A diabatic-states representation for scattering of triplet helium atoms, with $n\ge 3$, by the ground-state helium atom is presented. Ab initio calculations have been performed on the ${}_{}{}^3{}_{}{}^{}\Sigma _{g,u}^{+}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}\Pi _{g,u}^{}{}_{}{}^{}$, and ${}_{}{}^3{}_{}{}^{}\Delta _g^{}{}_{}{}^{}$ Rydberg states of ${\mathrm{He}}_2$. Matrix elements for radial and angular coupling between the orthogonalized diabatic states have also been evaluated. The angular couplings are quite weak, with the result that molecular states with different electronic symmetries are effectively uncoupled in low-energy collisions. The crossing by a repulsive diabatic-potential curve through the Rydberg series and into the continuum is shown to be an effective mechanism for associative ionization. Some qualitative observations on associative ionization and dissociative recombination are made.

A model has been derived to treat the multistate curve-crossing problem which often arises in diabatic representations of scattering processes. Transitions between diabatic states are assumed to occur only in the vicinity of curve crossings and the probabilities are evaluated semiclassically. Otherwise the nuclear trajectories are treated classically except for tunneling through long-range potential barriers taken into account by the JWKB approximation. Closed channels are included on an equal footing with the scattering channels. The model is applied to the problems of symmetric associative (and dissociative) ionization and excitation transfer in collisions of excited helium with the ground-state atom, using the diabatic representation for the Rydberg states of ${\mathrm{He}}_2$ presented in the preceding paper. The electronic coupling of the repulsive diabatic states in the continuum yields much larger ionization cross sections than does direct vibronic coupling of low-lying adiabatic states to the continuum. The diabatic-states model is shown to be a valid interpretation of associative ionization in helium. Ionization has a significant effect on some excitation-transfer cross sections. Integrated cross sections are presented for ionization of helium in the $n{}_{}{}^3{}_{}{}^{}S$, $n{}_{}{}^3{}_{}{}^{}P$, and $n{}_{}{}^3{}_{}{}^{}D$ states, $n=3, 4$, and for excitation transfer from the $n=3\mathrm{}$ states to states with $2\le n\le 4$ for collision energies from thermal to 100 eV. The thermally averaged associative ionization cross sections, in units of ${10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$, obtained at 300°K are as follows: ${\overline{Q}}_i\left(3\mathrm{}{}_{}{}^3{}_{}{}^{}S\right)=0.06\mathrm{}$, ${\overline{Q}}_i\left(3\mathrm{}{}_{}{}^3{}_{}{}^{}P\right)=2.0\mathrm{}$, ${\overline{Q}}_i\left(3\mathrm{}{}_{}{}^3{}_{}{}^{}D\right)=2.9\mathrm{}$, ${\overline{Q}}_i\left(4\mathrm{}{}_{}{}^3{}_{}{}^{}S\right)=0.6\mathrm{}$, ${\overline{Q}}_i\left(4\mathrm{}{}_{}{}^3{}_{}{}^{}P\right)=1.7\mathrm{}$, and ${\overline{Q}}_i\left(4\mathrm{}{}_{}{}^3{}_{}{}^{}D\right)=4.3\mathrm{}$. The results are in quite satisfactory agreement with recent experiments.

The expansion method for the nonrelativistic Hartree-Fock theory by Roothaan is extended to relativistic cases for general open-shell atoms. Numerical applications of the theory to the first transition elements are carried out. It is shown that the conventional approximation of the exchange potential shifts the energy value considerably, whereas the off-diagonal Lagrange multiplier has little effect on the energy. Some problems in the application of the theory to open-shell atoms are discussed.

A simple variational approach to perturbation theory is presented which contains as special cases the Goldhammer-Feenberg refinement of the Brillouin-Wigner series, the corresponding refinement of the Rayleigh-Schrödinger series, as well as the continued-fraction expansion of the resolvent matrix element. The approach is very effective and works for arbitrary coupling strengths. The method is applied to one-electron atoms, taking the whole Coulomb attraction $-\frac{Z}{r}$ as a perturbation. Already with the first-order wave function, one obtains 99.3% of the exact ground-state energy for all $Z$. With the second-order wave function one obtains 99.96% of the exact ground-state energy and 99.6% of the exact energy of the first excited state, again for all $Z$. In the present example the numerical effort is comparable to the one needed to perform the Rayleigh-Schrödinger expansion to the corresponding order of the wave function.

The kinetic-energy operator derived by De Celles and Darling with center-of-mass motion separated is used to derive an orthonormal set of functions for the spherically symmetric states of a three-body bound-state system of arbitrary masses and charges. This approach allows the calculation of the energy $E$ to be reduced to the solution of a coupled system of differential equations in one variable.

