An arbitrariness of virtual orbitals in the Hartree-Fock theory is discussed, and it is explicitly demonstrated that the energy spectrum of the virtual orbitals can be manipulated so that the convergence property may be improved in the perturbation theory and the configuration interaction calculation based on the Hartree-Fock equation.

A weak depolarized Rayleigh wing has been observed for liquid xenon and for several liquids composed of spherical-top molecules. The intensity contours for these liquids decrease exponentially with increasing frequency shift. This behavior is similar to that predicted for spectra of collision-induced scattering from gases.

Electron $g$ factors have been measured for 33 low-lying states of Ce i with the atomic-beam magnetic-resonance technique.

The radiative lifetimes of the ten fine-structure levels of the $2p^{5}3p$ configuration of singly charged sodium, and eight of the ten fine-structure levels of the $2p^{5}3p$ configuration of doubly charged magnesium, have been measured by the beam-foil excitation technique. Systematic trends of the $2p^{5}3p$ levels in the Ne i sequence are discussed.

In the previous paper, the Biberman-Holstein integral equation was solved for a slab and for all line shapes of interest in the limit of high optical depth. The eigenfunctions and eigenvalues obtained are used for the general calculation of the stationary number density in the excited state, when collisional excitation and deexcitation takes place, when excitation takes place by absorption of external radiation, or by both combined. General expressions are given for the line shape of a spectral line emitted by an optically dense slab, showing typical broadening and self-reversal.

An iterative method for determining bound-state eigenvalues and properties of the radial Schrödinger equation is appraised. The method stems from iterating the integral equation $\psi =\mu {(T+\frac{1}{2}{\gamma }^2)}^{-1\mathrm{}}(-V\psi )$, where $T$ and $V$ are the kinetic- and potential-energy operators. The basic theory is briefly reviewed, and calculations are performed for the Coulomb and screened-Coulomb potentials. The lowest three $\mu$ eigenvalues, together with the expected values of ${\left(\gamma \mathcal{r}\right)}^{-1\mathrm{}}$, $\gamma \mathcal{r}$, and ${\left(\gamma \mathcal{r}\right)}^2$, are obtained from a single iterated eigenfunction sequence. Convergence is rapid for eigenvalues but slow for expected values. There is some sensitivity to the choice of the numerical integration formula. Regarded as a numerical method, this approach may be most competitive for the determination of zero-energy potential-strength eigenvalues. Its disadvantages are listed. Analytical improvements to eigenfunctions can be easier to obtain by iteration than by perturbation, and some success has been achieved. A simple example suggests that the rate of convergence of an iterated eigenfunction sequence is less than that of a related perturbation sequence unless the choice of starting function is bad.

The possibility of an observable field dependence of the nuclear magnetic shielding in NMR experiments is discussed. It is shown that in most molecules such an effect should be unobservably small. However, in some molecules, especially where the magnetic shielding is abnormally large, it is possible that the field dependence of the shielding could be large enough to be observable. In NMR experiments of high accuracy in strong magnetic fields, the proportionality of the resonance frequency to the magnetic field should therefore not be automatically assumed, but should be experimentally verified.

This paper is an addendum to a paper by the author which presented a relation between the gradient of the electric field at the nucleus and dipole matrix elements. A scheme is suggested to take advantage of the result.

Accurate eigenvalues and radiative lifetimes are obtained for the bound $2p^2{}_{}{}^3{}_{}{}^{}P$ and $2p3p{}_{}{}^1{}_{}{}^{}P$ states of the heliumlike ions He i to Ne ix. A high-order $Z$-expansion perturbation procedure is used that does not require the explicit solution of perturbation equations. The results are compared with variational calculations. The predicted wavelengths of the $1s2p{}_{}{}^3{}_{}{}^{}P-2\mathrm{}{p}^2{}_{}{}^3{}_{}{}^{}P$ and $1s3p{}_{}{}^1{}_{}{}^{}P-2p3p{}_{}{}^1{}_{}{}^{}P$ transitions in helium are, respectively, 320.31 and 308.97 Å. Lines have been observed in helium at 320.39 and 309.04 Å.

By means of the atomic-beam-magnetic-resonance method we have measured the ${}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ level hfs and the ${}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ level $g_J$ factors of the ${\mathrm{Bi}}^{209}$ ${\mathrm{}\left(6p\right)\mathrm{}}^3$ ground configuration. Our experimental results are: $A\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\right)=491.028\mathrm{}\left(1\right)\mathrm{}$ MHz, $B\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\right)=978.639\mathrm{}\left(9\right)\mathrm{}$ MHz, $C\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\right)=0.0193\mathrm{}\left(5\right)\mathrm{}$ MHz, $g_J\left({}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\right)=0.6654\mathrm{}\left(2\right)\mathrm{}$ and $g_J\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\right)=1.26083\mathrm{}\left(12\right)\mathrm{}$. In order to interpret these measurements, we have examined the fine structure and the hfs of all the ${\mathrm{}\left(6p\right)\mathrm{}}^3$ levels. Our calculations show that the inclusion of spin-spin, orbit-orbit, and spin-other-orbit interactions do not improve the theoretical fit to the observed fine structure intervals. The effects of core polarization and relativity can be estimated by writing the dipole hfs Hamiltonian in the form $\mathcal{H}=2\mathrm{}\mu {\mu }_B\overrightarrow{\mathrm{I}}·\Sigma \stackrel{}{i}[{〈{\mathcal{r}}_l^{-3\mathrm{}}〉}_i{\overrightarrow{\mathrm{I}}}_i-\sqrt{10}{〈{\mathcal{r}}_{\mathrm{sC}}^{-3\mathrm{}}〉}_i{\left(s{C}^2\right)}_l^{\mathrm{}\left(1\right)\mathrm{}}+{〈{\mathcal{r}}_s^{-3\mathrm{}}〉}_i{\overrightarrow{\mathrm{s}}}_i]$ This analysis of the dipole hfs yields a value ${〈{\mathcal{r}}^{-3\mathrm{}}〉}_{6p}\left(\mathrm{hfs}\right)=1.02\mathrm{}\times {10}^{26}$ ${\mathrm{cm}}^2$ compared to the fine structure result ${〈{\mathcal{r}}^{-3\mathrm{}}〉}_{6p}\left(\mathrm{fs}\right)=1.25\mathrm{}\times {10}^{26}$ ${\mathrm{cm}}^2$. With these results we find $Q=-0.383\mathrm{}\left(40\right)\mathrm{}$ b and $\Omega =0.55\mathrm{}\left(3\right)\mathrm{}{\mu }_N\mathrm{b}$.

