The spontaneous ionization of molecular Rydberg states by vibrational-translational or rotational-translational conversion results in electrons ejected with close to zero energy. This condition means that, without serious initial-final trajectory discrepancies, the Rydberg electron can be treated to a good approximation as a classical object, moving in a closed Kepler orbit, calculated in the Coulomb potential located at the center of mass of the molecular ionic core. The interactions which produce ionization are the monopole contributions of the nuclei in the region between the inner turning point and the radius of the ionic core, $R_I\le R\le \frac{m_2}{(m_1+m_2)X}$, and the multipole moment and polarization potentials in the region between the core radius and the outer turning point, $\frac{m_2}{(m_1+m_2)X}\le R\le R_0$, plus additional interactions from the short-ranged part of the Hartree potential in both regions. The rate of ejection is given by the probability of ejection per period times the inverse period, $P(i\to j){\tau }^{-1\mathrm{}}$, for $i$, $j$ rotational-vibrational states. ${\tau }^{-1\mathrm{}}$ is proportioned to $n^{-3\mathrm{}}$, where $n$ is the principal quantum number of the Rydberg electron; thus the product $P{\tau }^{-1\mathrm{}}$ obeys the "propensity" rule for multi-quantum-mechanical vibrational transitions and the $n^{-3\mathrm{}}$ rule deduced from the normalization of Rydberg states in fully quantum-mechanical treatments. Rates for ${\mathrm{H}}_2$ are calculated in the Born and two-state approximations and compared with experimental and other theoretical results.

Line spectra at the $3d$ photoabsorption edges of La, Er, and Tm are interpreted as transitions of the type $3d^{10}4f^N\to 3d^{9}4f^{N+1\mathrm{}}$. Calculated relative $\mathrm{gf}$ values for these transitions are compared with published absorption curves.

The total energy of an accurate beryllium ground-state wave function is decomposed into energies of its natural-spin geminals. These are further subdivided into kinetic, electron-nuclear attraction, and electron-electron interaction energy contributions. Similar subdivisions for a Hartree-Fock wave function are presented. It is pointed out that it is unreasonable to expect natural-spin geminals to be populated on the basis of their energies. The important energy implications of singlet-triplet degeneracies in a two-matrix are also discussed. The correlation energy is analyzed at the geminal level, and it is shown that the electron-nuclear attraction part of the correlation energy is negative and not negligible. For the beryllium ground state the Coulomb correlation energy is about four times greater than the Fermi correlation energy. It is pointed out that the Hartree-Fock wave function for beryllium contains too much Fermi correlation.

The atomic-beam magnetic-resonance technique has been used to extend the hyperfine-structure and $g$-value results of Smith and Spalding to all five members of the $4f^{4}6s^2{}_{}{}^5{}_{}{}^{}I$ ground multiplet of ${}_{}{}^{143,145}{}_{}{}^{}\mathrm{Nd}$. The eigenvectors derived by Conway and Wybourne by fitting the excitation energies are reasonably consistent with the new data. The extent to which the hfs is affected by small components in the eigenvectors is investigated. Empirical values for the radial integrals that appear in the hyperfine structure are compared with predictions based on nonrelativistic Hartree-Fock and on relativistic Hartree-Fock-Slater calculations. The empirical value found for the dipole integral least susceptible to distortion by configuration interaction is within (1-3)% of the value calculated relativistically. Differences between theory and experiment are substantial for some of the other integrals, however. The values of the electric-quadrupole moments of ${}_{}{}^{143,145}{}_{}{}^{}\mathrm{Nd}$ are found to be -0.56(6) b and -0.29(3) b, respectively.

The usefulness of Hylleraas-type wave functions for variational calculation of three-electron excited $S$ states is investigated. The four lowest $\left(1\mathrm{}{s}^{2}ns\right){}_{}{}^2{}_{}{}^{}S$ wave functions for lithium are obtained in the same 57-term basis. The calculated energies are - 7.477 82, - 7.353 92, - 7.318 37, and - 7.303 39 a.u. compared to the corrected experimental results - 7.478 07, - 7.35410, - 7.318 53, and - 7.303 55 a.u. The largest part (0.000 15 a.u.) of the discrepancies is due to approximations in the core function. For the lowest ${}_{}{}^4{}_{}{}^{}S$ state for lithium a 44-term basis was used with the resulting energy - 5.212 40. The wave function was analyzed in terms of natural orbitals and found to exhibit strong $2s3s-2p3p$ "near degeneracy." The density for the Li state is significantly correlation contracted in the $3s$ region.