The method of numerical differentiation is applied to solve the coupled set of ordinary differential equations for a three-body system derived in the previous paper. This is done for the ground state of the He atom. The matching of the forward and backward numerical solutions of the coupled system of differential equations is achieved by repeated applications of Newton's correction method. The matching of up to 16 coupled curves is carried out successfully, but the results obtained show that a very large number of coupled terms are required to obtain a good convergence of the ground-state energy.

A procedure for obtaining a correlated multiconfiguration Hartree-Fock (MCHF) wave function, similar to the one used for the $1s^{2}2s^2$ groud state of Be, is applied to the $1s^{2}2s2p{}_{}{}^1{}_{}{}^{}P$ state. With ${\Phi }^{\mathrm{MCHE}}=a_1|1\mathrm{}{s}^{2}2s2p〉+a_2|1\mathrm{}{s}^{2}2p3d〉$ as a zero-order approximation, a first-order correction is derived which includes the effects of one- and two-electron replacements. Contributions from replacements from a particular pair of electrons were determined separately by a numerical MCHF procedure. The "pair correlation functions" were combined in a final configuration-interaction calculation with 52 configurations. The approximation yielded a total energy of -14.47085 a.u. which accounts for 97.5% of the correlation energy. A further first-order correction improved the correlation energy to 98.0%.

We present a systematic study of self-consistent symmetries (SCS) for the ten-electron atom ${\mathrm{O}}^{--}$ in an $s+p$ basis. The investigation of each SCS required a detailed study of the character of the iterative process. Convergency is ensured by a minimum-energy selection principle. Six deformed Hartree-Fock solutions for ${\mathrm{O}}^{--}$ have been obtained.

The determination of shielded diatomic orbitals (SDO's) which are eigenfunctions in prolate-spheroidal coordinates, of the two-center problem with an effective (cylindrically symmetric) parametric potential, is extended to many-electron systems. The simultaneous iterative resolution of the separated equations is significantly improved by the consideration of a simple electrostatic model and of the connection with a Mathieu equation. Calculations are carried out for three- and four-electron diatomic systems, i.e., HeH, Li${\mathrm{H}}^{+}$, ${\mathrm{He}}_2^{+}$, LiH, and ${\mathrm{He}}_2$ in their ground states. From our results, the SDO's appear to be comparable to the self-consistent field orbitals. They should be also a good basis set for "diabatic" molecular states. Possibility of an extension, without supplementary difficulties, of the effective potential is investigated.

The elastic scattering amplitude obtained from an open pseudostate calculation employing the algebraic variational method for partial waves of angular momentum $0<L<\mathrm{}3\mathrm{}$, is combined with phase shifts obtained in the extended polarization approximation for $4<L<\mathrm{}99\mathrm{}$ to obtain elastic differential cross sections for electron-hydrogen scattering for electron energies in the range 10-30 eV. Elastic angular distributions have been measured using a crossed-electron modulated-atom-beam method and absolute cross-section values determined using a phase-shift analysis method. The measurements include energies from 10 to 30 eV and angles from 10° to 140° with estimated experimental error limits of ± 17%. Within these error limits they are in agreement with the theoretical predictions except at 20 to 26 eV for angles greater than or equal to 120°, where the theoretical values are significantly higher than allowed by these error limits.

The total and differential elastic scattering cross section of electrons by atomic oxygen and nitrogen is calculated for energies between 1 and 500 eV in the distorted-wave (DW) approximation. The DW method has been modified by the inclusion of a polarization potential with an adjustable cutoff parameter which is determined variationally. The relative importance of polarization and exchange effects are compared for a variety of energies. These calculations show that the DW method is reliable above about 10 eV and with the inclusion of the polarization potential improves the estimates even below this energy.

We have calculated the projectile and target $K$-excitation cross sections and their ratio for a series of ion-target combinations using a basis of exact molecular eigenstates for a model system of one electron in the field of two effective nuclear charges. Except for very asymmetric high-velocity collisions, close agreement is found with Meyerhof's formula, which is based on an atomic two-state expansion. In this range, Meyerhof's formula may be considered an analytical fit to our ab initio results. We indicate that the apparent accuracy of Meyerhof's formula is due to mutual cancellation of effects arising from approximations and model assumptions used in his deduction of the cross-section ratio. The significant deviation found at high velocities results from interference between the coupling processes that govern $K$-vacancy production. This possibility was ruled out in Meyerhof's theory.

