Multifractal aspects of the perimeter-size distribution function $g_{\mathrm{st}}$ specifying the number of $s$ size clusters with perimeter $t$ have been examined and multifractal exponents determined numerically by the exact series method. In the percolation and compact-clusters region, multifractal exponents are also expressed analytically. In the lattice-animal region we show that the multifractal exponent describing the scaling behavior of the kth moment of the distribution function is directly connected to the growth parameter of the lattice.

A dilute nonsymmetric ferromagnetic neural network with multispin interactions and a version of the Hopfield model with multispin interactions are solved in the limit where the average number of inputs per spin is finite. It is found that in such asymmetric systems the critical exponents of local quantities can take a few different values, instead of just one.

Our previous work on setting up a systematic approach to approximations of many-body eigenvalues and radiative transition matrix elements for atomic systems is extended from the one-particle (one-hole) excitation states to two-particle (2P)–two-hole (2H) states. The formulation is based on the field-theoretic definition of one- and two-body propagators and corresponding vertex parts. The guiding principle in the approximation scheme is the gauge invariance (GI) of radiative transition matrix elements. At the level of the one Coulomb exchange approximation in the one- and two-body propagator kernels, we obtain a new approximation which we call the two-particle–two-hole approximation (2P-2H-A). We show that, in this approximation, the five-point vertex function which is generated by photon insertion in all possible ways in the two-body propagator (in order to obtain a GI radiative transition amplitude) reduces to the product of the three-point vertex and the inverse of the one-body propagator, discussed in previous work. The present analysis applies to states and energies, and transitions between states, of all atoms which consist of a nondegenerate core plus or minus two electrons. Some very preliminary numerical results on 2H energy levels, which include correlation corrections to the 2P-2H-A, are briefly touched upon.

The Schrödinger equation is solved directly for the ground state and excited 2 ${}_{}{}^1{}_{}{}^{}S$ state of the helium atom by using a rapidly convergent hyperspherical method which involves no adjustable parameters. The double and triple coalescence points are taken into account analytically. The center-of-mass motion is treated nonperturbatively, and the cases of infinite and finite nuclear masses are considered. The inclusion of 169 hyperspherical functions yields the precision of a few parts in ${10}^9$ and ${10}^8$ for the expectation value of the Hamiltonian operator and for all other expectation values, respectively, for the ground state, with only slightly less accuracy for the excited state.

A spectral method previously developed for solving the time-dependent Schrödinger equation in Cartesian coordinates is generalized to spherical polar coordinates. The solution is implemented by repeated application of a unitary evolution operator in symmetrically split form. The wave function is expanded as a Fourier series in the radial coordinate and in terms of Legendre functions in the polar angle. The use of appropriate quadrature sets makes the expansion exact for band-limited functions. The method is appropriate for solving explicitly time-dependent problems, or for determining stationary states by a spectral method. The accuracy of the method is established by computing the Stark shift and lifetime of the 1s state in hydrogen, the low-lying energy levels for hydrogen in a uniform magnetic field, and the 2p-nd dipole transition spectrum for hydrogen.

The open-shell coupled-cluster (CC) method is applied to the excited 2$p^2$ states of Be. With the 2$s^2$ and 2$p^2$ configurations included in the model (P) space, the CC equations prove very difficult to converge. When they do converge, very large (>5) excitation amplitudes are observed, and the second ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ corresponds to the intruder 2s3s configuration, rather than the desired 2$p^2$. The inclusion of 2s3s (but not 3$s^2$) in the model space, which thereby becomes incomplete, improves convergence significantly, and gives energies in very good agreement with values known from other sources.

Extensive variational calculations on the ${}_{}{}^2{}_{}{}^{}\mathrm{S}$ ground state of ${\mathrm{Be}}^{+}$ are reported. Using a 401-term expansion of Hylleraas-type functions, the nonrelativistic ground-state energy is determined to be -14.324 760 a.u., which is the lowest upper bound to the ground-state energy reported to date for this system. The Fermi contact interaction is determined to be 12.4980 a.u., which is in very close agreement with the experimental value of 12.503 88 a.u. The nuclear-magnetic shielding factor and the molar diamagnetic susceptibility are evaluated to be σ=1.415401 ×${10}^{\mathrm{-}4}$ a.u. and χ=-5.154 43×${10}^{\mathrm{-}6}$ ${\mathrm{cm}}^3$ ${\mathrm{mol}}^{\mathrm{-}1}$, respectively. The electron density at the nucleus, the moments 〈$r_i^n$〉, n=-1, 1, 2, and 3, and 〈$r_{\mathrm{ij}}^n$〉, n=-1, 1, and 2, are also reported. The specific mass shift (mass polarization contribution) is determined to be 2.758×${10}^{\mathrm{-}5}$ a.u. The rate of convergence of the calculation is discussed. The role of one versus two spin eigenfunctions in the basis set is discussed with respect to its bearing on the calculation of the Fermi contact interaction.

