An explicit form of quasiprojection operator $\hat{Q}$ is given for the calculation of resonances in the domain of inelastic scattering of electrons from atoms and atomic ions. This operator allows the calculation of the resonant spectrum through the diagonalization of $\hat{Q}H\hat{Q}$ with as many inelastic states projected out as desired. The technique gives rise to a limited number of spurious states, but they can be unambiguously identified and thus eliminated. The elimination procedure is illustrated in the case of three-electron (i.e., heliumlike target) ions whereby we explicitly determine the number of spurious states that arise in inelastic scattering below the $3{}_{}{}^3{}_{}{}^{}S$ state of the target.

A time-of-flight, atomic-beam magnetic-resonance method was used to compare the $g_J$ factor of $2p^{3}3s{}_{}{}^5{}_{}{}^{}S_2^{}{}_{}{}^{}$ atomic oxygen with the $g_J$ factor of metastable $2{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ atomic helium in magnetic fields of 3209 and 4228 G. The metastable oxygen and helium atoms were produced in a pulsed radio-frequency discharge source and after traversing the apparatus were detected by electron emission from a tungsten surface. By collecting data only in a given interval of the time-of-flight distribution the $\mathrm{O}\left({}_{}{}^5{}_{}{}^{}S\right)$ and $\mathrm{He}\left({}_{}{}^3{}_{}{}^{}S\right)$ states were isolated from other discharge products. From 158 pairs of resonances taken with four relative field and hairpin orientations the atomic magnetic $g$ factor is $g_J({}_{}{}^5{}_{}{}^{}S_2^{}{}_{}{}^{}, {}_{}{}^{16}{}_{}{}^{}\mathrm{O})=\frac{{\mu }_J}{J}=-2.002\mathrm{}0910\left(10\right)\mathrm{}$.

The effect on low-energy atomic inner-shell Coster-Kronig and super-Coster-Kronig transitions that is produced by relaxation and by exchange between the continuum electron and bound electrons has been examined and illustrated by specific calculations for transitions that deexcite the [$3p$] vacancy state of Zn. Taking exchange and relaxation into account is found to reduce, but not to eliminate, the discrepancies between theoretical rates and measurements.

We propose a new relativistic two-body formalism which reduces to a nonrelativistic Schrödinger theory for a single effective particle. The formalism is equal in rigor to that of Bethe and Salpeter, and considerably simpler to apply. We illustrate its use by computing $O\left({\alpha }^6\right)$ terms in the ground-state splitting of muonium and positronium involving infinite Coulomb exchange.

General formulas are obtained for the reduction of a wide class of two-electron matrix elements in Hylleraas coordinates to finite sums of radial integrals for states of arbitrary angular momentum. Multipole transition integrals and the Breit interaction are treated as special cases. A number of recurrence relations are derived for radial integrals containing $P_L\left(cos\theta \right)$ (where $cos\theta ={\hat{r}}_1·{\hat{r}}_2$) in terms of radial integrals containing lower-order Legendre polynomials. The results are well suited to computer implementation.

We present a general method, based on the diagrammatic techniques of spin algebras, for the calculation of permutation matrices of two-rowed (two-columned) irreducible representations of the symmetric group ${\mathcal{S}}_N$ relative to a basis (or bases) adapted to the subgroup(s) ${{\mathcal{S}}_N}_A\otimes {\mathcal{S}}_N\subset {\mathcal{S}}_N$ or to (a) subgroup chain(s) which may be generated by a recursive application of the above-given chain. These matrix elements are needed in spin-adapted configuration interaction calculations. This general technique is applied to the Serber coupling scheme, and general and explicit closed formulas are obtained for the matrix elements of an arbitrary transposition. An extension of this formalism to cases with an arbitrarily large frozen core is also outlined using the same technique. A computer program based on these derivations was written and its effectiveness compared with that of other approaches is discussed. An extension of these applications to more complex permutations than transpositions, as well as to other coupling schemes, is also briefly discussed.

