The formalism of Møller wave operators is shown to provide a stable basis for computation of bound-free transition amplitudes for both short- and long-range potentials without the direct calculation of scattering wave functions. This method, which relies on the techniques of expansion in finite $L^2$ bases and rotation of the coordinates into the complex plane, is applied to both an exponential potential and one that behaves asymptotically as $-\frac{1}{r^4}$. It is demonstrated that one obtains not only accurate magnitudes of the matrix elements, but accurate phases (i.e., the scattering phase shift) as well. Some relevant theoretical results with regard to the application of wave operators are also presented. Although couched in terms of potential scattering, the procedures are readily extendible to multichannel problems.

Pseudopotential local-spin-density calculations of the bond lengths, vibrational energies, and dissociation energies for the ${}_{}{}^3{}_{}{}^{}\Sigma _g^{-}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$, and two ${}_{}{}^1{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$ states of the silicon dimer have been performed. The bond lengths are within 1% and the vibrational frequencies are within 5% of experimental values. The dissociation energy of the ground state is larger than experiment by ∼1.25 eV. These results are near the limits of accuracy of the local-spin-density approximation.

We show that to any Lie algebra of observables in quantum mechanics there corresponds a system of densities, of which the one-body densities are just one example. These densities may correspond to a single pure state or to an ensemble of states. Like the one-body densities, they support a Hamiltonian dynamics of the time-dependent Hartree-Fock type, which can be applied, for example, to collective motions where the collective states of interest may be either more or less general than single Slater determinants. The dynamics can also be applied to the average dynamics of an ensemble of states.

Explicit Hilbert-space techniques are reported for construction of the discrete and continuum Schrödinger states required in atomic and molecular photoexcitation and/or photoionization studies. These developments extend and clarify previously described moment-theory methods for determinations of photoabsorption cross sections from discrete basis-set calculations to include explicit construction of underlying wave functions. The appropriate Stieltjes-Tchebycheff excitation and ionization functions of $n\mathrm{th}$ order are defined as Radau-type eigenstates of an appropriate operator in an $n$-term Cauchy-Lanczos basis. The energies of these states are the Radau quadrature points of the photoabsorption cross section, and their (reciprocal) norms provide the corresponding quadrature weights. Although finite-order Stieltjes-Tchebycheff functions are $L^2$ integrable, and do not have asymptotic spatial tails in the continuous spectrum, the Radau quadrature weights nevertheless provide information for normalization in the conventional Dirac $\delta$-function sense. Since one Radau point can be placed anywhere in the spectrum, appropriately normalized convergent approximations to any of the discrete or continuum Schrödinger states are obtained from the development. Connections with matrix partitioning methods are established, demonstrating that $n\mathrm{th}$-order Stieltjes-Tchebycheff functions are optical-potential solutions of the matrix Schrödinger equation in the full Cauchy-Lanczos basis. The energies at which the $n\mathrm{th}$-order optical potential vanishes identically correspond to generalized Gaussian quadrature points, in which case the associated Stieltjes eigenfunctions provide an optimal $L^2$ representation of the Schrödinger spectrum of discrete and continuum wave functions. The spectral contributions of individual Schrödinger states to $n\mathrm{th}$-order Stieltjes-Tchebycheff functions are obtained in closed form, indicating the latter are spectrally localized in the neighborhoods of the Radau quadrature energies, and spatially localized in accordance with the extent of the target eigenstate. Illustrative studies of the dipole spectra in atomic and molecular hydrogen clarify the nature and convergence of the method. Finally, procedures are indicated for construction of photoemission anisotropies and for performing coupled-channel calculations employing Stieltjes-Tchebycheff functions, and descriptive intercomparisons are made of the present development with more conventional computational procedures.

