For quantum-mechanical problems defined at arbitrary values of the spatial dimension D, dramatic simplifications occur for special values of D. These provide the basis for new expansions which are quite accurate in cases where more traditional expansions fail. This high accuracy originates in the fact that no dynamical parameters are modified, making the essential features of the solutions to the approximate problem the same as in the problem of interest. For example, the wave functions in the special dimensions have the same asymptotic properties and satisfy the same cusp conditions as the true wave function in the dimension of interest. We show how to identify those dimensions which may be useful and illustrate the expansion using the Yukawa potential and some anharmonic oscillators as examples. In the Yukawa problem, the leading correction to the energy in this expansion yields accuracy equal to second order in the usual perturbation theory and to sixth order in the 1/D series. For the oscillator problems, the accuracy of the zero-order wave function and first-order energy are nearly independent of the strength of the anharmonicity. The expansion is only developed for ground states but we conclude with a discussion of the special dimensions associated with excited states.

The semiclassical large-D expansion, with D the number of spatial dimensions, is very effective for some quantum-mechanical problems but not for those with singular potentials. We offer two ways to identify which systems will be poorly approximated by low orders of the 1/D series and show how to repair the difficulty in singular potential problems. The Yukawa potential and $r^2$+${\mathrm{gr}}^4$ anharmonic oscillator are presented as examples. We also discuss the large-order behavior of the 1/D series for the Yukawa problem and argue that this series is divergent.

The generators of the Lie algebra of totally symmetric spin-dependent operators are considered. An iterative method for calculating their matrix representatives in the spin-adapted basis is presented. The iterative method is applied for calculating the matrix elements of the spin-own-orbit operator.

A bivariational coupled-cluster method is advocated for the calculation of static electronic properties for closed-shell atomic and molecular systems. The proposed method uses a perturbed form of the Hamiltonian H(λ) which includes the field with which the system interacts, giving rise to the response in various orders. Higher-order properties can be calculated without the use of excited states. A novel linked energy functional ${\mathrm{scrJ}}_H^{\lambda }$[T,T’’] is used with two different types of cluster parameters T(λ) and T’’(λ), each of which is a power series in λ. Unlike the Euler energy functional proposed in one of our earlier works, this functional is a terminating series in cluster parameters. It is shown that the set of stationarity equations of ${\mathrm{scrJ}}_H^{\lambda }$ with respect to arbitrary variation of the cluster parameters of T(λ) and T’’(λ) corresponding to any power of λ is the same for certain conditions and furnishes all the relevant values of cluster parameters. The equations are derived for calculating static properties to any order with these stationary optimum values.

The time-dependent arrangement-channel quantum-mechanics (ACQM) equations are numerically solved for the collinear exchange reaction H+${\mathrm{H}}_2$. The method employs the fast-Fourier-transform scheme of Kosloff and Kosloff and an efficient time propagator involving a Chebyshev polynomial expansion due to Tal-Ezer and Kosloff. The time-dependent ACQM formalism, which is based on the use of the time-dependent arrangement-channel-component states as introduced by Kouri, Kruger, and Levin, involves a non-Hermitian matrix Hamiltonian in arrangement-channel space. We employ the channel-permuting array-coupling sequence of Baer, Kouri, Levin, and Tobocman. The remarkable property of the channel-component states that, below the dissociation threshold, only the component in channel j gives rise to outgoing waves in that arrangement channel, is explicitly seen in this time-dependent calculation. The calculated reaction probability is in reasonable agreement with the time-independent calculation result. The question of whether the numerically generated time-dependent solutions contain spurious solutions (which can arise in principle, due to the matrix nature of the Hamiltonian in arrangement-channel space) is discussed. It is found computationally that such spurious solutions do not create any difficulties in the calculation of reactive probabilities within the time-dependent ACQM formalism.

The remainder $R_{\left(t\right)}$ of the Born series 〈u‖v〉+〈u‖K‖v〉+...+〈u‖$K^{t\mathrm{-}1}$‖v〉 +$R_{\left(t\right)}$ is given an analytic continued-fractional form. The resulting formula is tested on a few schematic examples and recommended as a combined perturbative and algebraic method applicable to various scattering problems.

A parametrized phase-shift analysis of the e-He scattering data of Andrick and Bitsch for energies below the first excitation threshold is carried out. This approach does not assume that the higher-order phase shifts are given by the Born approximation and allows a rigorous analysis of the statistical errors. We obtain better ${\chi }^2$ values for our fits to these data than do Andrick and Bitsch. As a by-product of our phase-shift analysis we can obtain the interaction potential with associated confidence limits. The tail of these potentials is consistent with the assumption of adiabatic polarization at energies below the first excitation threshold.