A derivation and discussion of Armstrong's $N$-electron zero-momentum energy expression is presented. This functional $E_{\mathrm{zm}}\left(\chi \right)$ is linear in the wave function $\chi$, and thus is easier to compute and more sensitive to wave-function error than the variational upper bound $E_{\mathrm{var}}\left(\chi \right)$. The smallness of the quantity $\Delta \left(\chi \right)=E_{\mathrm{zm}}\left(\chi \right)-E_{\mathrm{var}}\left(\chi \right)$ provides an a priori criterion for assessment of the accuracy of an approximate wave function. Nontrivial numerical examples are given to illustrate the utility of this criterion.

The Casimir-Polder $R^{-7\mathrm{}}$ retarded potential between two neutral (polarizable) atoms at a separation $R$, which replaces the nonrelativistic $R^{-6\mathrm{}}$ van der Waals potential for $R$ sufficiently large, has been derived both by standard techniques and by assuming the atoms to be within a large volume with conducting walls and considering vacuum-fluctuation effects. The latter technique is here used to provide simple derivations of the $R^{-5\mathrm{}}$ potential between a charged point particle and a polarizable system—an addition to the usual $R^{-4\mathrm{}}$ polarization potential—and the $R^{-3\mathrm{}}$ two-photon contribution to the potential between two charged point particles. It also suggests a new, $R^{-2\mathrm{}}$ potential—an addition to the usual $R^{-1\mathrm{}}$ image potential—between a charged point particle and a conducting wall.

A modification of the Fermi contact interaction is proposed in which the nuclear moment is represented by a uniformly magnetized spherical shell of radius $r_0$. In effect, the delta function $\delta \mathrm{}\left(r\right)\mathrm{}$ in the Fermi Hamiltonian is replaced by $\delta (r-r_0)$. The Schrödinger equation for a hydrogenlike system thus perturbed is exactly solvable in terms of the Coulomb Green's function. Negative energy eigenvalues have the form $E_{\nu }=-\frac{Z^2}{2{\nu }^2}$, with $\nu$ a nonintegral quantum number. An asymptotic formula is derived for the quantum defect $\delta =\nu -n$. The $l=0\mathrm{}$ eigenfunctions are multiples of the Whittaker functions: $M_{\nu ,}^{\frac{1}{2}}\left(\frac{2Zr}{\nu }\right)$ for $r<\mathrm{}{r}_0$ and $W_{\nu ,}^{\frac{1}{2}}\left(\frac{2Zr}{\nu }\right)$ for $r>\mathrm{}{r}_0$. Explicit forms are given by expansion of the Whittaker functions to first order in quantum defect. In the limit $r_0\to 0$ results pertaining to the original Fermi Hamiltonian are approached. It is shown that a repulsive delta function maintains the unperturbed Coulomb energy while an attractive delta function pulls all bound state energies to $-\infty$. Perturbation expansions are discussed and comparisons made with earlier calculations. It is shown that second-order and higher perturbation energies diverge as $r_0\to 0$.

The ground-state wave function of a hydrogenic atom is approximated by ${\psi }_0\left(\mathrm{H}\right)=\exp \{-{\left[k{(x^2+y^2+s{z}^2)}^{\frac{1}{2}}\right]}^{\frac{1}{p}}\}$, and the expectation value of the Hamiltonian, including diamagnetic and spin-orbit terms in a magnetic field $H_z$, is minimized with respect to $k$, $s$, and $p$ so as to determine these $H_z$-dependent parameters, and the energy is evaluated. In the presence of an infinitesimal electric field ${\mathcal{E}}_t$, in the $t$ direction, the wave function is taken to be ${\psi }_0\left(\mathrm{H}\right)(1+bt+c\rho t)$, where $\rho ={(x^2+y^2+s{z}^2)}^{\frac{1}{2}}$. The energy is minimized with respect to $b$ and $c$, and the coefficient of ${{\mathcal{E}}_t}^2$ is identified as $-\left(\frac{1}{2}\right){\alpha }_t$, the polarizability component, which contains diamagnetic and weakly spin-dependent terms. Comparison with experiments of Castner and Lee on donors in Si is briefly discussed.

