The concept of exchange hole plays an important role in Slater's approximation to the Hartree-Fock equations. In the present work use is made of analytical variational solutions of the Thomas-Fermi equation for neutral atoms and ions, and an expression is derived for the radius of the exchange hole that depends on the position of the electron considered.

Difficulties in the theoretical calculation of possible resonance states above the pickup but below the first hydrogen excitation threshold in the positron-hydrogen scattering system are discussed. A series of tests are proposed which should help avoid certain conceptual and practical difficulties one encounters in the analysis. Existing theoretical calculations claiming the resonance states are critically examined, and it is concluded that there is as yet no evidence for the presence of such resonance states in this particular energy region.

A procedure is given for comparing the accuracy of the length and velocity oscillator strengths for electric dipole transitions between bound states. Based upon qualitative use of upper- and lower-bound formulas, the procedure applies to electronic systems of any size, and emphasizes the importance of such factors as the transition energy, the magnitude of the transition moments, spatial diffuseness, momentum distributions, and quality of the wave functions. The method is illustrated with a set of 36 trial transitions for the two-electron atoms He and ${\mathrm{Li}}^{+}$, where a variety of simple functions of varying quality are available to describe states of different nuclear charge and multiplicity, and where accurate $f$ values are available for comparison. The procedure proves to have high predictive power for cases where one $f$ value is clearly to be preferred, and it accurately identifies other cases where the $f$ values are of comparable quality. In addition, the method gives some qualitative assessment of the degree of preference for each formula.

Measurements are presented for the blue-wing satellite bands of the Rb first resonance line (780 nm) emitted in the presence of the noble gases Ne, Ar, Kr, and Xe. The continuum intensity per spectral interval and perturber density is expressed as a fraction of the total Rb (780 nm) fluorescence from optically thin vapor. Over the ∼${10}^{18}$-${10}^{19}$ ${\mathrm{cm}}^3$ density range investigated this normalized satellite-peak intensity is found to scale linearly with perturber density and approximately exponentially in $\frac{-1\mathrm{}}{T},T$ being the absolute temperature. In the classically forbidden region beyond the satellite peak the intensity drops exponentially with frequency for Xe perturbers, but departs from exponential behavior in the Ar and Kr data. We observe undulations in the spectra between line center and the satellite. This is expected due to the possibility of interference between contributions to the transition probability at two internuclear separations, the separations at which the difference potential equals the photon energy.

The behavior of a two-level system interacting with a bichromatic classical field is studied. Analytic expressions are found for the Green's-function operator for the system if the field is weak and near resonance, and a systematic procedure is developed to obtain the Green's-function operator in a more general case. The results are used to calculate the probability of a transition from the lower to the upper state; resonance conditions occur when the level spacing of the system, ${\omega }_{ba}$, is near $n{\omega }_{\lambda }-(n-1\mathrm{})\mathrm{}{\omega }_{\mu }$, where $n$ is a positive or negative integer, and ${\omega }_{\lambda }$ and ${\omega }_{\mu }$ are the frequencies of the components of the field. The resonance frequency depends on the intensity of the field components. In addition, the cross section for scattering photons out of the field has been calculated. It is found that the spectrum of the scattered radiation consists of lines at frequency ${\omega }_k={\omega }_{\mu },{\omega }_{\lambda }$, and odd harmonics of these, combinations of the fundamental and the harmonics with $\Delta ={\omega }_{\lambda }-{\omega }_{\mu }$, and satellites shifted from all of these features by an amount which depends on the intensity of the field and ${\omega }_{ba}$. The cross section is intensity dependent particularly insofar as the location of resonance peaks is concerned. It is suggested that this effect may be exploited in transferring intensity modulation of one component of the field to phase modulation in the other component.

A new method is presented for cascade-free high-precision lifetime measurements of highly excited atomic states. Experimental examples are given for the lifetimes of the ${\mathrm{Ba}}^{+}$ $6p {}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ level and the Ne $3p {}_{}{}^{\prime }{}_{}{}^{}\left[\frac{3}{2}\right]_2^{}{}_{}{}^{}$ level. The experimental arrangement uses a fast ion beam which is excited in a gas cell and, after a definite time of flight, is further excited by a continuous-wave intracavity dye-laser beam. Further possible applications of the method are mentioned.

