The Slater $X\alpha$ method and its local-potential modifications are examined with reference to the many-electron exchange-correlation effects in the beryllium atom and its isoelectronic series. The linear coupled-cluster method and a hierarchy of approximations to it are employed for this purpose. The role of the exchange parameter $\alpha$ in providing an accurate description of the exchange-correlation effects is analyzed in the light of the electron-gas model. It is found that for Be atoms an $\alpha$ value of 0.768, that which causes the local potential to mimic the Hartree-Fock potential, is the best suited reference state for many-body calculations. The impact of the.$\alpha$ variation on the exchange-correlation corrections in the Be isoelectronic series is assessed. With increase in the nuclear charge $Z$, exchange-correlation corrections favor the use of $\alpha$ values closer to $\frac{2}{3}$, the Gaspar-Kohn-Sham limit, in the $X\alpha$ model. The instabilities in the cluster equations induced by ringdiagram terms are also noted. The futility of using gradient corrections to the $X\alpha$ model to account for exchange-correlation effects is brought out in the calculations. It is found that a simple scaling of the electron-gas potential results in excellent single-particle reference states for many-body calculations.

In this series of papers we will present a new approach to quantum-chemical methods in which the two-electron integrals are absent. In this first paper we discuss the basic ideas and their conceptual and technical advantages. The electronic interactions are reduced to a classical electrostatic problem by writing the Coulomb and exchange matrices in terms of the Fock potentials. The separation of short- and long-range contributions in these potentials is presented as the key for the development of both new calculational methods and realistic models for complicated chemical problems. These ideas are developed in the following paper.

Experimental energy-loss spectra for excitation of the $5{}_{}{}^2{}_{}{}^{}S\to 5{}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ transition in Cd II by electrons have been observed for the first time. Differential scattering cross sections are reported from spectra obtained at 75-eV electron energy. Comparisons are made to experimental results for the ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ integral excitation cross section from absolute emission measurements on the 214.4-nm (${}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$) transition, to theoretical results for the analogous $4{}_{}{}^2{}_{}{}^{}S\to 4{}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ transition in Zn II, and to semiempirical calculations in the Gaunt-factor approximation. A detailed description of the crossed (90°) electron-beam—ion-beam apparatus is given, and the method of conversion of scattering intensity to absolute cross section is discussed.

Absolute cross sections for multiple ionization (${\sigma }_q$) of He, Ne, Ar, and Kr are presented for proton-impact energies between 10 and 4000 keV. Charge states up to 6+ are observed for Kr. The results are obtained by measuring relative yields of multiply charged ions with a time-of-flight spectrometer and normalizing to total ion-production cross sections ${\sigma }_T=\Sigma q{\sigma }_q$. The results are compared with existing proton- and electron-impact data. Relative multiple- to single-ionization cross-section ratios are also presented for ${\mathrm{He}}^{+}$ impact.

Linear-and circular-polarization measurements have been made for the $3{}_{}{}^3{}_{}{}^{}P\to 2{}_{}{}^3{}_{}{}^{}S$ transitions in helium resulting from the collision processes, ${\mathrm{He}}^{+}+\frac{{\mathrm{H}}_2}{{\mathrm{D}}_2}\to \mathrm{He}\left(3\mathrm{} {}_{}{}^3{}_{}{}^{}P\right)+\frac{{{\mathrm{H}}_2}^{+}}{{{\mathrm{D}}_2}^{+}}$. This was carried out at an incident-ion lab energy of 1.5 keV and for laboratory scattering angles ranging from 0.50 to 2.33 deg. The behavior of linear polarization for ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ targets as a function of scattering angle was found to be different, contrary to what one would expect from simple models that depend only upon the electronic structure of the system. Some correlation of the circular polarization between the two isotopes was found.

Electron capture to $n\ge 2$ levels of ${\mathrm{F}}^{7+}$ $\left(1s2s\right){}_{}{}^3{}_{}{}^{}S$ projectiles lead to three-electron ions with an energetically allowed $K$-Auger decay channel. We have measured the F $K$-Auger emission spectra for collisions with thin gas targets of He and Ne with sufficiently high resolution to distinguish capture to several of the low-lying $n$ values. The $K$-Auger production cross sections are reported as a function of the $n$ level into which the electron is captured. The $n$-level dependence is measured for projectile energies of 6, 9, 12, and 15 MeV and is compared to the Oppenheimer-Brinkman-Kramers OBK model for charge transfer and to a $\frac{1}{n^3}$ function. The measured $K$-Auger electron-production cross sections closely follow the $\frac{1}{n^3}$ function which differs from the predicted OBK $n$ dependence, even though the cross sections for the higher $n$ levels agree with the predicted OBK energy dependence. The effects of cascading upon the calculated $n$ dependence are also studied.

