The matrix element for electric dipole transitions is correctly given by the length formula in the Hartree-Fock, configuration-interaction, and related approximations, which involve the diagonalization of an approximate, but nonlocal, Hamiltonian.

A theoretical investigation of the low-field vibronic Zeeman effect in octahedral molecules with odd-electron central atoms is presented. Such systems are of interest because the vibronic $g$ values differ from those of the parent electronic level and provide a means of studying the vibronic interaction. Only ${\Gamma }_8$ parent electronic states are of interest, and the vibronic multiplet splitting is derived for these states in terms of a minimum number (two) of vibronic interaction parameters. The Zeeman splitting of the vibronic multiplet is determined in terms of the parent electronic Zeeman effect, and for ${\Gamma }_8$ vibronic states, in addition, in terms of the vibronic interaction parameters. It is shown that that in spite of anisotropy in the Zeeman splitting of ${\Gamma }_8$ electronic or vibronic levels and the tumbling of free molecules, the observed Zeeman splitting should appear as four resolvable bands or lines except in certain limiting cases, but in all cases the information on $g$ values can in principle be determined from experiment.

Analytic calculations of the expectation values of $r^n$, $n=-2,-1,1,2\mathrm{}$, the kinetic energy, and the charge density at the origin, have been performed for two-electron atoms with nuclear charge $Z$ using a variational principle first proposed by Delves. The trial wave function ${\psi }_{0T}$ was taken as a product of single-particle hydrogenic states with a single variational parameter $\overline{Z}$. In each case the complementary function ${\psi }_{1T}$ was obtained by exactly solving a differential equation relating ${\psi }_{1T}$ to both ${\psi }_{0T}$ and the operator whose expectation value was to be calculated. For $Z\ge 2$, the least accurate results obtained are for $〈r^2〉$ which are 1.9 and 0.7% lower than the highly accurate values obtained by Pekeris for He and ${\mathrm{Li}}^{+}$, respectively. In all cases the reduction in the error obtained using a one-parameter wave function given by the variational principle for minimizing the energy increases as $Z$ increases, and the results are, in many cases, comparable to those given by the Hartree-Fock treatment. A condition proposed by Aranoff and Percus involving the minimization of an auxiliary expression was used to renormalize ${\psi }_{1T}$. This method left unchanged or improved the results for all cases considered.

Minima in continuum generalized oscillator strengths for atoms are found to exist, and their position as a function of energy is discussed. It is shown that at increased energy loss they occur at larger momentum transfer. Calculations are performed for the Na $3s\to \epsilon p$, K $4s\to \epsilon p$, and Ar $3p\to \epsilon d$ transitions.

The self-energy of $K$ electrons in atoms with atomic numbers in the range $70\le Z\le 90$ is determined numerically. Detailed theoretical evaluations of $K$-electron binding energies including all effects of $O\left(\alpha m{c}^2\right)$ are given for the four heavy closed-subshell atoms - W, Hg, Pb, and Rn - in the range considered. With the present values of the electron self-energy, the theoretical $K$-shell binding energies for these elements are found to agree with experimental determinations to better than 1 part in ${10}^4$.

A semiclassical discussion is given of a simple case of the Hanle effect when the population of the level of interest varies with time because of decays to it from higher levels excited by collisions. It is found that the width of the intensity-vs-applied-field curve is no longer simply related to the lifetime of this single level, but depends on the relative populations and lifetimes of all other participating levels. It is shown by examples that, unless these effects are taken into account, the lifetimes deduced from the measurements can be greatly in error.

Accurate numerical Hartree-Fock (HF) self-consistent-field (SCF) wave functions have been calculated for ${\mathrm{Pr}}^{+++}$, ${\mathrm{Ce}}^{++}$, ${\mathrm{Nd}}^{+++}$, ${\mathrm{Pr}}^{++}$, ${\mathrm{Pm}}^{+++}$, ${\mathrm{Nd}}^{++}$, ${\mathrm{Sm}}^{+++}$, ${\mathrm{Pm}}^{++}$, ${\mathrm{Eu}}^{+++}$, ${\mathrm{Sm}}^{++}$, ${\mathrm{Gd}}^{+++}$, ${\mathrm{Eu}}^{++}$, ${\mathrm{Tb}}^{+++}$, ${\mathrm{Gd}}^{++}$, ${\mathrm{Dy}}^{+++}$, ${\mathrm{Tb}}^{++}$, ${\mathrm{Ho}}^{+++}$, ${\mathrm{Dy}}^{++}$, ${\mathrm{Er}}^{+++}$, ${\mathrm{Ho}}^{++}$, ${\mathrm{Tm}}^{+++}$, and ${\mathrm{Er}}^{++}$ rare-earth ions having an $f^N$ electron configuration in their ground-state manifolds. Calculations, in general, have an accuracy of seven significant digits in the total energy. Values of several Hartree-Fock parameters have also been obtained from these wave functions.