Thermalization of hot tμ atoms formed in ${\mathrm{D}}_2$-DT-${\mathrm{T}}_2$ mixtures and relaxation of the triplet hyperfine state of tμ are treated by Monte Carlo simulation. The cross sections for collisions with molecular targets are derived from cross sections for collisions with the component atoms. The statistical histograms for the time-dependent velocity distributions are fitted by sums of two Maxwellian functions, and the coefficients and temperatures are tabulated for tritium fractions of 10%, 50%, and 90% and mixture temperatures of 30, 100, and 300 K. The results provide a basis for analyzing some previously puzzling aspects of the observed cycling rate in recent muon-catalyzed-fusion experiments. In particular, thermalization is found to be very incomplete for some low-temperature targets. Also, the relative rates of kinetic thermalization and hyperfine-state relaxation depend strongly on the target tritium fraction, and hyperfine effects may be observable. The velocity distribution of tμ atoms at times of molecular dtμ formation is shown to depend only rather weakly on the initial velocity distribution, which is not well known.

Thermalization of hot tμ atoms formed in ${\mathrm{D}}_2$-DT-${\mathrm{T}}_2$ mixtures and relaxation of the triplet hyperfine state of tμ are treated by Monte Carlo simulation. The cross sections for collisions with molecular targets are derived from cross sections for collisions with the component atoms. The statistical histograms for the time-dependent velocity distributions are fitted by sums of two Maxwellian functions, and the coefficients and temperatures are tabulated for tritium fractions of 10%, 50%, and 90% and mixture temperatures of 30, 100, and 300 K. The results provide a basis for analyzing some previously puzzling aspects of the observed cycling rate in recent muon-catalyzed-fusion experiments. In particular, thermalization is found to be very incomplete for some low-temperature targets. Also, the relative rates of kinetic thermalization and hyperfine-state relaxation depend strongly on the target tritium fraction, and hyperfine effects may be observable. The velocity distribution of tμ atoms at times of molecular dtμ formation is shown to depend only rather weakly on the initial velocity distribution, which is not well known.

Measurements of cross sections for positronium formation in collisions of positrons of energies up to 251.4 eV with helium atoms and hydrogen molecules are reported. In both gases the cross sections are more than twice as large as the only other experimental values published to date. At low energies the measured values agree quite well with available calculations.

Absolute cross sections for proton and helium-ion impact on neon, sodium, and magnesium are presented. By using coincidence techniques cross sections for pure single- or double-electron capture are separated from those events where one or more ionizations accompany the capture process. In addition, information about the direct-ionization channel is presented. The present data are combined with previous experimental information to identify specific channels which contribute to the cross sections presented here.

Approximate expressions for inelastic elements of the S matrix are given in terms of integrals over vibrational functions and R-matrix electron states. The magnitude and symmetry of these integrals determine the vibrations that are excited by various resonances, and the present analysis successfully predicts the spectra measured for benzene, ethylene, and formaldehyde. The recently observed s-wave component in the rotational excitation of ${\mathrm{N}}_2$ is also explained by the treatment. The theory shows how vibrational excitation can give information about the symmetries of resonant electronic states. This can be particularly useful when angular distribution measurements are unavailable.

The shape of the distribution of electrons ejected into the forward direction in the ${\mathrm{He}}^{+}$-He and ${\mathrm{He}}^{2+}$-He collisions in the 0.6–1.6-MeV impact-energy region are investigated. In both cases the anisotropies observed in the measured double-differential electron spectrum are analyzed by the help of a series expansion and a fitting procedure for a better characterization of the cusp. The theoretical peak shapes are also analyzed by this method. In the case of ${\mathrm{He}}^{2+}$ the agreement between the measured and the calculated cusp shapes is especially poor.

Experimental ratios for branching into the fine-structure levels of the Na 3p multiplet, as a consequence of an optical collision with He, Ne, Ar, Kr, or Xe, are reported. The process studied is Na(3s ${}_{}{}^2{}_{}{}^{}S_{1/2}^{}{}_{}{}^{}$)+scrR+nhν→ Na(3p ${}_{}{}^2{}_{}{}^{}P_j^{}{}_{}{}^{}$)+scrR+(n-1)hν , where scrR represents a rare-gas atom and where the laser frequency ν is tuned in the wings of the Na resonance transitions. The branching ratios are defined as I(D1)/I(D2) where I(D1) and I(D2) are measured intensities of the atomic Na D1 and D2 lines. The ratios are determined for detunings ranging from about 650 ${\mathrm{cm}}^{\mathrm{-}1}$ in the blue wing to 170 ${\mathrm{cm}}^{\mathrm{-}1}$ in the red wing of the Na 3p multiplet. The branching is found to be strongly detuning dependent in the vicinity of the NaAr, NaKr, and NaXe near-red-wing satellites. The blue-wing branching ratios show a detuning-dependent approach to a recoil, or sudden statistical, limit of 0.5, irrespective of the rare gas. Fine-structure changing cross sections have also been measured for resonant excitation of the Na 3p ${}_{}{}^2{}_{}{}^{}P_j^{}{}_{}{}^{}$ state; the results are consistent with cross sections obtained from wing excitation.