The relativistically corrected Schrödinger (RS) equation [($m^2$$c^4$+$p^2$$c^2$${)}^{1/2}$ +V-E]Ψ=0 is sometimes used to describe charged fermions, which are subject to strong Zitterbewegung. Since, however, equations of this type describe bosons with weak Zitterbewegung effects, it is mandatory to screen the Coulomb potential V in an r range of order h/mc, and also to add a screened spin-orbit potential. Thereby the singularity problems with Coulomb potentials are avoided. While the Pauli Hamiltonian must not be used in variational or eigenvalue equations, the RS Hamiltonian may be. An effective potential for the RS equation is derived, which yields relativistically corrected energies and wave functions for fermion systems with Coulomb potentials. Accurate numerical solutions for an unshielded single-center Coulomb potential are presented.

The classical adiabatic motion of charged particles in strong magnetic fields, otherwise known as guiding center motion, exhibits an anholonomic phase similar to Berry’s phase and the phase discovered in a general context by Hannay. Analysis of this effect reveals that there often is no best way to define the gyrophase when magnetic field lines are curved. Instead, a change in the definition of the phase is a kind of gauge transformation, the study of which leads to new insight into guiding center theory. The path-dependent phase that occurs in this problem is coupled with the metrical structure of physical space, giving rise to a transport process for vectors and frames which is similar to parallel transport in non-Euclidean geometry. Strong analogies with Fermi-Walker transport and Thomas precession in special relativity are pointed out.

This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel’fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.

A new set of atomic scattering factors derived from relativistic Hartree-Fock wave functions has been calculated for publication in the new edition of the International Tables for Crystallography, rd ed. (Reidel, Dordrecht, in press). Presented here is a discussion of the numerical stability of the results and a discussion of the spin-polarization function, S(θ,φ), including a comparison of the new calculations with experimental data. After the calculated S(θ,φ) functions are normalized to the Coulombic values, the difference between theory and experiment leads to some interesting conclusions.

Perturbation theory is a standard method used to calculate electronic correlation effects in atomic and molecular systems. It is able to provide a size-consistent description of electronic energies. Perturbation theory for excited states is often only applicable in a restricted way due to convergence problems (intruder-state problems). The starting point for a perturbation treatment is the decomposition of the total Hamiltonian H into $H_0$+V. The present study is devoted to proper definitions of ${\mathrm{H}}_0$ for a convergent treatment of ground and excited states. We use the scaled Hartree-Fock operator as an unperturbed Hamiltonian. It contains scaling factors to modify the Coulomb and exchange terms. These factors can be chosen in order to improve the convergence properties for the treatment of ground and excited electronic states. As a simple illustrative example, second-order perturbation calculations are reported in complete model spaces for ${}_{}{}^1{}_{}{}^{}\mathrm{\Sigma }_{\mathrm{g}}^{+}{}_{}{}^{}$ states of the hydrogen molecule. The total energies obtained show an agreement of better than 0.23 eV with the exact results.

We investigate the performance of an adiabatic separation of electronic and nuclear motion in the presence of a magnetic field and find that the diagonal term of the nonadiabatic coupling elements must be added to the nuclear equation of motion in the Born-Oppenheimer approximation. As an alternative to the Born-Oppenheimer approximation we introduce the so-called screened Born-Oppenheimer approximation which is particularly suited to describe the adiabatic separation of electronic and nuclear degrees of freedom in a magnetic field. A new interpretation of the well-known gauge-centering phases is given. Furthermore, we provide a qualitative discussion of some of the effects of the off-diagonal nonadiabatic coupling elements.

Formulas for the phase and the normalization of the continuum wave function pertaining to the scattering problem with at most one turning point are obtained in a general form based on a kind of arbitrary-order phase-integral approximation. The accuracy of the formulas is illustrated by application to the Yukawa potential. The phase-integral expressions are analytically simple and have a wider range of applicability than the perturbation-theory formulas for the normalization obtained by J. McEnnan, L. Kissel, and R. H. Pratt [Phys. Rev. A 13, 532 (1976)] and for the phase shift obtained by A. Bechler and R. H. Pratt [Phys. Rev. A 28, 1190 (1983)]. The perturbation-theory phase-shift formula, e.g., for the Yukawa potential, is, however, seen to have some advantages due to its simple structure.