The effect of the electromagnetic polarization of the nucleus is considered on the energy levels of the high orbital states in a muonic atom. A simplified calculation is carried out to estimate the approximate dependence of the shifts of the energy levels on the atomic number $Z$ for the orbital states $3s$, $3p$, and $3d$. To do this, a formulation is proposed in which the energy shift is regarded as the amount of energy which flows into the system when the approximated second-order electromagnetic interaction between muon and nucleus is adiabatically switched on. The many-body property of the nucleus is formally expressed as the correction term to the photon self-energy part which represents the dielectricity of nuclear matter. Then the formal parallel holds between the present treatment of nuclear polarization and that of vacuum polarization in field theory. The applicability of the theory is restricted to the high orbital state, in which the bound-state wave function is less sensitive to the nuclear polarization effect, and its energy fluctuation is much less than the energy attributed to the muonic state.

A study is made of the completeness of the basis sets used by Clementi and Nesbet for the calculation of atomic Hartree-Fock and correlation energies. A good criterion for completeness is found to be related to the overlap between these sets and the hydrogenic functions that contribute most in many-body perturbation theory. Thus the experience gained in perturbation-theory calculations is shown to be helpful in choosing the best basis sets for the matrix method. Moreover, perturbation calculations can be simplified in those cases where the function space can be adequately approximated by a set of easily integrable functions.

The distinguishable electron method is used to compute the spin density at the nucleus of a lithium atom in its lowest ${}_{}{}^2{}_{}{}^{}S$ and ${}_{}{}^2{}_{}{}^{}P$ states. For the initial approximation, we take an open-shell product of Slater-type orbitals with optimized exponents. The required perturbation functions are determined entirely by the exact solution of a single one-electron differential equation. A final mixing of spin states, which significantly affects the spin density, can be carried out perturbatively due to the weak correlation interaction. Our final results are comparable in accuracy with those determined by Brueckner-Goldstone perturbation theory, but do not involve the calculation of correlated wave functions.

Measurements are presented of the total cross sections for formation of metastable hydrogen atoms by charge transfer as protons traverse targets of helium, argon, nitrogen, and oxygen. Projectile energies range from 4 to 26 keV. An ${\mathrm{H}}^{+}$ projectile beam was directed into a cell containing the target gas and emerged into an evacuated region where the metastable-state content was determined. The $\mathrm{H}\mathrm{}\left(2s\right)\mathrm{}$ flux was monitored by electric field mixing of this state with the $2p$ level and detection of the resulting Lyman-$\alpha$ photon. The relative variation of cross section with projectile energy for targets of He and Ar is in agreement with previous work. For oxygen the cross section increases with energy from 4 to 10 keV and remains constant from 10 to 26 keV; for nitrogen the cross section increases monotonically with increasing energy.

Using the atom-beam recoil technique, absolute total cross sections were measured for the scattering of electrons in the energy range 0.3-9 eV by potassium, rubidium, and cesium. The results of these measurements are in substantial disagreement with the experimental values of Brode, obtained more than 40 years ago. On the other hand, our potassium cross sections agree extremely well with the recent calculations of Karule, and Karule and Peterkop, as well as with the experimental values of Collins et al. For cesium and rubidium, the existing theoretical calculations are not in as good agreement with our results as in the case of potassium.

Ab initio calculations are presented for several states of ${\mathrm{He}}_2^{+}$ and ${\mathrm{Li}}_2^{+}$ that are relevant to collision problems. In ${\mathrm{He}}_2^{+}$, qualitative agreement is found with the predictions from the analysis of the oscillations and anomalous thresholds in ${\mathrm{He}}^{+}$-He collision cross sections. For ${\mathrm{Li}}_2^{+}$, the results suggest that the oscillations in ${\mathrm{Li}}^{+}$-Li differential scattering cross sections are due to $\Sigma -\Pi$ transitions caused by the coupling of electronic and nuclear rotations.

It is shown how the generalized random-phase approximation, previously used to develop a self-consistent theory of elastic scattering and frequency-dependent polarizability, can be extended to the calculation of inelastic scattering cross sections from the ground state of the target. It is also shown that when calculating electron-atom scattering cross sections, one can simultaneously calculate without much additional work elastic and some specific inelastic scattering cross sections for an electron scattering off a positive-ion target. Improvements to the approximation scheme, using self-consistent perturbation methods, are briefly discussed.