Recently measured cross sections for electron loss and charge transfer from laser-excited Na Rydberg targets are compared with experimental and theoretical cross sections for capture to all states in p+H(1s) collisions at intermediate velocities. For projectile velocities greater than 1.2 times the Bohr velocity of the target initial state the classical $n^4$ scaling rule correctly relates ground-state and Rydberg charge-transfer cross sections. However, for projectile velocities between 0.27 and 1.2 times the Bohr velocity a significant departure from $n^4$ scaling occurs, and after scaling, the capture cross section from Rydberg targets is up to 4 times greater than that from H(1s). The effect is the same for l=0 and for l-mixed Na targets and is evident also in independently obtained data on capture from hydrogen Rydberg atoms. A significant difference between the dynamic response of Rydberg and H(1s) states is discussed, and theoretical investigation is called for.

For the study of collision-induced dipole moments, the collisional ${\mathrm{H}}_2$-He complex is treated as a molecule in the self-consistent-field and size-consistent coupled electron pair approximations. The basis set accounts for 95% of the correlation energies and separates correctly at distant range. The vibrational average (v=v’=0) of the induced dipole components is obtained and recast in a simple but accurate analytical representation. It is used for computations of the spectral moments and spectral profile of the collision-induced rototranslational spectra of ${\mathrm{H}}_2$-He. The computations are in close agreement, on an absolute intensity scale, with Birnbaum’s measurements at 77, 195, and 292 K of such spectra. The fundamental theory is thus seen to predict these spectra reliably even when the anisotropy of the potential is neglected, probably with an accuracy of better than 5%, over the frequency band from 0 to ∼2000 ${\mathrm{cm}}^{\mathrm{-}1}$.

Nitrous oxide was bombarded with protons with an energy range of 100–325 keV. The photon emission from collisional excitation of the ${\mathrm{N}}_2$O in the wavelength range of 2000–7000 Å was examined. The only spectral features found were from transitions from the A ${}_{}{}^2{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ state to the X ${}_{}{}^2{}_{}{}^{}\Pi$ state of ${\mathrm{N}}_2$${\mathrm{O}}^{+}$. The emission cross sections for the four most prominent bands were measured. The resulting cross-section data do not make a distinction between the Bethe-Born theory and the impact-parameter model.

A correlation-polarization potential originally introduced by O’Connell and Lane [Phys. Rev. A 27, 1893 (1983)] is used in e-${\mathrm{H}}_2$ scattering calculations in which the vibrational motion of the target is taken into account. Eigenphase sums (as a function of internuclear separation) and cross sections for elastic scattering and for rovibrational excitations are compared to their counterparts calculated using the ab initio nonadiabatic model polarization potential of Gibson and Morrison [Phys. Rev. A 29, 2497 (1984)]. At low energies, these scattering quantities are found to be quite sensitive to the treatment of polarization. To assess these model potentials, theoretical total, momentum transfer, and rotational- and vibrational-excitation cross sections are compared to experimental data.

Cross sections for elastic electron scattering by the ${}_{}{}^1{}_{}{}^{}\mathrm{\Sigma }_{\mathrm{g}}^{+}{}_{}{}^{}$ ground state of ${\mathrm{H}}_2$ are computed at fixed internuclear distance 1.402$a_0$, for incident energies up to 13.6 eV. Variational R-matrix calculations are supplemented by asymptotic distorted-wave calculations for higher partial waves and by a residual Born-approximation sum for the polarization potential. Polarization response is treated dynamically, using ${}_{}{}^1{}_{}{}^{}\mathrm{\Pi }_{\mathrm{u}}^{}{}_{}{}^{}$ and ${}_{}{}^1{}_{}{}^{}\mathrm{\Sigma }_{\mathrm{u}}^{+}{}_{}{}^{}$ polarization pseudostates obtained by variational calculations. Pseudostate closed channels are included in the coupled equations integrated outside an R-matrix sphere.

A method has been developed for the evaluation of the Coulomb integrals containing the product of the Coulomb wave function with Slater-type orbitals which serve as the basis set in constructing the wave function of multielectron atoms within the self-consistent-field approximation, in terms of the Gauss hypergeometric function for an arbitrary nlm state. Analytical expressions are derived for the integrals in closed form irrespective of either the triad nlm or of the quantization axis. This method has been applied in the framework of continuum distorted-wave approximation to obtain the charge-transfer cross section in various collision cases involving heavy stripped ions and multielectron targets. The calculated results are found to be in good agreement with the recent experimental findings and the existing theoretical calculations in the high-energy region.