A novel probability approach is utilized to solve the coupled nonlinear partial differential equations describing multiphase and multicomponent fluid flow in a porous medium. The approach generalizes diffusion-limited aggregation (DLA) to finite two-fluid mobility ratios and to more complete descriptions of physical systems. The limits of validity of the usual DLA method are determined. Two-dimensional saturation and concentration profiles are determined as a function of the mobility ratio. Solutions are obtained for the full range of flow regimes (stable to highly unstable).

We show that the Reply of Weglein [Phys. Rev. A 31, 4025 (1985)] to our previous Comment [Phys. Rev. A 31, 4023 (1985)] is not valid.

We report new results on the direct measurement of the associative ionization (AI) cross section in collisions between velocity-selected and spin-oriented Na(3p) atoms. Improvements in the Doppler-shift velocity-selection technique permit measurement over an energy range spanning more than two orders of magnitude from subthermal to suprathermal regions. Spin orientations, parallel and antiparallel, enable determination of the excitation function (velocity dependence of the AI cross section) for the separate singlet and triplet manifolds of ${\mathrm{Na}}_2$ states contributing to the AI process.

The high-density solution of the hypernetted-chain equations for charge-cluster plasmas, as mapped on the Onsager charge-smearing optimization, defines ‘‘Onsager molecules.’’ These are natural generalizations of the ion-sphere (Thomas-Fermi) ‘‘atom.’’ The self-energy of these molecules and their dissociation properties provide the key for the first accurate calculation of the screening potentials in fluid classical plasmas.

We present simulation results on phase transitions in a fluid system whose interactions depend on internal quantum states: they are ‘‘turned on’’ when the molecules are in a hybrid state. Surprisingly, a simple effective temperature-dependent classical potential describes well many features of the transition.

We numerically discuss the asymmetric map & (i=1,2 respectively correspond to x>0 and x<0). Severe differences appear with respect to the Feigenbaum scenario (${\varepsilon }_1$=${\varepsilon }_2$=0, $a_1$=$a_2$, $z_1$=$z_2$), the strongest corresponding to simple discontinuity (${\varepsilon }_1$≠${\varepsilon }_2$, $a_1$=$a_2$, $z_1$=$z_2$) in which case many inverse cascades are observed. The whole set of these cascades can be seen as a new road to chaos.

The absolute specific mass shifts for the ${}_{}{}^2{}_{}{}^{}\mathrm{S}$ ground state of the lithium atom are calculated using a 352-term Hylleraas-type wave function. The specific mass shifts for ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ are 5.180 and 6.042 ${\mathrm{cm}}^{\mathrm{-}1}$, respectively, using first-order perturbation theory, and 5.177 ${\mathrm{cm}}^{\mathrm{-}1}$ ${(}^7$Li) and 6.039 ${\mathrm{cm}}^{\mathrm{-}1}$ ${(}^6$Li) from a nonperturbative approach. With use of Pekeris’s accurate values for the specific mass shifts of ${\mathrm{Li}}^{+}$, the transition isotope shift of Li relative to ${\mathrm{Li}}^{+}$ is calculated to be 1.101 GHz, using first-order perturbation theory, and 1.071 GHz from a nonperturbative procedure. The calculated shift is in very close agreement with the experimental value of 1.111±0.006 GHz obtained by Lorenzen and Niemax.

We present the result of a simple superposition of configuration calculation of the term values for the ${}_{}{}^1{}_{}{}^{}\mathrm{S}$, ${}_{}{}^1{}_{}{}^{}\mathrm{P}$, and ${}_{}{}^1{}_{}{}^{}\mathrm{D}$ singly excited states of the Mg atom. In particular, we have examined quantitatively the importance of the configuration interaction between the doubly excited perturber and the singly excited 3snl series. The numerical result of this calculation is in excellent agreement with the experimental data, including the highly excited states.

We present a theoretical study of the 3pns and 3pnd ${}^1$P doubly excited autoionization series above the first-ionization threshold of magnesium atom. The effect of the configuration interaction on the width and the energy position of the autoionization state is examined in detail. In particular, we have shown that the width of the narrow 3pnd ${}^1$P series is strongly affected by the configuration interaction. The results of the present calculation are also in good agreement with the existing theoretical and experimental data. In addition, the present theoretical approach is shown to be effective in the quantitative calculation for higher excited states in the autoionization series.

