Electron correlation energy in confined two-electron systems

Radial, angular and total correlation energies are calculated for four two-electron systems with atomic numbers Z=0-3 confined within an impenetrable sphere of radius R. We report accurate results for the non-relativistic, restricted Hartree-Fock and radial limit energies over a range of confinement radii from 0.05-10a₀. At small R, the correlation energies approach limiting values that are independent of Z while at intermediate R, systems with Z?1 exhibit a characteristic maximum in the correlation energy resulting from an increase in the angular correlation energy which is offset by a decrease in the radial correlation energy.     

Exact wave functions for concentric two-electron systems

We show that the exact solution of the Schrödinger equation for two electrons confined to two distinct concentric rings or spheres can be found in closed form for particular values of the ring or sphere radii. In the case of rings, we report exact polynomial and irrational solutions. In the case of spheres, we report exact polynomial solutions for the ground and excited states of S symmetry.     

Modern methods for calculating photoionization and electron-impact ionization of two-electron atoms and molecules

A number of recently developed methods for calculating multifold differential cross sections for photoionization and electron impact ionization of atoms and molecules with two active electrons are reviewed. The methods are based on unique approaches to the calculation of three-body Coulomb wave functions. The exterior scaling method and the driven Schrödinger equation formalism are considered. The effectiveness of the time-dependent approaches to the scattering problem, such as the paraxial approximation and the time-dependent scaling, is demonstrated. A novel numerical method is formulated, which has been developed by the authors to solve the six-dimensional Schrödinger equation for an atom with two active electrons on the basis of the Chang-Fano transformation and the discrete variable representation. The threshold behavior manifested by the angular distributions of the two-electron photoionization of a negative hydrogen ion and a helium atom and the multifold differential cross sections for the electronimpact ionization of hydrogen and nitrogen molecules are analyzed on the basis of numerical simulations. The Wannier law for the angular distribution of double ionization is demonstrated to be incorrect even at very low energies.     

A two-electron group model theory for radio-frequency ionization ofnoble gases with turbulent flow

Equations are derived for predicting the current-voltage characteristic curves of axial RF discharges in noble gases, with turbulent flow. The electrons are considered to be made up of two Maxwellian groups: bulk and tail electrons. The bulk electrons are described by a temperature T_b, and have kinetic energies (1/2 mv²=eV) from 0 to eV _l (eV_l=the threshold energy of the first dominant inelastic collision process). The electrons of the depressed tail of the distribution function are described by another temperature, T_t<T_b, and have (eV>eV_l). The terms in these equations correspond to the prevailing processes occurring inside the noble gas discharge. The rate coefficients given are derived, based on the two-electron group model. The effect of the high velocity flow is accounted for by the terms giving the divergence of the flux of particles in the redirection of flow in each of the continuity equations for the primary species and by adding a diffusion coefficient due to turbulence to the static discharge diffusion coefficients of the ions and metastables     

First observation of two-electron one-photon transitions in single-photon K-shell double ionization

Experimental evidence for the correlated two-electron one-photon transitions (1s(-2)→2s(-1)2p(-1)) following single-photon K-shell double ionization is reported. The double K-shell vacancy states in solid Mg, Al, and Si were produced by means of monochromatized synchrotron radiation, and the two-electron one-photon radiative transitions were observed by using a wavelength dispersive spectrometer. The two-electron one-photon transition energies and the branching ratios of the radiative one-electron to two-electron transitions were determined and compared to available perturbation theory predictions and configuration interaction calculations.     

Correlated two-electron momentum spectra for strong-field nonsequential double ionization of He at 800 nm

We report on a kinematically complete experiment on nonsequential double ionization of He by 25 fs 800 nm laser pulses at 1.5 PW/cm;{2}. The suppression of the recollision-induced excitation at this high intensity allows us to address in a clean way direct (e,2e) ionization by the recolliding electron. In contrast with earlier experimental results, but in agreement with various theoretical predictions, the two-electron momentum distributions along the laser polarization axis exhibit a pronounced V-shaped structure, which can be explained by the role of Coulomb repulsion and typical (e,2e) kinematics.     

Monte Carlo optimisation of correlated wave functions for normal two-electron systems

A simple, new type of correlated wave function is proposed for the studies of normal two-electron atomic systems: ψ(r₁, r₂) = Σc_mΦ_m(r₁, r₂) with Φ_m(r₁, r₂) = exp[−(r₁ + r₂)]/(br₁₂ + a)^m, where , a, b are non-linear variational parameters. A notable feature of this basis function is that only three terms are required within the framework of the Raleigh-Ritz variational principle to obtain fairly accurate energy eigenvalues and satisfactory cusp conditions. The non-linear variational parameters are optimised by using the Monte Carlo technique.     

The equivalent potential for relativistic confined systems

In previous papers a method of obtaining bound states and wavefunctions for confined relativistic systems was presented. The input is the asymptotic expansion of the two-point functions. Confinement is imposed by systematic removal of the two-particle cut. We extend this method by developing an equivalent (angular momentum dependent) potential, which gives the correct wavefunction to a given order of R, the infrared scale parameter. We prove the uniqueness of the wavefunction by requiring that there are no CDD poles and by the connection of our moment conditions to the requirement that the residues of the bound-state poles must be positive. Finally we test the bound-state approximation for a system defined by an equivalent potential $\text{V(r) = λ}{}^2\text{tanh}{}^2\text{(}\text{g}{}^2\text{r}\text{λ}\text{)}$. Although in this case there is a threshold we still find excellent results when $\text{λ}{}^2\text{g}{}^2$ is large, i.c., when there are many bound states.     