Configuration-interaction wave functions are calculated for all intrashell doubly excited states of the $N=2\mathrm{}$ shell for He and isoelectronic ions ${\mathrm{Li}}^{+}$, ${\mathrm{Be}}^{2+}$, and ${\mathrm{Ne}}^{8+}$ with the use of a basis of Sturmian functions. The density function $\rho (r_1,{\theta }_{12}|r_2=\alpha )$, which is the conditional probability that one electron will be found at distance $r_1$ from the nucleus with interelectronic angle ${\theta }_{12}$ given that the other electron is at distance $\alpha$ from the nucleus, is computed for each of the states and for various values of $\alpha$. Density plots for the ${}_{}{}^1{}_{}{}^{}S_{}^e{}_{}{}^{}$ states of He are compared with those from Hylleraas-Kinoshita wave functions to assess the quality of the Sturmian basis. Sequences of plots are then examined to investigate the possibility of collective motion in doubly excited states. For He the form of the densities $\rho (r_1,{\theta }_{12}|\alpha )$ is in qualitative accord with the molecular model proposed by Kellman and Herrick in that the collective rotational and bending vibrational states are readily identifiable. With increasing nuclear charge a transition occurs to less collective behavior, corresponding to superpositions of hydrogenic configurations associated solely with the $N=2\mathrm{}$ shell. Some states, e.g., ${}_{}{}^1{}_{}{}^{}D_{}^e{}_{}{}^{}$, correspond to single independent-particle configurations. Electron density distributions are also presented for the lowest $S$ and $P$ states of the He $n_1=2\mathrm{}$, $n_2=3\mathrm{}$ intershell manifold with a view to extension of the molecular interpretation to this case. Results are compared with those of calculations of the quantum states of model problems such as particles on concentric spheres.

The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell $n=3\mathrm{}$ in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\rho \left({\theta }_{12}\right)$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.

We report a measurement of the Lamb shift in hydrogenlike argon based on the electric-field-quenching method, and a measurement of the unperturbed lifetime of the $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state. We find the lifetime of the unperturbed $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state to be 3.487(0.036) ns in agreement with the theoretical value of 3.497 ns. This is the first measurement of the $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state lifetime of sufficient accuracy to observe the contribution to the total decay rate of the single-photon magnetic dipole decay. Our measured value of the Lamb shift is $\delta E(2\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-2\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{})=37.89\mathrm{}\left(0.38\right)\mathrm{}$ THz, lower than, but in agreement with, Mohr's value of 38.25(0.025) THz, and 2.7 standard deviations below Erickson's value of 39.01(0.16) THz. Both the quenched and natural lifetimes were measured by the beam-foil time-of-flight technique. In this paper we emphasize the treatment of the systematic effects in the beam-foil time-of-flight method including cascades from higher excited states, interference from the spectra of heliumlike ions, collisions in the residual gas, and perturbations from a highly excited extra electron, as well as systematic effects more unique to electric field quenching.

For the computation of convolution integrals of $B$ functions [Eq. (2.14) of E. Filter and E. O. Steinborn, Phys. Rev. A 18, 1 (1978)] which are a special class of exponential-type functions, different analytic representations are available. We have investigated the numerical properties of these convolution formulas and found out that results of sufficient accuracy can be obtained for molecular calculations. We have also investigated the numerical properties of the corresponding one-center integrals. Some new mathematical properties of $B$ functions that are essential for the understanding of the numerical properties of the convolution formulas are derived.

The problem of the ${\mathrm{H}}_2$ molecular ion is reinvestigated with the use of a method of truncated Hill determinants. Modified Hill determinants are found such that the truncations thereof form a convergent sequence. The wave functions obtained from the present formulation are expected to be more reliable than those reported earlier and in fact are shown to be orthonormal to one part in ${10}^6$. Equilibrium bond lengths ($R_e$), stiffness constants, and the wave functions at $R_e$ for various states are presented.

The isotope shifts of long series of spectral lines in the optical spectra of the alkaline-earth elements Ca and Sr have been measured by Doppler-free, two-photon laser spectroscopy with a thermionic diode. The variation of transition isotope shifts with transition energy yields the level isotope shifts with reference to the ionization limit and how these shifts vary with the binding energy of the optically active electron. The largest isotope shifts of levels in these elements appear to arise from the specific-mass effect in the presence of state mixing.

A general method which combines hypervirial relations with the Hellmann-Feynman theorem and perturbation theory is applied in order to calculate the rotational eigenvalues of the hydrogen molecular ion in a magnetic field. Analytical expressions as well as numerical results are presented for both low and high field strengths.

We consider making quantum nondemolition measurements on a harmonic oscillator by coupling it to an oscillator readout system using a four-wave-mixing interaction. The quantum nondemolition variable of interest appears as a nonlinear function of the number operator. It has the form ${\hat{D}}_1\mathrm{}\left(t\right)=cos\left(\chi a^{†}at\right)$. A sequence of measurements of this operator may be made with completely predictable results. The possible use of this system to detect gravitational radiation is suggested.