A short review of Hermann Weyl's theory for singular second-order differential equations is given and its numerical aspects are discussed. It is pointed out that this method is suitable for the treatment of perturbations which make the spectrum continuous. The Stark effect on the ground state of the hydrogen atom is taken as an example. The spectral density, the imaginary part of Weyl's "$m$ function," is calculated numerically using Runge-Kutta integration and Airy integrals for the asymptotic region. Showing $\delta$-function-like behavior with poles of $m$ on the real axis for the discrete levels, the spectral density involves approximate Lorentzians for the metastable states of the continuous spectrum, corresponding to poles of $m$ in the complex plane. Trajectories of these poles for electric fields up to 0.25 a.u. are shown for the one-dimensional as well as for the full three-dimensional problem.

Separate relativistic Hartree-Fock solutions for atoms in their initial and final states are used for calculating the radiative decay rates of $L$-shell vacancy states. The matrix-element calculations include the exchange effects of the nonzero overlap of wave functions from different subshells. Results are presented for a set of elements from $Z=18\mathrm{}$ to 94. The rates for producing multiple vacancy states in the decay process are included in the calculation. The corrections to single-potential Hartree-Slater values of the $L$ x-ray emission ratios are less than the present accuracies and spread in the experimental data.

The utility of variational principles for the estimation of the expectation values of certain operators, so far established only for helium, has been tested for atoms with $Z$ from 2 through 18 for the operators ${\overline{r}}^q\equiv \Sigma i^{}{r}_i^q$ for $q=-1,1,2\mathrm{}$ using simple atomic wave functions. The trial wave function ${\Psi }_t$ is taken to be a Hartree product of hydrogenic orbitals with different effective charges fixed through phenomenological approaches such as Slater's empirical screening rules. This form for ${\Psi }_t$ is particularly useful for these exploratory calculations since all steps in the variational method can actually be carried out analytically. Results for $〈{\overline{r}}^{-1\mathrm{}}〉$ and $〈\overline{r}〉$ are obtained through neon, while for the more interesting case of $〈{\overline{r}}^2〉$ they are extended up through argon. The results for $〈{\overline{r}}^2〉$ are generally the least accurate and are especially sensitive to the choice of effective charges, since they depend almost entirely on the outermost electrons whereas the ${\Psi }_t$ used represents an average description of the atom. The variational estimates of $〈{\overline{r}}^2〉$ represent a substantial improvement over the zeroth-order estimates and agree to within about 3 to 10% with the Hartree-Fock results. The simplicity and relative accuracy of these calculations generally support the utility of the variational principle and suggest future calculations with a better ${\Psi }_t$ as a starting point.

Level crossings between the substates $2 {}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$, $m_j=-\frac{3}{2}$ and $2 {}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$, $m_j=\frac{1}{2}$ of atomic hydrogen have been observed for electric fields up to 250 kV/m. A nonlinear low-field Stark effect has been measured for the $2 {}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ state, in agreement with theoretical calculations which include Lamb shift and hyperfine structure.

Using a published correlated wave function computed by Poshusta and Matsen, we have obtained a vibrationally averaged first-order spin-spin constant of -0.04089 ${\mathrm{cm}}^{-1\mathrm{}}$ for the $a {}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ metastable state of molecular helium. This is in good agreement with the experimental value of -0.0367 ${\mathrm{cm}}^{-1\mathrm{}}$ recently reported by Lichten et al. Our results strongly suggest that second-order spin-orbit effects are small.

We have investigated the method of effective potentials for replacing the core electrons in molecular calculations. The effective potential has been formulated in a way which simplifies computations while producing wave functions of ab initio quality. The effective potential is expressed in an analytic form which (i) represents the actual ab initio nonlocal potential (as defined by the matrix elements for a given basis set) and (ii) permits efficient computations of the effective-potential integrals (by incorporating the properties of Gaussian basis functions). To minimize the number of basis functions required in the molecular calculations, we define a new ab initio effective potential derived from modified Hartree-Fock valence orbitals whose core character has been removed. The effective-potential method as formulated becomes a very strong and reliable tool in attempting calculations on very large molecules. Applications to Li, Na, and K are presented.