The Firsov formalism for calculating the electronic stopping cross section has been modified by (i) giving a precise quantum-mechanical definition of the electron flux across the Firsov plane in terms of the bound-state wave functions and (ii) by including the relative motion of the Firsov plane in the flux calculation. This formalism is then extended in a semiphenomenological manner to the velocity region where the relative velocity of the colliding atoms is of the same order of magnitude as the orbital velocities of their electrons. When hydrogenic $1s$ wave functions are used, three adjustable parameters result from this treatment, one from the modification of the Firsov formalism and two from the extension to higher velocities. It is found that adjustment of these parameters to give a best fit to experimental data yields an expression which accurately gives the electronic stopping cross section $S_e$ for any collision partners, and at all nonrelativistic velocities. The three parameters are considered as adjustable for the purposes of this paper, but it is shown here that the low-energy parameter $Z$ is calculable from first principles. In addition, one of the high-energy parameters $a$ is shown to be a linear function of the target atomic "radius."

The energy distribution of electrons has been measured for ionizing collisions of Ar with Ar in the center-of-mass energy range of 30-250 eV. The distributions display several prominent peaks which can be attributed to the autoionization of highly excited atomic and molecular states.

A projection-operator technique is used to investigate the resonance states of two-electron systems in the inelastic scattering energy region. Computations are carried out for the $P$ autoionization states for He and the ${\mathrm{H}}^{-}$ ion in the $n=2\mathrm{}$ to $n=3\mathrm{}$ energy region. The results are compared with previous calculations and experiments. For the ${}_{}{}^1{}_{}{}^{}P$ resonance of He, very good agreement is obtained as compared with the accurate measurement of Madden and Codling.

In variational solutions of differential or integral equations the variance integral and variance sum provide measures of the inadequacy of the trial function. For certain potential-scattering problems the variance integral can be used to derive rigorous upper and lower bounds to the phase shift. In more general scattering problems, the study of the variance sum or integral gives a check on the convergence of the standard variational calculations, and an alternative criterion for choosing the best trial function. Results are shown for an attractive exponential potential.

We present delayed-coincidence measurements of the average kinetic energy losses (${\overline{Q}}_{\mathrm{mn}}$) for ${\mathrm{O}}^{+}$-Ne collisions that produce ${\mathrm{O}}^{+m}$ and ${\mathrm{Ne}}^{+n}$ ions. The experiments cover the incident energy range from 50 to 200 keV, values of $m$ and $n$ from 1 to 4, and distances of closest approach (${\mathcal{r}}_0$) from 0.15 to 0.016 Å. The ${\overline{Q}}_{4n}$ and ${\overline{Q}}_{3n}$ functions are double valued for ${\mathcal{r}}_0\le 0.094 \mathrm{and} 0.062$ Å, respectively. We propose that this phenomenon is the result of $K$-shell excitation in the oxygen ion.

Numerical coupled-channel two-state calculations have been employed to yield a reducedtotal-cross-section curve as a function of reduced velocity for the case where the potential curves lie parallel to one another and there is an exponential coupling between them. This plot may be used to estimate the energy dependence of the charge-transfer inelastic total cross sections for transitions that occur at large internuclear distances due to coupling between two close-lying states. We have specifically applied the above plot to several alkali-ion-alkali-atom systems and find reasonable agreement between theory and experiment. The velocity at which the total cross sections reach their maximum is obtained for the general case and is found to be in fair agreement with experiment. A comparison between the results of Demkov and these calculations is also made.

Clusters of condensed matter which are produced under some particular conditions in supersonic molecular jets can be ionized by electrons. Measured ionization cross sections show a sharp dependence on $N$, the mean number of molecules per cluster. For small clusters ($\overline{N}<50\mathrm{}$) the cross sections increase as $N$; for larger clusters as about ${\overline{N}}^{\frac{2}{3}}$. Furthermore the electron initial energy $W_{\mathrm{eM}}$, for which the cross section is maximum, increases with $N$. In this paper we present a model for the computation of the cluster ionization cross sections which includes the energy losses inside the cluster of both primary and secondary electrons. The escape probability for secondary electrons is given as a function of their initial position and energy. This latter is related to the primary-electron energy. Results of these computations for ${\mathrm{H}}_2$, C${\mathrm{O}}_2$, and ${\mathrm{N}}_2$ clusters are in good agreement with experimental data.

A many-body optical potential $\Sigma$ which has been demonstrated to give excellent electron-helium elastic scattering cross sections is analyzed. It is shown to encompass, yet be more general than, almost all previous model potentials used in electron scattering and Rydberg-state calculations. A partial semiempirical form of this potential is achieved. It is shown how $\Sigma$ can be computed by methods of variation-perturbation theory.

Several methods have recently been proposed for avoiding computational anomalies in the multichannel variational formalism of Kohn. The proposed methods themselves have characteristic difficulties, discussed in detail here, leading to ambiguous or discontinuous results. Two new methods are proposed. One, the optimized minimum-norm (OMN) method, is a development of the minimum-norm method of Harris and Michels. The other, the optimized anomaly-free (OAF) method, applies the idea of a preliminary unitary transformation of the open scattering channels to the anomaly-free method of Nesbet. Model calculations are reported that show both OMN and OAF methods to be superior to earlier methods. The OAF method has the additional advantage of avoiding anomalies that occur in the OMN method.