The first measurements of the temperature dependence of the intrinsic dipolar relaxation time $T_{1B}$ due to binary collisions in dilute ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ gas are reported. Experimental conditions sufficiently pure to observe $T_{1B}$ were obtained by coating bare Pyrex sample cells with clean neon gas. The experiments were performed at low temperatures (1.7 to 19.0 K) where the colliding atoms have correspondingly low energy, so the effect of the long-range attractive forces is strongly felt and the measurements are sensitive to the depth and nature of the helium attractive well. Calculations of $T_{1B}$ using the best available helium potentials have been fitted to the experimental results to test the parameters which describe the potential. The data favor a potential of the Bruch-McGee form, but having a slightly deeper attractive well of 11.5 K.

The scattering of electrons from carbon monoxide has been studied in the framework of a previously proposed pseudopotential method. It is found that the single-center expansion of the continuum electron orbital about the center of mass of the molecule is very well converged when the two noncoincident nuclear singularities and the nonlocal exchange interactions are properly taken into account in the scattering equations. An analytical proof has been given to show that unlike the total cross section, the momentum-transfer cross section in the fixed-nuclei approximation is finite even for polar molecules. The calculated results of the momentum-transfer cross section for $e^{-}$-CO scattering in the energy range between 0.1 and 10.0 eV are in good accord with the experimental measurements.

The energy and intensity of scattered electrons resulting from excitation of the $\left(2\mathrm{}{s}^2\right){}_{}{}^1{}_{}{}^{}S$ and $\mathrm{}\left(2s2p\right)\mathrm{}{}_{}{}^3{}_{}{}^{}P$ autoionizing states of helium have been measured as a function of incident electron energy. For incident energies only slightly higher than the threshold for $\left(2\mathrm{}{s}^2\right){}_{}{}^1{}_{}{}^{}S$ excitation, post-collision interactions between the scattered and ejected electrons cause the slowly moving scattered electron to lose an amount of energy proportional to $E_{\mathrm{ex}}^{-1.25\mathrm{}\pm 0.05}$ (where $E_{\mathrm{ex}}$ is the energy the scattered electron would have had in the absence of a post-collision interaction). This energy dependence compares well with an energy gain of the ejected electron proportional to $E_{\mathrm{ex}}^{-1.2\mathrm{}}$, obtained by previous workers. Measurements of the intensity of scattered electrons show that both ${}_{}{}^1{}_{}{}^{}S$ and ${}_{}{}^3{}_{}{}^{}P$ total cross sections peak at threshold and have magnitudes about (2.0 and 4.0) × ${10}^{-20\mathrm{}}$ ${\mathrm{cm}}^2$, respectively, then rise to broad maxima of magnitudes about (2.5 and 5) × ${10}^{-20\mathrm{}}$ ${\mathrm{cm}}^2$ about 4.0 eV above their thresholds. A subsidiary peak at 59.0 eV in the ${}_{}{}^3{}_{}{}^{}P$ cross section is attributed to decay of the previously unidentified $\left(2s2\mathrm{}{p}^2\right){}_{}{}^2{}_{}{}^{}S$ ${\mathrm{He}}^{-}$ state into the ${}_{}{}^3{}_{}{}^{}P$ channel. The magnitude, energy, and width of this resonant structure in the ${}_{}{}^3{}_{}{}^{}P$ channel agrees with theoretical predictions.

A study of the low-energy elastic $e$-${\mathrm{H}}_2$ scattering information contained in multiconfiguration variational wave functions for a low-lying (positive) energy of the ${\mathrm{H}}_2^{-}$ system is described. The wave function is calculated using standard bound-state techniques, employing the modified valence-bond method with a basis set of Slater-type orbitals, and from it are obtained starting values at moderate electron-molecule separations for a simple single-channel scattering calculation, which is carried out beyond the region where exchange and coupling of partial waves are thought to be important. Results are presented for $s_{\sigma }$ elastic scattering for incident-electron energies ≤ 10 eV in the static-exchange approximation with polarization included by means of an adiabatic polarization potential. The study shows that the standard ${\mathrm{H}}_2^{-}$ variational function does indeed provide a good approximation to the elastic scattering wave function in the region close to the molecule, but that care must be taken in choosing the basis set. Comparisons are made with other more traditional calculations.