The charge-transfer processes occurring in collisions of Li + ${\mathrm{Na}}^{+}$ and Na + ${\mathrm{Li}}^{+}$ have been studied theoretically using the molecular-wave-function approach. The wave functions and Born-Oppenheimer breakdown terms were evaluated using rigorous methods. The six lowest molecular states (dissociating to the $2s$ and $2p$ atomic states on Li and to the $3s$ and $3p$ atomic states of Na) were included in the coupled equations. The transition probabilities were calculated using linear trajectories for a variety of impact parameters and ion velocities. We find that the over-all transition processes are well represented as a succession of simple two-state transition processes ($\Sigma -\Sigma$, $\Sigma -\Pi$, and $\Pi -\Pi$). The $\Sigma -\Sigma$ two-state process can be described in terms of three steps involving (i) a coupling region as the atoms come together [$\mathrm{}(10-20)\mathrm{}{a}_0$], (ii) an uncoupled phase changing region for shorter separatons ($<10\mathrm{}{a}_0$), and (iii) a decoupling region as the atoms depart [$\mathrm{}(10-20)\mathrm{}{a}_0$]. On the other hand, in the molecular—wave-function formulation, the $\Sigma -\Pi$ two-state transition process involves continuous coupling (for $R<\mathrm{}7\mathrm{}{a}_0$). As a result the transition probabilities for $\Sigma -\Pi$ coupling differs from that of $\Sigma -\Sigma$ coupling, leading to rather different forms for the cross sections.

We present a convenient numerical method for the calculation of scattering phase shifts and wave functions. In this procedure the wave function is chosen to be an expansion in a convenient basis set for radial distances within the range of the short-range interaction. Outside this range, the wave function is a linear combination of the correct asymptotic solutions. Matching these two solutions determines the phase shift and overall normalization of the wave function. This method is an application of the theory first proposed by Bloch. The method is similar in some respects to $R$-matrix theory. The computational procedure can also be shown to be equivalent to the Kohn variational principle. In this method the choice of basis states is not as restricted as in previous applications and, in particular, the artificial boundary condition of the usual $R$-matrix approach is discarded. The freedom thus obtained in the choice of basis states leads to a rapidly convergent expansion for the wave function and the phase shifts. Furthermore, this freedom enables the method to be easily applied to a wide variety of problems including the case where there is a long-range Coulomb interaction in addition to short-range interactions. Several numerical examples are presented to illustrate the simplicity, flexibility, accuracy, and rate of convergence of the method.

This paper describes experimental studies of the multiple scattering of a number of different ions with $1<\mathrm{}{Z}_1<18\mathrm{}$ and 500 keV incident energy, transmitted through thin films of Se, Pd, Ag, Te, ${\mathrm{Ta}}_2$${\mathrm{O}}_5$, and Bi, and foils consisting of a mixture of C and Au atoms. We found that the targets fell into two groups with respect to structure. The first group, in which the targets were amorphous or had very small crystallites, showed multiple-scattering distributions of heavy ions, in agreement with current theory. In the second group, where the targets had grain sizes larger than approximately 100 Å, the distributions were consistently narrower than those predicted theoretically. Possible mechanisms responsible for the deviations were investigated.

The nonlinear absorption of intense laser light is used to obtain information about the times required for spectral cross relaxation ($T_3$) and collisional dephasing ($T_2$) in organic-dye solutions. The steady-state transmission experiment of Huff and DeShazer indicates that $T_2$ is in the range 0.1-1 psec for cryptocyanine-methanol, while $T_3$ is in the range 1-10 psec. For DDI-glycerin (1, 1′-diethyl-2, 2′-dicarbocyanine-iodide-glycerin), measurements of ultrashort-pulse transmission as a function of incident energy and optical polarization suggest a value $T_2=0.4\mathrm{}\pm 0.2$ psec for that system. These results are based on a new set of rate equations for absorption by inhomogeneously broadened organic-dye solutions. The equations display the effects of spectral cross relaxation and are not limited to light intensities less than those at which coherent interaction effects become important. For the case of ultrashort incident pulses, the equations are extended to account for excited-state absorption, the dependence of absorption on optical polarization, and the existence of a multiplicity of interacting excited states.

The zero-temperature dispersion curve in liquid ${\mathrm{He}}^4$ is calculated assuming anomalous dispersion, $\omega \mathrm{}\left(k\right)=c k(1+\gamma {\hslash }^2{k}^2)$, in zeroth order and including three-phonon interactions within the continuum model. It is shown that the damped-phonon mode changes to a collective-type excitation at higher momenta. Estimates of $\gamma$ at saturated vapor pressure and 10 atm are obtained from experimental data on the onset of the long-lived excitation in liquid ${\mathrm{He}}^4$.

The spin orientation of photoelectrons and its angular distribution are studied by applying the Jacob-Wick helicity formalism. Applications are presented for photoelectrons ejected from rare-gas atoms, Ar and Xe, in their autoionization spectra. The spin polarizations and their angular distributions show pronounced variations with photon energy across autoionization resonances.