A Green's-function approach is used to solve the Schrödinger equation in an effective potential ($V_0$), which is the sum of independent-atom static potentials. The equation for the Green's function is conveniently solved in momentum space (MS), where the MS "potentials" (Fourier transforms of the atom-centered potentials) have translational symmetry. The Green's function is then used to construct the solution to the Schrödinger equation for scattering in the potential $V-V_0$ (where $V$ is the $e$-molecule static potential plus a local exchange potential) relative to scattering in $V_0$. This solution is found in coordinate space using single-center expansions about the internuclear midpoint. These are more rapidly convergent for $V-V_0$ than for $V$ or $V_0$ alone. The sum of the amplitudes for scattering in $V_0$ and in $V-V_0$ relative to $V_0$ then represents the amplitude for scattering from the molecule. This method is intended to combine the dynamical methods best suited for each type of potential (multicenter for $V_0$ and single center for $V-V_0$). It also exposes the shortcomings of the use of $V_0$ alone.

We have generated a highly polarized ($P=0.48\mathrm{}\pm 0.02$) intense (5×${10}^5$/sec) beam of "slow" ($\Delta E\sim 2$ ev) positrons ($e^{+}$) and shown that it is possible to achieve polarization as high as $P=0.69\mathrm{}\pm 0.04$ with reduced intensity. The measured polarization of the slow $e^{+}$ emitted by five different positron moderators showed no dependence on the moderator atomic number (2s$Z$). This allows us to conclude that only source positrons with final kinetic energy below 17 keV contribute to the slow-$e^{+}$ beam, in disagreement with recent yield functions derived from low-energy measurements. Measurements of polarization and yield with absorbers of different $Z$ between the source and moderator show the effects of the energy and angular distributions of the source positrons on $P$. The depolarization of fast positrons transmitted through high-$Z$ absorbers has been measured. Applications of polarized slow- $e^{+}$ beams are discussed.

Energy levels and fine-structure splittings are calculated for the ground-state configuration of the oxygen isoelectronic sequence using the multiconfiguration Hartree-Fock method with Breit-Pauli corrections. Electric quadrupole and magnetic dipole transition probabilities are reported for transitions between several of these levels. Our results are compared with those from other theories and, in the case of energy levels, also with observation. For degrees of ionization up to about 12 ($Z=20\mathrm{}$), our energy levels are in better agreement with observation than other theories, but for ${}_{}{}^1{}_{}{}^{}\mathrm{D}_2^{}{}_{}{}^{}$ a relatively larger, systematic discrepancy remains.

For the $\frac{1}{Z}$ perturbation theory of atoms, a partial-wave method is presented for determining first-order pair wave functions. It rests on the fact that the Hylleraas variational principle decouples for the individual partial waves ($l=0,1,2,\dots$) and that all partial waves for $l\gg 1$ are easily representable (Schwartz, 1962). The $l\mathrm{th}$ partial wave is approximated in basis functions obtained by projecting the well-known Hylleraas functions (containing the powers of $r_{12}$ onto $P_l\left(cos{\theta }_{12}\right)$. Results for the ${1s}^2$ ground state show rapid convergence. The variational value for the (total) second-order ${1s}^2$ energy, which would be provided by 45 Hylleraas functions, is achieved with 10, 12, 9, 8, 5, and 5 basis functions for $l=0,1,2,3,4\mathrm{}$, and $5$, respectively. For any $1\ge 6$, one function is sufficient. Also good convergence is found for three-electron integrals (parts of the second-order lithium energy).

The determination of the most standard effective Hamiltonians is obtained by means of a simple general similarity transformation. The wave operator is shown to be a solution of an operator equation which is the analog of the Møller equations of scattering theory and which generalizes those previously established by Bloch, Löwdin, Jørgensen, and Lindgren. The wave-operator equation is solved by efficient iteration or perturbation-iteration methods which exhibit good convergence properties for degenerate systems and/or in presence of intruder states. In the following paper the method is applied to the theoretical determination of transferable effective-spin interactions.

Starting from a Pariser-Parr-Pople Hamiltonian, exact effective-spin interactions are determined in a significant set of $\pi$ molecules by the means of the wave-operator method presented in the preceding paper. Comparison with the quasidegenerate perturbation theory (QDPT) demonstrates the technical efficiency of the wave-operator formalism which exhibits regular convergence properties while the QDPT diverges for actual physical Hamiltonians. The determination of accurate spin-exchange interactions between bonded and nonbonded carbon atoms makes possible a careful study of their transferability. Then it is shown how effective-spin Hamiltonians of large systems can be constructed from Hamiltonians of small pattern molecules. Finally, the wave-operator formalism is applied to effective Hamiltonians arising from higher ionic atomic states which correspond to the lowest dipolar-allowed states and may be considered as an extension of the family of the Heisenberg-type effective Hamiltonians.