It is shown that the effective potentials of the adiabatic approximation, which depend on the magnetic field parameter $\beta =\frac{B}{B_0}$ ($B_0\cong 4.70\times {10}^5$ T) and the quantum number $s=-m\ge 0$ of the $z$ component of the angular momentum, asymptotically (i.e., for large $s$) can be traced back to one single potential function which solely depends on the ratio $p=\frac{\beta }{(s+\frac{1}{2})}$. For this asymptotic potential, numerical solutions of the Schrödinger equation are determined in the range ${10}^{-3\mathrm{}}\le p\le {10}^{+3\mathrm{}}$ for $0\le \nu \le 20$ ($\nu$ being the number of nodes of the longitudinal wave function). Exploiting the concept of quantum excesses, the asymptotic energies are extrapolated to $\nu >20\mathrm{}$. It is found that the asymptotic energies provide the energy values of the real physical problem of hydrogenic atoms in magnetic fields $\beta \gtrsim 1$ within an accuracy of ≲1% for every $s\ge 3$ and arbitrary $\nu$, with the accuracy improving rapidly, as $\beta$, $s$, or $\nu$ is increased. Thus our results ideally complement those for $s<\mathrm{}3\mathrm{}$ for which accurate results have been tabulated in the literature.

Angular distributions of photoelectrons from resonant two-photon ionization via the $2p$ ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ or $2p$ ${}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ state of lithium have been measured. The quantum beats of hyperfine levels strongly affect the angular distribution. Three microscopic parameters are extracted: the ratio of the radial matrix elements from $2p$ excited states to $l=0\mathrm{} \mathrm{and} 2$ final states, the difference of the phase shifts of the continuum electrons in final channels, and a time-dependent parameter for hyperfine interaction. The experimental results agree with theoretical quantum-defect calculation and extrapolated bound-state data.

The perturbed-stationary-state method, appended with electron translation factors, has been applied to charge transfer in ${\mathrm{Li}}^{+}$ +Ca collisions for energies from 0.1 to 20 keV/amu. The Born-Oppenheimer wave functions and eigenvalues were generated using the pseudopotential technique, which reduced the many-electron system to a simpler two-electron problem. The molecular ground-state $X{}_{}{}^1{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ is calculated to be bound and has the potential-well parameters $R_e=6.20\mathrm{}{a}_0$, $D_e=1.11\mathrm{}$ eV, ${\omega }_e=235\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, and $B_e=0.263\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$. From the scattering computations, a representative value for the charge-transfer cross section is 4×${10}^{-15\mathrm{}}$ ${\mathrm{cm}}^2$ at 5 keV/amu. The cross section decreases rapidly as the energy is reduced below 1 keV/amu. At all energies studied, the dominant electron-capture reaction product is the ground-state Li atom.

Plane-wave Born-approximation calculations are presented for electron ionization of triply ionized noble-gas ions, and compared with recent experiment. Excellent agreement with experiment is found for ${\mathrm{Ne}}^{3+}$, ${\mathrm{Ar}}^{3+}$, and ${\mathrm{Kr}}^{3+}$. The calculated cross section for ionization of ${\mathrm{Xe}}^{3+}$ is larger than the measurement by almost a factor of 2. The discrepancy is entirely due to the large cross section for excitation of the ${\mathrm{}\left(4d\right)\mathrm{}}^{10}\to {\mathrm{}\left(4d\right)\mathrm{}}^9\mathrm{}\left(4f\right)\mathrm{}$ transition followed by autoionization. It is suggested that configuration interaction will redistribute this transition with some strength transferred to exciting bound levels.

Parameters for the Bethe stopping-power theory for Ar, Kr, and Xe are obtained both via integration over optical oscillator strengths and from explicit plane-wave Born-approximation calculations. The former procedure produces a Bethe mean excitation energy ($I$) significantly lower than that measured, while the latter procedure produces $I$ values within 10 eV of the measured values for Ar and Kr. For Xe the experimental $I$ value lies midway between the two calculated $I$ values. Inner-shell corrections are obtained from the explicit calculations. The $K$-shell corrections ($C_K$) are a factor of 2 larger than those obtained from hydrogenic calculations. The total inner-shell correction ($\Sigma {\mathrm{nl}}^{}{C}_{\mathrm{nl}}$) is in qualitative agreement for Ar, Kr, and Xe with results from a statistical model. However, the quantitative difference between the two sets of inner-shell corrections leads to significant increases in $I$ values inferred from experimental measurements below 1 MeV proton energy.