A rigorous-bound calculation is performed of positron-hydrogen scattering below the positronium pickup threshold. The correlation function is a generalized Hylleraas function with three nonlinear parameters which includes a virtual-positronium factor explicitly. Phase shifts are calculated as a function of the number of linear parameters using up to 84 terms. Converged phase shifts can be extrapolated to four significant figures. At three of the seven values of $k$ computed, the results are found to disagree slightly but significantly from Schwartz's results. Final phase shifts are estimated to be accurate to ± 2 × ${10}^{-4\mathrm{}}$ rad.

The $S$-wave scattering parameters have been calculated for the $e^{+}-\mathrm{H}$ and $e^{+}-\mathrm{He}$ systems below the positronium-formation threshold. The phase shifts were computed using the Harris variational method, while at zero energy the scattering lengths were obtained by the Kohn variational method. For hydrogen, a generalized Hylleraas wave function with three nonlinear parameters was used, giving an upper-bound scattering length $a\le -2.10278\mathrm{}$, agreeing with Schwartz's result $a\le -2.10\mathrm{}$. An extrapolation procedure gives an estimate of $a=-2.1036\mathrm{}\pm 0.0004$. At nonzero energies, agreement with the lower-bound results of Bhatia, Temkin, Drachman and Eiserike, to better than 0.004 rad is achieved. Since the exact atomic-helium wave function is not known, the situation for helium is less clear; various model-target wave functions are examined using a generalized Hylleraas function with two nonlinear parameters. For the exponential model used by Drachman, the scattering is in close agreement with previous results, but the annihilation-rate parameter $Z_{\mathrm{eff}}$ is higher by 15% in poorer agreement with experiment at zero energy.

Sum rules for the asymptotic Born cross sections have been applied to the evaluation of various inelastic-scattering cross sections of metastable He ($2{}_{}{}^1{}_{}{}^{}S$ and $2{}_{}{}^3{}_{}{}^{}S$). The ionization cross sections for the two species are similar in magnitude, and each constitutes only a small part of the total cross section for inelastic scattering. Although the first Born approximation is inappropriate for the scattering of slow charged particles, the energy dependence exhibited in the recent experiment by Long and Geballe for the ionization of He ($2{}_{}{}^3{}_{}{}^{}S$) by low-energy electrons (≤ 16 eV) is compatible with the high-energy behavior predicted by the theory. The exchange correction evaluated from the Mott formula has been included in the ionization cross section by electron impact. Optical oscillator strengths for the first three transitions to doubly excited states [$2{}_{}{}^1{}_{}{}^{}S\to \left(2\ln l^{\prime }\right){}_{}{}^1{}_{}{}^{}P$] are also given.

Cross sections for Penning detachment, $X^{*}+Y^{-}\to X+Y+e$, and collision-induced photon emission, $X^{*}+Y^{-}\to X+Y^{-}+h\nu$ have been calculated for collisions of metastable neutral atoms and negative ions at low energies ($E\sim 1$ eV). A semiclassical treatment based on the impact parameter method and adiabatic electronic interactions was employed. Collisions involving ${\mathrm{H}}^{*}$ ($2{}_{}{}^1{}_{}{}^{}S$), ${\mathrm{Ca}}^{*}$ ($3{}_{}{}^1{}_{}{}^{}D$), and ${\mathrm{He}}^{*}$ ($2{}_{}{}^1{}_{}{}^{}S$) with ${\mathrm{H}}^{-}$, ${\mathrm{O}}^{-}$, ${\mathrm{Cl}}^{-}$, and ${\mathrm{C}}^{-}$ were investigated. In all collisions not involving ${\mathrm{H}}^{*}$ ($2{}_{}{}^1{}_{}{}^{}S$), Penning detachment predominates and exhibits cross sections on the order of ${10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$.

It is demonstrated that the pure-Coulomb $t$ matrix satisfies a modified unitarity condition and that its discontinuity is not zero, as has been asserted elsewhere.

A strong-coupling model for the simple exchange reaction $A+BC\to AB+C$ is developed by considering the atom-diatom inelastic problems $A+BC$ and $AB+C$ separately and then introducing coupling between these configurations to allow for reaction. The inelastic systems are developed in the form of self-coupled differential-integral equations and are recoupled by the matrix element over internal states of the resonance energy between the adiabatic surfaces for the separate configurations. The model is applied to the D + ${\mathrm{H}}_2$ → DH + H system and its isotopes in a two-level approximation. At the threshold energy (0.33 eV) for the D + ${\mathrm{H}}_2$ → DH + H system, the calculated reactive total cross section is 1.62 ${\mathrm{\AA }}^2$ and the DH product is backscattered in the center-of-mass system.