Matrix variational calculations of electron scattering by atomic N have been carried out, including target configurations $2s^{2}2p^3$, $2s2p^4$, $2p^5$, and all possible single virtual excitations of $2s$ and $2p$ orbitals. Total and differential elastic and inelastic cross sections are reported for energies below the first $n=3\mathrm{}$ excitation threshold. In the lowest energy range, scattering is dominated by a ${}_{}{}^3{}_{}{}^{}P$ shape resonance whose exact location is sensitive to small differences of electronic correlation energy. Results were obtained for several values of a net correlation energy parameter that produces a systematic shift in the resonance position. Detailed cross sections are given for the value of this parameter in best accord with existing low-energy scattering data. A ${}_{}{}^3{}_{}{}^{}P_{}^o{}_{}{}^{}$ shape resonance is found in the ${}_{}{}^2{}_{}{}^{}D_{}^o{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ excitation cross section just below the $n=3\mathrm{}$ threshold. Estimates are given of the rate constants for deactivation of the ${}_{}{}^2{}_{}{}^{}D_{}^o{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ states by thermal electrons. The locations of the ${}_{}{}^1{}_{}{}^{}D$ and ${}_{}{}^1{}_{}{}^{}S$ states of ${\mathrm{N}}^{-}$, which do not couple to continuum states, are estimated.

Matrix variational calculations of electron scattering by atomic C have been carried out, including target atom configurations $2s^{2}2p^2$, $2s2p^3$, and $2p^4$ and all possible single virtual excitations of the $2p$ orbital. Total and differential elastic and inelastic cross sections are reported for energies below the first $n=3\mathrm{}$ excitation threshold. In the lowest energy range, scattering is dominated by a ${}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ shape resonance. The position of this resonance, which is sensitive to small differences of electronic correlation energy, is parametrized by a net correlation energy parameter whose value is chosen to place the ${}_{}{}^2{}_{}{}^{}D_{}^o{}_{}{}^{}$ state of ${\mathrm{C}}^{-}$ at its experimental position, just below the neutral ground state. Estimates are given of the rate constants for deactivation of the ${}_{}{}^1{}_{}{}^{}D$, ${}_{}{}^1{}_{}{}^{}S$, and ${}_{}{}^5{}_{}{}^{}S_{}^o{}_{}{}^{}$ states by thermal electrons.

Concentration distributions of 800-keV ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}_{}^{+}{}_{}{}^{}$ ions implanted in metallic targets from titanium ($Z_2=22\mathrm{}$) through germanium ($Z_2=32\mathrm{}$) were determined from $\gamma$-ray-yield distributions using the ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}(p, \gamma ){}_{}{}^{15}{}_{}{}^{}\mathrm{O}$ resonance reaction at 1061.0 keV. Comparison of the first and second moments of these distributions with those predicted by the theory of Lindhard, Scharff, and Schiott (LSS) shows that the measured distributions are generally shallower and narrower. In addition, the measured distributions are strongly skewed such that their concentrations fall off more rapidly for deeper target layers. Values for the third moment of these distributions are also presented. We have attributed the discrepancies between the measured and predicted distributions to the behavior of the electronic stopping power, $S_e$. Values of $S_e$ for the targets studied here were determined by altering the LSS value of $S_e$ for each target by a multiplicative constant such that the predicted projected range is equal to the first moment of the measured distribution. With this adjustment, the predicted range straggling agrees closely with the measured second moments. The adjusted values of $S_e$ exhibit a $Z_2$ dependence which is not accounted for in either the LSS or Firsov pictures of electronic stopping power. Increases in the LSS values by as much as 60% are needed to describe the measured distributions. The $Z_2$ dependence of $S_e$ observed here suggests that there is an oscillatory behavior similar to the $Z_1$ dependence previously reported by Ormrod, MacDonald, and Duckworth, and by Hvelplund and Fastrup.

$L$-shell x-ray production cross sections for the $\mathrm{Ll}$, $L{\alpha }_{1,2}$, $L{\gamma }_1$, and $L{\gamma }_{2,3,\left(6\right)\mathrm{}}$ have been measured for ${}_{}{}^1{}_{}{}^{}\mathrm{H}$, ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, and ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ on Yb and Pb and for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ on Sm. The energy ranges for the incident-ion species were 0.3 to 2.4 MeV/amu for ${}_{}{}^1{}_{}{}^{}\mathrm{H}$, 0.15 to 4.4 MeV/amu for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, and 0.9 to 3.0 MeV/amu for ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$. Comparisons of the $L{\alpha }_{1,2}$, $L{\gamma }_1$, and $L{\gamma }_{2,3,\left(6\right)\mathrm{}}$ x-ray production cross-section data have been made to the predictions of the plane-wave Born approximation (PWBA) and the PWBA with corrections for Coulomb deflection and/or binding-energy effects. The comparisons show that for the individual x-ray production cross sections, there are problems associated with the binding-energy correction calculations for the $L{\gamma }_1$ and $L{\gamma }_{2,3}$ x-ray data. Furthermore, measurements of the ratio of $L{\alpha }_{1,2}$ to $L{\gamma }_{2,3}$ x-ray intensities do not support the use of an increased effective binding energy in the PWBA formulation.