Results of exact numerical calculations of differential and total $1s-1s$ electron-capture cross sections evaluated in the second Born approximation are presented for targets and projectiles of various charges $Z_T$ and $Z_P$ at velocities between 10 and 200 MeV/amu. For symmetric systems with $Z_P=Z_T=Z$ the Thomas peak in the differential cross section, characteristic of a free-wave second Born-approximation process, appears at velocities above $Z^2$×(5 MeV/amu), where $Z$ is the nuclear charge of the target (or projectile). The shape of this Thomas peak contains information about real and virtual intermediate states of the system. For total cross sections at velocities below $Z^2$×(2 MeV) the second Born-approximation cross section is larger than the first Born-approximation cross section indicating a breakdown of the second Born approximation using the free-wave Green's function. Results using the peaking approximation of Drisko converge to our exact second Born-approximation results only at velocities well above $Z^2$×(10 MeV/amu). For systems asymmetric in $Z_P$ and $Z^T$ no exact scaling is found, although the systematics are qualitatively similar to the symmetric case using $Z=\frac{1}{2}(Z_P+Z_T)$. For $p$ + Ne at 100 MeV, the exact Born-approximation results lie somewhat above exact impulse-approximation calculations. It is found that the peaking approximation of Briggs and Simony converges to exact second Born-approximation results as the asymmetry of the projectile and target charges increases. At very high velocities the peaking approximation of Drisko also converges slowly to the exact second Born-approximation result.

A study of electron-impact excitation of highly charged ions in the helium isoelectronic sequence is reported. Nonresonant, intermediate-coupling collision strengths are computed for transitions up to and including the $n=3\mathrm{}$ states. With the 13 fine-structure levels, i.e., $1s^2\left({}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}\right)$, $1s2s({}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{};{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{})$, $1s2p({}_{}{}^3{}_{}{}^{}P_{0,1,2}^{}{}_{}{}^{};{}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{})$, $1s3s({}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{};{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{})$, and $1s3p({}_{}{}^3{}_{}{}^{}P_{0,1,2}^{}{}_{}{}^{};{}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{})$, there are 78 different inelastic transitions. Main calculations are carried out in the nine-state distorted-wave approximation including partial waves $l\le 15$. Relativistic effects are taken into account in the Breit-Pauli approximation. Nine-state close-coupling calculations are also carried out for ${\mathrm{Fe}}^{24+}$ with $l\le 4$ at a few energies for comparison (and for later work in paper II). Higher partial waves ($l>\mathrm{}15\mathrm{}$) are treated by the Coulomb-Bethe method using accurate relativistic eigenenergies and oscillator strengths for the target ions. The $Z$ dependence of the departure from $\mathrm{LS}$ coupling is investigated. Present results are in good general agreement with previous calculations where available, but some discrepancies are noted or clarified. Collision strengths are tabulated at energies from near the excitation energy of the $n=3\mathrm{}$ complex to sufficiently high energies to enable rate coefficients to be computed. For transitions to the $n=3\mathrm{}$ levels and for transitions where resonances are not important, the present values should yield reasonably accurate excitation rates (resonances are considered in paper II). The present data are required particularly for the analysis of observed spectra from high-temperature, high-density plasmas where the line intensities may depend on collisional redistribution among a number of excited $n=2\mathrm{}$ and $n=3\mathrm{}$ states.

The role of resonances in electron-impact excitation of highly charged ions is examined and some methods are described in order to take account of autoionization and dielectronic recombination. The intermediate-coupling calculations of paper I are extended to include these processes and effective collision strengths are obtained. Results are presented to illustrate particular features in a number of transitions in ${\mathrm{Fe}}^{24+}$ and ${\mathrm{Mo}}^{40+}$. Resonance analysis is carried out employing numerical interpolation techniques and multichannel quantum-defect theory, employing reactance matrices in nine-state close-coupling and distorted-wave approximations. It is shown that all three effects considered in the present study, departure from $\mathrm{LS}$ coupling, resonance enhancement due to autoionization, and reduction in autoionization enhancement due to dielectronic recombination, can be of considerable importance in the cross sections of highly ionized systems. In the high-$Z$ ions of the helium isoelectronic sequence the contribution of autoionizing resonances remains large even after all probable radiative channels are included, although the net autoionization effect decreases with $Z$.