It is shown that Compton profiles (CP) can be measured by inelastic ion-electron scattering. Within the impulse approximation the binary-encounter peak (BEP) reflects the CP of the target atom whereas the electron-loss peak (ELP) is given by projectile CP's. Evaluation of experimental data reveals that inelastic ion-electron scattering might be a promising method to supply inelastic electron or photon scattering for the determination of target CP's. The measurement of projectile CP's is unique to ion scattering since one gains knowledge about wave-function effects because of the high excitation degree of fast heavy-ion projectiles.

Electronic stopping cross sections for ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ projectiles (840-175 keV) have been measured with the inverted Doppler-shift attenuation method in liquid ethane (${\mathrm{C}}_2$${\mathrm{H}}_6$) at two different densities. The density of the target has been varied by changing the temperature, and measurements have been performed at 0.525 g/${\mathrm{cm}}^3$ (199 K) and 0.362 g/${\mathrm{cm}}^3$ (287 K). At the higher density the stopping cross section is about 2% smaller. This result agrees with a calculation of the stopping cross section of liquid ethane, applying Lindhard's theory in the local-density approximation using a simple model of the liquid. It is also in agreement with various observations of the so-called physical-state effect, which show that the stopping cross section of the same substance is smaller in a condensed phase than in the gaseous one.

The measurements of $M$-shell x-ray-production cross sections induced by ${}_1{}^1{}_{}{}^{}\mathrm{H}_{}^{+}{}_{}{}^{} \mathrm{and} {}_2{}^4{}_{}{}^{}\mathrm{He}_{}^{+}{}_{}{}^{}$ ions are compared to the first-Born-approximation and ECPSSR (energy loss, Coulomb-deflection effects; perturbed-stationary-state approximation, with relativistic corrections) theories. Most of the reported experimental data were measured in our laboratory and the other measurements were taken from the literature. The data from our laboratory were for incident ${\mathrm{H}}^{+}$ and ${\mathrm{He}}^{+}$ ions in the energy range from 0.25 to 2.5 MeV. The $M$-shell x-ray-production cross sections were measured for the following thin targets: ${}_{59}{}^{}{}_{}{}^{}\mathrm{Pr}$, ${}_{60}{}^{}{}_{}{}^{}\mathrm{Nd}$, ${}_{63}{}^{}{}_{}{}^{}\mathrm{Eu}$, ${}_{64}{}^{}{}_{}{}^{}\mathrm{Gd}$, ${}_{66}{}^{}{}_{}{}^{}\mathrm{Dy}$, ${}_{67}{}^{}{}_{}{}^{}\mathrm{Ho}$, ${}_{68}{}^{}{}_{}{}^{}\mathrm{Er}$, ${}_{70}{}^{}{}_{}{}^{}\mathrm{Yb}$, and ${}_{72}{}^{}{}_{}{}^{}\mathrm{Hf}$. The data from the literature were for protons and ${\mathrm{He}}^{+}$ ions in the energy range from 30 keV to 40 MeV. These data were for the following elements: ${}_{54}{}^{}{}_{}{}^{}\mathrm{Xe}$, ${}_{59}{}^{}{}_{}{}^{}\mathrm{Pr}$, ${}_{60}{}^{}{}_{}{}^{}\mathrm{Nd}$, ${}_{62}{}^{}{}_{}{}^{}\mathrm{Sm}$, ${}_{63}{}^{}{}_{}{}^{}\mathrm{Eu}$, ${}_{64}{}^{}{}_{}{}^{}\mathrm{Gd}$, ${}_{65}{}^{}{}_{}{}^{}\mathrm{Tb}$, ${}_{66}{}^{}{}_{}{}^{}\mathrm{Dy}$, ${}_{67}{}^{}{}_{}{}^{}\mathrm{Ho}$, ${}_{68}{}^{}{}_{}{}^{}\mathrm{Er}$, ${}_{70}{}^{}{}_{}{}^{}\mathrm{Yb}$, ${}_{72}{}^{}{}_{}{}^{}\mathrm{Hf}$, ${}_{73}{}^{}{}_{}{}^{}\mathrm{Ta}$, ${}_{74}{}^{}{}_{}{}^{}\mathrm{W}$, ${}_{78}{}^{}{}_{}{}^{}\mathrm{Pt}$, ${}_{79}{}^{}{}_{}{}^{}\mathrm{Au}$, ${}_{80}{}^{}{}_{}{}^{}\mathrm{Hg}$, ${}_{82}{}^{}{}_{}{}^{}\mathrm{Pb}$, ${}_{83}{}^{}{}_{}{}^{}\mathrm{Bi}$, and ${}_{92}{}^{}{}_{}{}^{}\mathrm{U}$. The first-Born-approximation calculations of the ionization cross section were made using the plane-wave Born approximation for direct ionization and the Oppenheimer-Brinkman-Kramers approximation of Nikolaev for electron capture. The ECPSSR theory of Brandt and Lapicki [Phys. Rev. A 23, 1717 (1981)] goes beyond the first Born approximation and accounts for the energy loss, Coulomb deflection, and relativistic effects in the perturbed-stationary-state theory. The first Born approximation overpredicts all measurements. The ECPSSR theory predicts the $M$-shell production cross sections correctly for $Z_2>70\mathrm{}$ and energies per $\mu >0.25\mathrm{}$ MeV/μ. In the rare-earth region the ECPSSR results lie above the data at higher projectile energies and fall off below the data at lower energies.

