R. Krivec

Boundary solutions of the two-electron Schrödinger equation at two-particle coalescences of the atomic systems

The limit relations for the partial derivatives of the two-electron atomic wave functions at the two-particle coalescence lines have been obtained numerically using accurate correlation function hyperspherical harmonic method wave functions. The asymptotic solutions of the proper two-electron Schroedinger equation have been derived for both electron-nucleus and electron-electron coalescence. It is shown that the solutions for the electron-nucleus coalescence correspond to the ground and singly excited bound states, including triplet ones. The proper solutions at small distances R from the triple coalescence point were presented as the second order expansion on R and ln R. The vanishing of the Focks logarithmic terms at the electron-nucleus coalescence line was revealed in the frame of this expansion, unlike the case of electron-electron coalescence. On the basis of the obtained boundary solutions the approximate wave function corresponding to both coalescence lines have been proposed in the two-exponential form with no variational parameters.

Analytic presentation of a solution of the Schrödinger equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into the nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.

Wavefunctions of Helium-Like Systems in Limiting Regions ¶

We find an approximate analytic form for the solution ψ(r1, r2, r12) of the Schrödinger equation for a system of two electrons bound to a nucleus in the spatial regions r1 = r2 = 0 and r12 = 0, which are of great importance for a number of physical processes. The forms are based on the well-known behavior of ψ(r1, r2, r12) near the singular triple coalescence point. The approximate functions are compared to the locally exact ones obtained earlier by the correlation function hyperspherical harmonic (CFHH) method for the helium atom, light helium-like ions, and the negative ion of hydrogen H−. The functions are shown to determine a natural basis for the expansion of CFHH functions in the considered spatial region. We demonstrate how these approximate functions simplify calculations of high-energy ionization processes.

Calculation of the photoionization with deexcitation cross sections of he and helium-like ions

We discuss the results of the calculation of the photoionization with deexcitation of excited He and helium-like ions Li+ and B3+ at high but nonrelativistic photon energies θ. Several lower 1S and 3S states are considered. We present and analyze the ratios Rd+* of the cross sections of photoionization with deexcitation, σ(d)+*(θ), and of the photoionization with excitation, σ+*(θ). The dependence of Rd+* on the excitation of the target object and the charge of its nucleus is presented. In addition to theoretical interest, the results obtained can be verified using long-lived excited states such as 23S of He.

Two‐electron processes in atom‐photon interactions

Two‐electron processes in atom‐photon interactions have been studied. Various mechanisms of double photoionization, angular correlations, total cross sections, energy distributions and results for non‐Helium like systems are discussed.

High Energy Limit of the Two‐Electron Photoionization Processes in the Helium‐Like Systems

We present the analysis of double photoionization and of ionization accompanied by excitation of the helium atom and helium‐like ions. The high‐energy limit corresponding to the ultrarelativistic energies transferred to the atom is considered. The cross‐sections ratios are expressed through certain integral parameters of the wave function of the initial state. The calculations with the wave function obtained earlier by Correlation Function Hyperspherical Harmonic Method are carried out. The dependence of the relative role of the quasifree mechanism on the nuclear charge Z in ionization of the ground state and of the excited n′s states is analyzed.

Two-Electron Photoionization Cross Sections at High Energies

Double and single electron photoionization cross sections and their ratios at high and ultra-relativistic energies are calculated for H−, He and helium-like ions in ground and excited states including triplet states. The ratios contain shake-off and quasi-free terms. A high precision non-variational wave function is used. The quasi-free mechanism increases the ratios impressively: for He we get 0.0762 instead of 0.0164 in the non-relativistic case. Ratios are inversely proportional to Z 2, with a factor increasing from 0.094 in the non-relativistic to 0.595 in the ultra-relativistic limit.