The double-scattering mechanism of Thomas represents the dominant means in an ion-atom collision at high velocities for the electron-projectile subsystem to rid itself of its large internal kinetic energy and thus allow the electron to be captured. The plane-wave second Born approximation to the single-electron capture amplitude calculated in a linearized-propagator approximation gives the lowest-order quantum estimate of the Thomas mechanism by employing initial and final bound-state momentum values around the center of these distributions along with a neglect of the electronic part of the momentum transfer in the two collisions of the mechanism. It is shown that additional contributions to the amplitude deriving from values of the bound-state momentum in the wings of the distributions are non-negligible and in fact explain the large differences found between exact and linearized-propagator approximation results. Consequently, the second Born approximation is seen to suffer from the same problem as does the first Born approximation, namely, too dominant a contribution of the large bound-state momentum components.

Relativistic-configuration-interaction (RCI) and complete-active-space (CAS) multiconfiguration self-consistent-field (MCSCF)/CI calculations are performed for the low-lying anion states of ${\mathrm{HBr}}^{\mathrm{-}}$ and ${\mathrm{HI}}^{\mathrm{-}}$. At both the RCI and CASSCF/CI levels of theory, the ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$ anion states are found to be repulsive in the Franck-Condon regions of the HX ${}_{}{}^1{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ states and bound relative to the H+$X^{\mathrm{-}}$ asymptotes at r>$r_e$ of the parent neutral states. The HI molecule is predicted to have a slightly positive adiabatic electron affinity, and electron attachment in this system is considered in a Langevin model. Electronic transition moments between the ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{1/2}^{+}{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}\mathrm{\Pi }_{1/2}^{}{}_{}{}^{}$ states of ${\mathrm{HBr}}^{\mathrm{-}}$ and ${\mathrm{HI}}^{\mathrm{-}}$ are calculated in order to examine the origins of nonadiabatic ${\mathrm{H}}^{\mathrm{-}}$ fragment production observed in dissociative-attachment experiments on HBr.

Total cross sections and collision strengths for the 5s-5p, 4d-5p, and 5s-4d transitions in ${\mathrm{Sr}}^{+}$, calculated in the Coulomb distorted-wave, Coulomb-Born, and Coulomb-Bethe approximations, are presented. The calculations have been carried out using a unitarized three-state approximation (5s-4d-5p) and in the LS coupling scheme, for the incident energy range 3–300 eV. The model for alkali-metal ions, of a single outer electron, has been considered, and exchange between the optical and incident electron neglected. The collision strengths have been processed using a new method for interpolating and assessing collision strengths, which allows data to be extrapolated correctly up to high energies. This new approach confirms the high-energy behavior shown in the present data.

We report on an experimental study of Lyman-α emission from H(2p) capture processes in collisions between ${\mathrm{H}}^{+}$ ions and Na(3s) as well as laser-excited aligned Na(3p) atoms. The impact energy lies between 0.5 and 10 keV. We observe a strong enhancement of capture from Na(3p) over capture from Na(3s), in agreement with recent calculations by R. Allan et al. and W. Fritsch, but at variance with earlier experimental and theoretical results (V. S. Kushawaha et al. and M. Kimura et al.).

The pseudopotential calculations of Plenkiewicz et al. on electrons colliding with argon are extended to consider the elastic electron-krypton scattering system. Phase shifts, differential, total, and momentum-transfer cross sections are obtained for incident electron energies in the range 0–20 eV. Our results are found to be in very good agreement with available experimental data and other theoretical results.

The differential cross section for ionization of atoms and molecules by proton impact is modeled over a wide range of incident-proton and ejected-electron energies by an analytical equation. This equation is based on the molecular promotion model at low energies and on the classical binary-encounter approximation, modified to agree with the Bethe theory of ionization, at high energies. Three adjustable parameters are sufficient to fit the experimental electron energy distribution at any given impact energy. Equations are given which fit two of these parameters as a function of proton energy over the range of 5 keV to 5 MeV. The third parameter is found to be a constant, independent of proton energy, but its value varies somewhat from target to target. The relation of one of these parameters to the optical oscillator strength is discussed as is the extension of the model to multishell targets. As examples, the model is applied to fit data on molecular hydrogen, helium, and argon. Quantities such as the stopping cross section, the average ejection energy, and the fraction of electrons ejected with energies greater than a given value are easily calculated using this model.