Ionization extrapolated ranges have been measured for N ions (0.3-2.0 MeV) in ${\mathrm{N}}_2$ and air, for Na ions (0.3-2.0 MeV) in air, and for Ar ions (0.3-1.0 MeV) in Ar, ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, and air. A magnetically analyzed beam of ions was brought through a differential-pumping system into a gas stopping cell. A single-grid ionization chamber, movable along the beam axis, was used to obtain the data necessary to plot a Bragg curve. The observed range-energy relations are compared with previous data where possible. The accuracy of the measurements is ± 3%.

The transition probability for the multiphoton ionization of the ground-state hydrogen atom by an intense electromagnetic field has been calculated following a nonperturbative method developed by Reiss. One distinctive feature of the calculation is a minor reduction in the power-law dependence of the interaction parameter in the transition probability.

In a recent paper we briefly reported on the possibility of producing spin-polarized hydrogen atoms through the charge-exchange reaction ${\mathrm{H}}^{+}+\mathrm{Xe}\to \mathrm{H}\left({}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)+{\mathrm{Xe}}^{+}\left({}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\right)$. At proton energies of 15 eV, spin polarization is implied by the preferential population of ${\mathrm{Xe}}^{+}\left({}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\right)$ over ${\mathrm{Xe}}^{+}\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\right)$ and the preliminary two-state calculations presented earlier indicated that a high degree of polarization could be obtained. In this paper we shall give a more complete discussion of the earlier work. In particular we shall describe the model upon which the calculations have been based, and the validity of this model will be examined. Three-state calculations, which allow for the population of ${\mathrm{Xe}}^{+}\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}\right)$, have been performed and these results will be compared with the earlier two-state results. We have also calculated the polarization as a function of scattering angle. It is found that the polarization varies rapidly with angle, owing to diffraction effects, and at certain angles almost 100% polarization is achieved.

A general form for the projector onto open channels for rearrangement processes in molecular collisions is given. This form is investigated for two special cases of the three-atom process $A+BC\to AB+C$ on a single potential-energy surface: (a) the rearrangement to and from the ground vibrational and rotational states of $\mathrm{BC}$ and $\mathrm{AB}$ for arbitrary total angular momentum and (b) the semiclassical limit of this form. The projector for special case (a) can be found exactly under either of two limiting conditions, and for case (b) we find the projector to be a local operator, such that in the semiclassical limit the process $A+BC\to AB+C$ occurs only when the distance from $A$ to the center of mass of $\mathrm{BC}$ equals the distance of $C$ from the center of mass of $\mathrm{AB}$.

Backscattering cross sections for the electron-impact $K$-shell ionization of C, N, O, Na, and Mg adsorbed on a tungsten (100) surface are reported. The ionization event studied is the scattering of $K$-shell electrons to unfilled states within a few eV of the Fermi level of the W substrate. These data are obtained through a new technique which analyzes the currents in ionization-loss features by comparing them with the currents in the elastic peaks. This technique involves numerical simulation of the ionization features using distribution functions. Measured cross sections in the backscattered region are compared with the Burhop theory over an energy range extending from one to four times the ionization energy. The measured and calculated cross sections decrease with increased atomic number and with primary energy. Although some improvement in shape agreement is noted with increased atomic number, the experimentally derived backscattering cross sections exceed the calculated values for the entire energy range investigated. It is shown that the disagreement noted, even for the higherenergy range studied, cannot be attributed to double-scattering processes. The simple theory based on the first Born approximation, although reasonable in total-cross-section determination, is not accurate enough to account for behavior in the backscattered region.

A form of semiclassical theory for molecular inelastic or reactive scattering is developed. In such a theory we achieve a separation of the dynamics into two parts, elastic motion and inelastic or reactive motion, by the following procedure. The isotropic part of the intermolecular potential, $V_0$, is expanded in a Taylor series about the equilibrium internuclear geometries of the interacting molecules. This separates $V_0$ into a part which is not dependent on the internal coordinates of the molecules, for which only elastic scattering can occur, and a part which is dependent on the internal coordinates. This latter part, along with the unexpanded anisotropic parts of the intermolecular potential, can act as a "coupler" to produce inelastic or reactive scattering. The constant term in the expansion for $V_0$, is used as the potential in which the classical trajectory governing the elastic motion is calculated. This determines the "distance of closest approach" and, relative to the range of the couplers, defines an "energy of activation" or effective threshold, if relevant. The time-dependent Schrödinger equation is solved to produce inelastic or reactive probabilities. The theory is applied to ${\mathrm{H}}^{+}$ -${\mathrm{H}}_2$ vibrational excitation in the two- and three-state approximations for energies $15<E<\mathrm{}1000\mathrm{}$ eV. Discrepancies with the measurements of Herrero and Doering probably can be attributed to the omission of the electron-capture channel. The agreement with the 0 → 2 measured cross section above 200 eV is excellent.