For the theoretical assessment of the $2{}_{}{}^3{}_{}{}^{}P$ helium fine structure to become comparable to the precision measurements that have been made, it is necessary that the theory be calculated through order ${\alpha }^{6}m{c}^2$. In particular, the second-order contribution from the Breit and mass-polarization operators must be evaluated to an accuracy of 1% or so. In this work, for each of five possible intermediate state symmetries the Dalgarno-Lewis method is used to obtain the first-order perturbed wave function, from which the second-order energy follows by integration. Both the perturbed and unperturbed wave functions are expanded in Hylleraas-type series with a progressively larger number of terms, the second-order energies being computed at each stage; up to 455 terms are used for ${}_{}{}^3{}_{}{}^{}P$ intermediate states and up to 286 for ${}_{}{}^1{}_{}{}^{}P$, ${}_{}{}^3{}_{}{}^{}D$, ${}_{}{}^1{}_{}{}^{}D$, and ${}_{}{}^3{}_{}{}^{}F$. The sequence of second-order energy results for each symmetry is extrapolated to the limit of an infinite number of basis functions to arrive at a final result. The ${}_{}{}^3{}_{}{}^{}P$, ${}_{}{}^1{}_{}{}^{}P$, and ${}_{}{}^3{}_{}{}^{}D$ states will contribute to both the larger and the smaller fine-structure intervals ${\nu }_{01}$ and ${\nu }_{12}$, respectively, while ${}_{}{}^3{}_{}{}^{}F$ and ${}_{}{}^1{}_{}{}^{}D$ states affect only ${\nu }_{12}$. The total theoretical result, up to order ${\alpha }^{6}m{c}^2$, for ${\nu }_{01}$ is much more accurate than that for ${\nu }_{12}$, allowing the fine-structure constant $\alpha$ to be determined very precisely by comparison of theory to experiment, with the result ${\alpha }^{-1\mathrm{}}=137.036\mathrm{}08\left(13\right)\mathrm{}$, good to 0.94 ppm.

The consequences of factorizing an exact wave function into the product of an antisymmetric part and an exponential part are discussed. The exponent is taken as a symmetric real function and satisfies the equations of infinite-order perturbation theory. Practical calculations to determine both factors are performed by minimizing two functionals and the roles these play are analyzed. The relation between this procedure and that suggested by Boys and Handy for the transcorrelated wave function is pointed out. Illustrative calculations on helium show that good wave functions are obtained by both procedures.

A high-energy electron-impact spectroscopy (HEEIS) apparatus has been constructed for high-precision Compton-scattering experiments. Electron-Compton-scattering experiments are performed by crossing a beam of high energy, but nonrelativistic, electrons with a beam of atoms or molecules and measuring the energy-loss spectrum of the scattered electrons over a range of scattering angles. The improvements of design and technique, the method of data analysis, and the theory used to convert cross sections to Compton profiles are discussed fully. It was found that the energy-loss spectra taken over a range of scattering angles do not reduce by means of the binary-encounter approximation (impulse approximation) to Compton profiles in agreement with theory. This disagreement is most apparent in a shift of the experimental Compton peak—the Compton defect—from the peak predicted by the binary-encounter theory. The Compton defect has been studied in detail for momentum transfers from 1.5-12 a.u. for both He and ${\mathrm{H}}_2$. Defect measurements for ${\mathrm{D}}_2$, ${\mathrm{N}}_2$, and Ne have also been made and it was found that the ${\mathrm{N}}_2$ and Ne defects were opposite in direction from the He and ${\mathrm{H}}_2$ defects. The ${\mathrm{D}}_2$ defect was identical to that for ${\mathrm{H}}_2$. The electron Compton defect is discussed in relation to other recent defect measurements using x-ray and ($e, 2e$) techniques as well as recent theoretical results. An evaluation of the theory used to convert cross sections to Compton profiles is presented and, on the basis of the defect measurements, it is suggested that, even when the binary-encounter conditions have been attained at large momentum transfers, the binary-encounter theory breaks down in the high accuracy (1%) limit. An explanation for this breakdown is given and recent theories, which at least qualitatively account for the Compton defect, are discussed.

