrm{LS}$-adapted coupled-pair many-electron theory (CPMET) and its linear version (LCPMET) are applied to the ground state of the Be atom in order to examine the reliability of the analytic coupled-pair approaches. Using CPMET within a ($7s,7p,4d,3f,3g$) set of orbitals, we obtain 97% of the experimental correlation energy. Special attention is paid to the convergence properties for both radial and angular correlation effects. By means of the partial-wave analysis of the various pair energies it is shown that the difference between LCPMET and CPMET is primarily due to the quasidegeneracy of the Be ground state. The role of different types of coupling terms for pair clusters is also examined yielding a basis for an understanding of various approximate coupled-pair theories. The results are compared with other calculations and the role of monoexcited and triexcited clusters is briefly discussed.

Various approximations to the orthogonally spin- and orbital-symmetry-adapted coupled-pair many-electron theory (CPMET) are suggested and discussed. These approximations include methods of the well-known coupled-electron-pair-approximation (CEPA) type and methods consisting of truncating the number of pair clusters considered in the nonlinear part of the CPMET. To enable assessment of the reliability of our approximate approaches in the presence of quasidegeneracy effects, calculations of the correlation energy are performed for the Be-atom ground state.

Cusp conditions are obtained for the coalescence of any number of particles interacting Coulombically. This is an extension of the results of Hoffmann-Ostenhof and Seiler for atoms and molecules in the infinite-nuclear-mass approximation in two aspects: (1) the center of mass is first removed, and (2) the masses and charges are all allowed to differ.

Sometimes it is useful and necessary to consider quantum-mechanical systems enclosed in "boxes" of different shapes with impenetrable or semipermeable walls, in contrast to the usual requirement that wave functions vanish at infinity. This kind of model has been widely applied to interpret, analyze, and discuss several physical phenomena. In this communication we describe and study the hypervirial relationships for finite systems in order to generalize some previous works related to the virial theorem. We apply our formal results to various simple models and discuss their potential applications to solid-state theory.

A simple derivation is given for the first quantum correction to the Thomas-Fermi kinetic energy. Its application to the total binding energy of neutral atoms exploits the technique for handling strongly bound electrons that was developed in a preceding paper, and justifies the numerical value of the second correction adopted there. A proposal is made for extrapolating this improved description to the outer regions of the atom.

We consider methods of deriving closed quadratures for the terms of the one-dimensional perturbation series, and develop a general procedure for regularizing divergent integrands in the case of excited states. Several applications are given: a closed formula for the effective mass (in the framework of the $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ expansion) is derived; two methods of improving the convergence of the perturbation series are analyzed; generalization to the symmetry-adapted perturbation theory is described.

The $1s\to 2s$ and $1s\to 2p$ excitations of the hydrogenic ions ${\mathrm{C}}^{5+}$ and ${\mathrm{O}}^{7+}$ are studied using a pseudostate expansion. A small ($3-s$, $3-p$) basis is used above the $n=3\mathrm{}$ threshold while in the resonance region between the $n=2\mathrm{}$ and $n=3\mathrm{}$ thresholds a larger ($6-s$, $5-p$, $2-d$, and $1-f$) set is employed. The coupled integrodifferential equations are solved by a variational approach. High-energy and high-$L$ results are found using the Coulomb-Born approximation with exchange. Positions and widths of resonances are given. An analytic fit to the calculated $K$ matrices is used to determine cross sections in the resonance region. Excitation rates are computed.