Total collisional depopulation rates for $\mathrm{He}\left(n{}_{}{}^1{}_{}{}^{}P\right)\left(4\mathrm{}\le n\le 13\right)$ in thermal collisions with $\mathrm{He}\left(1\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)$ have been measured using the transient-decay method. Related loss cross sections increase in proportion to $n^4$ in the limited range $4\le n\le 6$, reach a maximum of 2600±600 ${\mathrm{\AA }}^2$ at $n=10\mathrm{}$, and decrease approximately in proportion to $n^{-2.5\mathrm{}}$ for $11\le n\le 13$. The measurements were found to be inconsistent with a strong "selection rule," $\Delta L=2\mathrm{}$, for the $\mathrm{He}\left(n^{1}P\right)-\mathrm{H}\mathrm{e}$ collisions. A model in which $\Delta L$ for the collision is largely unrestricted provides a satisfactory interpretation of the observations, in agreement with recent $l$-mixing studies of atomic Rydberg levels. The experimental cross sections compare favorably with values calculated using an approximate scaling formula for collisional $l$ mixing and, for $n>\mathrm{}10\mathrm{}$, with predictions based on a simple perturbation treatment in the weak-collision approximation.

The stopping of Ag ions recoiling through Al, Ti, V, Fe, Ni, Zn, Zr, and Pd was measured at various recoil energies with a method based on the Doppler shift of $\gamma$ rays. The dependence of the electronic stopping cross section on the stopper atomic number and on recoil velocity is compared to the semiempirical predictions of Ziegler et al.

A calculation of the amplitude for inner-shell ionization accompanying the resonant scattering of a proton from the atomic nucleus is presented which accounts for higher-order correction terms not included in previous treatments of this problem. The correction terms arise from the use of a more accurate asymptotic representation of the scattering wave functions which appear in the distorted-wave Born approximation for the amplitude. The energy transferred to the electron plays the role of a small parameter in the analysis of the accuracy of the approximation. Some techniques used in a previous derivation of a low-frequency approximation for bremsstrahlung are directly applicable to the closely analogous problem considered here. Numerical estimates of the correction terms are obtained using a very simple model of the ionization process.

Using the parallel-plate-capacitor method and a capacitance manometer to determine pressures, total cross sections for the production of positive and negative charges were measured for 5-4000-keV-proton impact on He, Ne, Ar, Kr, ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, CO, ${\mathrm{O}}_2$, C${\mathrm{H}}_4$, and C${\mathrm{O}}_2$. From these, ionization and electron-capture cross sections were obtained and fitted to semiempirical equations describing the energy dependence in terms of a few parameters. At high energies very good agreement is obtained in the comparison of the ionization cross sections to earlier proton- and electron-impact measurements and with theoretical treatments where they are available, but discrepancies exist for some targets at low energy. Above 10 keV the electron-capture cross sections are in agreement with earlier work for all the targets except CO and C${\mathrm{H}}_4$ for which they are (20-40)% higher.

We present a new method for obtaining local, energy-dependent, single-channel or multichannel optical potentials. The optical potential is obtained by calculating and smoothing a potential that yields the correct scattering matrix for a sequence of truncated interaction-potential matrices. The new method is illustrated by applications to electron-hydrogen-atom scattering in a truncated Hilbert space. The resulting optical potentials accurately reproduce the original close-coupling calculations in the same Hilbert space, and they show clearly the effects of charge polarization and flux absorption. The coordinate dependence and energy dependence of the smoothed optical potentials are compared to models in the literature.