We consider the quantum decay from the locally stable ground state of a one-dimensional metastable potential. We consider the case where the metastable minimum is almost degenerate with the vacuum level. In this case, the quantum decay probability is influenced by backscattering from the continuum states; a feature which is not accounted for in standard WKB theory. We evaluate the decay rates, from the ground state and the higher-resonance states, by the use of complex-time path-integral methods. The decay rate from the low-lying states of the metastable well is characterized by two quasi-zero-modes and a transmission factor, which approaches zero proportional to the square root of the potential drop. Moreover, we present a useful set of rules for the complex-time path-integral phase factors. These rules considerably simplify the calculation of decay rates (i.e., the pole condition of the Fourier-transformed Green’s function).

A variation-iteration procedure is described for determining the amplitude for scattering in the presence of a laser field. The essential limitation of the method lies in the requirement that the interaction of the charged projectile with the external field be sufficiently weak, relative to its interaction with the target, to justify use of perturbation theory to account for the effect of the field in intermediate states of the collision process. This still allows for fields which are strong enough to significantly affect the motion of the projectile in initial and final states (leading, for example, to multiphoton transitions) and this strong interaction is treated nonperturbatively. The first two terms in the modified perturbation expansion are analyzed in detail. They are expressed in terms of those matrix elements which describe one- and two-photon free-free transitions in the absence of an external field. The method is not restricted to a consideration of fields of low frequency but the first-order amplitude obtained here does reduce, in that limit, to the known form of low-frequency approximation and the second-order amplitude provides a correction. The theory is described in the context of nonrelativistic potential scattering. A discussion is included of some of the special features of the second-order amplitude associated with potentials having a long-range Coulomb tail.

A new derivation of the Weizsacker-type gradient corrections to Thomas-Fermi (TF) kinetic energy functional is presented. The development is based on the first-order reduced density matrix as obtained from the one-body Green’s function in the mean-path approximation devised for the purpose, using the Feynman path-integral approach; the mean-path approximation turns out to be essentially equivalent to the eikonal approximation used in quantum collision theory for high-energy collisions. This derivation agrees with the conventional gradient expansion truncated at second order, in that it gives the kinetic energy functional of the TF-(1/9)W model, that is, the sum of the original TF kinetic energy and (1/9) of the Weizsacker gradient correction. However, in the present derivation, TF-(1/9)W results from a reduced density matrix of closed form; the original TF local relation between particle, density, and one-body potential is preserved; and the kinetic energy density contains a Laplacian of particle density with a factor half of that from the gradient expansion. Most significantly, the TF-(1/9)W kinetic energy functional is the consequence of representing both the diagonal and off-diagonal elements of the density matrix correctly to zero order through the mean-path approximation to the one-body Green’s function, whereas in the conventional TF approximation, the zero order of the gradient expansion, off-diagonal elements are not correct to the same order. Other results of the present approach include a nonlocal exchange energy functional of density, a one-body effective potential that contains a contribution from the kinetic energy functional derivative, and the construction of closed-form density matrices that give various kinetic energy functionals of TF-λW form (justifying various existing empirical λ values). Also presented are the results of numerical calculation for rare-gas atoms of TFD-λW models (TFD denotes Thomas-Fermi-Dirac) with λ=(1/3), 0.186, (1/6), and (1/9). .AE

Various approximate density functionals for the kinetic energy density of atoms and molecules are analyzed. These include the results of a gradient expansion to first and second orders and a form recently derived from a new Green’s function approximation [W. Yang, preceding paper, Phys. Rev. A 34, 4575 (1986)]. All the approximate functionals studied diverge to minus infinity at a nucleus, due to the ${\nabla }^2$ρ term that is in them, while the exact functional is positive and finite everywhere. Away from nuclei, however, the Hartree-Fock results are well reproduced, including the atomic shell structure. New functionals are proposed to correct the divergent behavior, and accurate total kinetic energy values are obtained from a new formula for kinetic energy density $t_{\mathrm{MP}\left(\mathrm{r}\right)=C_k\mathrm{\rho }{\mathrm{}\left(\mathrm{r}\right)\mathrm{}}^{5/3}}$ +(1/72)‖∇ρ(r)${\Vert }^2$/ρ(r)+( 1/12)${\nabla }^2$ρ(r), with a divergence correction.