Note on quantizing constrained systems

Some problems occurring in quantizing the constrained systems in which the elimination of non-physical variables is not unambiguous are illustrated by simple examples. The modification of the standard procedure of quantization is proposed in terms of path integral formalism.     

Electromodulation spectroscopy of confined systems

The convolution formalism of Aspnes and Rowe for the dielectric function in the presence of an electric field is applied to electromodulation spectroscopy of confined systems. It is shown that for isolated confined systems where excitonic effects are not important, electromodulation spectroscopy leads to first derivative like optical features of two dimensional critical points. For superlattices in weak fields (neglecting excitonic interactions), the familiar third derivative forms of Aspnes and Rowe are applicable. In contrast, when moderate and high electric fields are applied to superlattices, electromodulation spectroscopy will exhibit optical features characteristic of first derivative line shapes. This phenomenon arises because the electric field causes the envelope functions of the electron and hole states to become spatially localized. The importance of excitons and their inhomogeneous broadening to the experimentally observed line shapes are considered.     

Polarizabilities of confined two-electron systems: the 2-electron quantum dot,the hydrogen anion,the helium atom and the lithium cation

The static dipole polarizabilities of two-electron systems confined by a spherical harmonic-oscillator potential?ω?have been calculated by the coupled-cluster CCSD method. The combined effect of the confining potential?ω?and the central electrostatic field on the polarizabilities of the quantum dot, and the confined systems, H^?, He and Li⁺, respectively, have been investigated. The polarizabilities of the quantum dot can be calculated analytically. The polarizability?α?of the 2-electron quantum dot for ω?=?0.01 is calculated to be 19?996?au, in perfect agreement with the exact value, 20?000?au. Already medium confinement, ω?=?1.0, reduces?α?to 2.00?au. The decrease of the polarizability is smaller for H^? (α?=?216.1?au for ω?=?0.0 and 0.985 au for ω?=?1.0), and much smaller for He and Li⁺ (1.3819 and 0.3813?au for He for ω?=?0.0 and ω?=?1.0, respectively, and 0.1921 and 0.128?au for Li⁺). The theoretical polarizabilities for unconfined (ω?=?0.0) H^?, He and the Li⁺ cation are in very good agreement with the best published theoretical and/or experimental data. Our final polarizability for H^?, 216.0±0.5?au, appears to be one of the most accurate values published so far. The optimization procedures of basis sets applicable to calculations of polarizabilities of systems confined by a spherical harmonic-oscillator potential are presented.     

A monte carlo method for approximating critical cluster size in the nucleation of model systems

A formalism is presented for estimating critical cluster size as defined in classical models for nucleation phenomena. The method combines Bennett's Monte Carlo technique for determining free-energy differences for clusters containingn andn- 1 atoms with the steady state nucleation rate formalism. A simple form for the free energy of formation of then cluster [including a termA (n)n ^2/3] is used to predict critical cluster size and critical supersaturation ratio, S^*. This approach is applied to Lennard-Jones vapor clusters at 60 K. Results for free-energy differences for the 13, 18, 24, and 43 clusters predict a critical cluster size of 70 ± 5 atoms at a critical supersaturation ratio given bylnS ^*=2,45 0.15. This method is intended to provide estimates of critical cluster size for more ambitious attempts to calculate cluster free energies or for initializing conditions in microscopic simulations of nucleating systems.This material is based upon work supported by the National Science Foundation under Grant No. ATM80 15790 and the National Aeronautics and Space Administration under Grant No. NAS8-31150.     

Observation of inner electron ionization from radial rydberg wave packets in two-electron atoms

We have observed multiphoton ionization of the 5s core electron from a 5snd radial Rydberg wave packet of Sr atoms using a short optical pulse. When the outer nd electron is at its outer turning point the inner 5s electron is removed from the atom, and the outer electron is left in a Sr+ Rydberg state, but when the outer electron is at the inner turning point this does not occur. Analysis of the final Sr+ Rydberg states shows that the two electrons interact as the inner electron leaves, so that the outer electron is not simply projected onto the Sr+ Rydberg states.     

Ab initio calculations of lower resonant states of two-electron systems

The complex scaling and the stabilization methods are applied to calculating the parameters of the lowest resonance states of He and Li⁺. The results obtained by both methods are in a good agreement with the published data. It is shown that the wave functions constructed using the restricted configuration interaction approximation provide fairly accurate estimates of the resonance parameters. The possibility of using the stabilization method for calculating the phase shift function is also discussed.     

Note on the quantum displacement of the critical point

The quantum corrections to the law of corresponding states are studied by calculating the critical pressure, temperature, and density to first order in Planck's constanth on an exactly soluble model. The ratio of the critical parameters to the corresponding classical values are found to be (p _c/p _c ⁰)^1/2=_c/_c ⁰ = T_c/T_c ⁰ = 1–0.67, with=h _c ^1/3(mkT _c)^–1/2. The critical ratio is independent ofh to first order. The results are compared with critical data for noble gases and hydrogen isotopes.