The effect of frequency modulation on an inhomogeneously broadened line is studied when the modulation frequency or amplitude is larger than the homogeneous width but less than the inhomogeneous width. It is shown that the deviation from the instantaneous response for the polarization disappears as a result of the averaging effect when the field intensity is weak.

Atomic form factors and incoherent scattering functions for the helium isoelectronic sequence through $Z=10\mathrm{}$ have been calculated with correlated ground-state wave functions. The wave functions are 120-term configuration-interaction expansions which give total ground-state energies differing from the most accurate available values by 0.02% for He i and 0.0007% for Ne ix. Comparison with published accurate calculations for helium indicates that the scattering factors, both coherent and incoherent, for the other members of the isoelectronic sequence, should be correct to at least three, and probably four, significant figures.

A general approach for describing complex quantum-mechanical systems in terms of $c$-number variables is discussed on the basis of a formalism developed by Lax. We apply this formalism to the particular problem of noise fluctuations in parametric oscillators. From a microscopic Hamiltonian describing light interacting with optically active atoms in a cavity with loss, we arrive at an equation for the motion in terms of $c$-number variables. We then solve this system for the noise properties of the signal mode near threshold and above, when the pump and idler modes may be treated adiabatically, and compare our work with the results of Graham, who treated the signal and idler symmetrically.

An argument is given which shows that there is no undetermined constant in values of the exchange energy of solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ obtained from pressure measurements.

The formulation of a relativistic restricted multiconfigurational self-consistent-field theory for open-shell atoms is given. The relativistic Hamiltonian is the sum of the one-electron Dirac-Hamiltonian and the two-electron Coulomb-repulsion terms. The total wave function is assumed to consist of a linear combination of eigenfunctions of total operators $J^2$ and $J_z$, which are themselves assumed to be known linear combinations of Slater determinants of four-component one-electron orbitals. The applicability of the formulation derived in this first paper is limited to wave functions expanded in terms of a set of distinct Slater determinants $D_i$, which obey the three conditions. (i) Two or more "core shells" may belong to a same symmetry specy $k$ but their occupations must be similar. (ii) Distinct "peel shells" must belong to different symmetry species $k$. (iii) The symmetry species of the peel shells must be different from the symmetry species of the core shells. The most important cases are the calculations using wave functions expanded in terms of the distinct Slater determinants which arise from the ${\left(2\mathrm{}\overline{p}\right)}^n \mathrm{}\left(2p\right)\mathrm{}$, ${\left(3\mathrm{}\overline{d}\right)}^n {\mathrm{}\left(3d\right)\mathrm{}}^{n^{\prime }}$, and ${\left(4\mathrm{}\overline{f}\right)}^n {\mathrm{}\left(4f\right)\mathrm{}}^{n^{\prime }}$ configurations.

A method for obtaining electromagnetic many-body forces using canonical transformations is described. The Hamiltonian thus derived is more general than that of Primakoff and Holstein.

This paper presents a semiclassical theory of a single-mode gas laser valid for essentially arbitrary values of field intensity. The theory is formulated in terms of an ensemble-averaged form of the density-matrix equations of motion involving a single set of coordinates. The polarization and the population-inversion density are expanded in spatial Fourier components. One obtains a set of coupled difference equations in the Fourier coefficients $y_n$, which may be solved subject to appropriate boundary conditions on $y_n$ for large $n$. The solutions may be expressed in terms of continued fractions in the general case, and in closed form in important special cases. Quantities of interest are evaluated by computer. The steady-state laser intensity and frequency are obtained from the coupled electromagnetic field equations as functions of the cavity parameters and the linear gain of the amplifying medium. At moderate values of field intensity, the velocity distribution of the space-averaged population inversion density exhibits a fine structure for low-velocity atoms superimposed on the usual broader depletion. This structure is a manifestation of the coherent ringing of the amplifying medium. The results are compared with those obtained in the rate-equation approximation, and with the recent work of Stenholm and Lamb. Discrepancies with the latter paper are explained.

Bell's technique for the calculation of long-range dispersion-force coefficients is transformed to yield rigorous upper and lower bounds to these coefficients. The bounds involve in a systematic way the negative oscillator strength sums $S(-k)\mathrm{}$ of the two interacting atoms. By a slight modification, we are also able to bound the leading relativistic correction to the long-range potential. Application of the method to the interaction between two hydrogen atoms yields extremely tight bounds to all the relevant coefficients.

Recent high-energy measurements of $K$-shell $x$-ray yields produced by proton impact are compared with the binary-encounter-impulse approximation and with previous measurements. The scaling law derived from the binary-encounter model is found to be in good agreement with experiment.

The dissociation and ionization cross sections of the ${\mathrm{H}}_2^{+}$ molecular ion due to electron impact have been studied using the classical impulse approximation. The effect of the Coulomb field of the molecular ion is explicitly included in the calculation. The results are compared with other theoretical calculations.