Results for the lowest (Schulz) autodetaching state of ${\mathrm{He}}^{\mathrm{-}}$ [1s(2s${)}^2$] are reported. The calculation utilizes the full projection-operator formalism as explicitly developed by Temkin and Bhatia [Phys. Rev. A 31, 1259 (1985)]. Eigenvalues, scrE=〈Φ‖QHQ‖Φ〉, are calculated using projection operators Q depending on increasingly elaborate target wave functions going up to a 10-term Hylleraas-form, and a configuration-interaction total wave function Φ of 40 configurations. Results are well converged, but our best value is ∼0.13 eV above the experimental position at 19.37 eV. We conclude that the shift (Δ) in the Feshbach formalism gives a large contribution (relative to the width) to the position $E_r$ (=scrE+Δ). An appendix is devoted to the evaluation of the most complicated type of three-center integral involved in the calculation.

An exact solution is presented for the problem of a harmonic oscillator of time-dependent mass with an external driving force. The wave functions for both pseudostationary and quasicoherent states and the Green’s function are calculated. A solution is obtained in the Heisenberg picture. The amplitude for transition from a coherent state to a simple harmonic-oscillator coherent state is calculated.

An efficient method is proposed for evaluating first- and second-order nonadiabatic matrix elements of the form 〈${\psi }_i$(q;Q)‖δ${\psi }_j$(q;Q)/δQ〉 and 〈${\psi }_i$(q;Q)‖${\delta }^2$${\psi }_j$(q;Q)/δ$Q^2$ 〉, where ${\psi }_i$(q;Q) and ${\psi }_j$(q;Q) denote multiconfigurational self-consistent-field electron wave functions. The method is based on a finite-difference procedure and requires the numerical computation of symmetric overlaps of the type 〈${\psi }_i$(q,$Q_0$-x)‖${\psi }_j$(q,$Q_0$ +x)〉. It gives an accuracy which is quadratic in the nuclear displacement x for both the first- and second-order nonadiabatic coupling constants. The wave functions are separately optimized for each state and obtained through the direct second-order MCSCF method. The biorthogonal scheme of Malmquist is implemented that expresses ${\psi }_i$(q;Q) and ${\psi }_j$(q:Q) in an orthogonal common basis. The method is applied for the calculation of the nonadiabatic coupling elements and the Born-Oppenheimer corrections to the two lowest ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$ states of ${\mathrm{NaLi}}^{+}$, relevant for analyzing the asymmetric charge exchange in the ion-atom collision Na+${\mathrm{Li}}^{+}$→${\mathrm{Na}}^{+}$+Li. .AE

The spectrum-generating algebra for the problem of a particle in a potential well is shown to be su(1,1). Both the infinitely deep and finite square wells are considered. The generators can also be derived via a systematic procedure for determining the time-dependent constants of the motion. The coherent states are explicitly constructed.

An exact path-integral treatment of the s states for the Hulthén potential is presented. A procedure of the nontrivial change of variable accompanied by the local time rescaling is given in detail. The dimensional extension trick is applied to convert the radial path integral into a path integral in the SU(2) manifold. The exact energy spectrum and the normalized s-state eigenfunctions are obtained from the poles of the Green function and their residues, respectively. The Yukawa and the Coulomb limits are also discussed.

A local-spin-density-functional (LSD) scheme is developed using the normalization conditions for an electron gas around a nucleus of charge Z. The correlation factor for electrons of parallel spin, the Fermi hole, is decomposed into single-particle correlation factors. This leads to an orbital-dependent exchange-only density with no adjustable parameters such as the α of the Xα theory; the parameters $B_1$, $B_2$, and ${\alpha }^{\lim }$ are constant for all atoms once the shape of the Fermi hole is chosen. These parameters are rigorously calculated without assuming an approximate shape for the Fermi-hole correlation factor. The exchange density using this unspecified Fermi-hole correlation factor reduces exactly to the homogeneous free-electron-gas exchange density at the high electron-density limit. The theoretical relationship of this generalized exchange (GX) scheme to the Xα and free-electron-gas exchange densities is discussed, and it is found that the Xα scheme needs a classical approximation to justify the use of a variable α. These schemes are compared numerically using the total energies and eigenvalues for the atoms helium to krypton, and it is found that the LSD GX scheme gives the most reliable results.

We investigate the linear response of a nonrelativistic hydrogenlike atom to a harmonic uniform electric field, which is switched on adiabatically. For the electron initially in an arbitrary bound ‖nlm〉 state, the first-order correction to the wave function has been written in terms of a vector function ${\mathrm{w}}_{\mathrm{nlm}}$. We report an exact closed-form expression of the vector ${\mathrm{w}}_{\mathrm{nlm}}$ that we have derived using the Coulomb Green’s function. Our general analytic results prove to be useful in a systematic study of two-photon processes involving the hydrogen atom.