A formalism is developed for calculation of electron and positron scattering from atoms at low and intermediate energies in terms of appropriate matrix elements of the off-shell $T$ operator. The technique is based on an $L^2$ discretization of the full electronic Hamiltonian, initially avoiding all specification of channels and asymptotic boundary conditions. Scattering information relating to specific initial and final channels is obtained by taking appropriate off-shell matrix elements, followed by construction of the total real axis discontinuity of the off-shell elements via moment techniques. The formalism is applied to the problem of purely elastic $e^{+}$-H scattering, resulting in converged total cross sections and angular distributions below the pickup threshold.

Elastic amplitudes for $e^{+}$-H scattering are calculated for positron energies from 0.25-1.15 a.u. (∼7-34 eV), an energy range where pickup and impact-ionization channels are open. These amplitudes are used to calculate the elastic $e^{+}$-H cross section, and assuming unitarity, the total $e^{+}$-H scattering cross section.

We report absolute measurements of the total cross section, ${\sigma }_{10}$, for charge transfer of ${\mathrm{C}}^{+}$, ${\mathrm{N}}^{+}$, and ${\mathrm{O}}^{+}$ in ${\mathrm{N}}_2$ and ${\mathrm{H}}_2$ in the energy range of 5-100 keV. Maxima in all six cross sections occur in this energy range.

Double-$K$-vacancy sharing has been investigated for 30-MeV Cl on thick Ca, Sc, and Ti targets by measuring the ratios of the hypersatellite to the satellite $K$-x-ray yields in both collision partners. A simple extension of the single-$K$-vacancy-sharing model predicts a double-$K$-vacancy-sharing ratio equal to the square of the single-$K$-vacancy-sharing ratio. Our results are in agreement with this prediction.

The threshold behavior of the absolute total electron-detachment cross section for collisions of ${\mathrm{Cl}}^{-}$ and ${\mathrm{Br}}^{-}$ with rare gases has been determined in an ion-beam—gas-target apparatus. The relative collision energies investigated range from below the electron affinity of the halogen up to about 100 eV. For all of the reactants studied, the threshold for detachment is found to occur at about twice the electron affinity of the negative ion. The general features of measurements are in agreement with the ideas of a simple collision model. Upper limits to detachment-rate constants which are based upon the measurements are presented.

Several methods designed to avoid computational anomalies in the multichannel variational formalism of Kohn are considered. Two new methods are proposed. A new variant of the optimized anomaly-free method ensures that the computed $K$-matrix is symmetric. This method avoids spurious singularities but may give results that vary discontinuously with energy. A second new method, the restricted interpolated anomaly-free method, avoids both singularities and discontinuities, and produces a symmetric $K$ matrix. Two-channel model calculations show that both new methods have accuracy comparable to the earlier anomaly-free and optimized anomaly-free methods.

Results of a relativistic calculation of the cross section for the ionization from the $K$ and $L$ shells by high-energy electrons are presented. The calculations use the first-order Born approximation to treat the interaction between the scattered and atomic electrons. Plane waves are used to treat the high-energy scattered electron, while solutions of the Dirac equation in a Hartree-Slater central potential are used to describe the atomic electrons. Results are presented for a set of elements from $Z=18\mathrm{} \mathrm{to} 92$ and incident energies from 50 keV to 1 GeV. Calculated results are given for the correction due to the density effect arising from the polarizability of the medium. In the calculation of this correction, the medium is treated as composed of free electons.