States Of Two-electron Systems Research Articles

Full-scale analytical perturbation calculation of He-isoelectronic series with Hydrogenic bound states via Green's function expansion (monopole) of Coulomb interactions

Increasing number of carrier electrons and nuclear charge in Hydrogenic systems associated with bound and ionised (scattered) states invites long-standing divergence issue of Schrödinger's equations due to Coulomb interactions. In this regard, He-isoelectronic series represents the benchmark for atoms and molecules within non-relativistic quantum limit. Schrödinger equations for such complex systems being transformed into either Whittaker-M or Associated Laguerre Polynomial, furnish multipole operators of Green's function expansion of Coulomb interaction a robust framework of lower and upper incomplete Gamma functions. The resulting monopole integral gives terminating, finitely summed, simple and analytical forms of Lauricella functions. This novel form succinctly remedies the paradox of Energy Contribution or Correction occurring due to divergent length scales among Hydrogenic bound states. Both singly and doubly excited spherically symmetric states of two-electron systems ( ) are examined to register full-scale third-order perturbation for monopole factor within 0.2–8.2% deviation of ground-state energies from experimental values. The deviation urges for further analytical treatment to dipole factor. Moreover, exchange of electronic coordinates by virtue of exchanging quantum numbers exhibits invariant integrals of multipoles which confirms reciprocity condition of Green's function. This methodology has achieved a new paradigm of employing generic H-like orbitals for electronic structure calculations.

Calculations Energy of the (<i>nl</i><sup>2</sup>) <sup>1</sup><i>L<sup>π</sup></i> Doubly Excited States of Two-Electron Systems via the Screening Constant by Unit Nuclear Charge Formalism

In this work, the total energies of doubly excited states (ns2) 1Se, (np2) 1De, (nd2) 1Ge, (nf2) 1Ie, (ng2) 1Ke, and (nh2) 1Me of the helium isoelectronic sequence with Z ≤ 10 are calculated in the framework of the variational method of the Screening Constant by Unit Nuclear Charge (SCUNC). These calculations are performed using a new wavefunction correlated to Hylleraas-type. The possibility of using the SCUNC method in the investigation of high-lying Doubly Excited States(DES) in two-electron systems is demonstrated in the present work in the case of the (nl2) 1Lπ doubly excited states, where accurate total energies are tabulated up to n = 20. All the results obtained in this paper are in agreement with the values of the available literature and may be useful for future experimental and theoretical studies on the doubly excited (nl2) 1Lπ states of two-electron systems.

The “Fermi hole” and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function

The impact of the antisymmetrization is often addressed as a local property of the many-electron wave function, namely that the wave function should vanish when two electrons with parallel spins are in the same position in space. In this paper, we emphasize that this presentation is unduly restrictive: we illustrate the strong non-local character of the antisymmetrization principle, together with the fact that it is a matter of spin symmetry rather than spin parallelism. To this aim, we focus our attention on the simplest representation of various states of two-electron systems, both in atomic (helium atom) and molecular (H2 and the π system of the ethylene molecule) cases. We discuss the non-local property of the nodal structure of some two-electron wave functions, both using analytical derivations and graphical representations of cuttings of the nodal hypersurfaces. The attention is then focussed on the impact of the antisymmetrization on the maxima of the two-body density, and we show that it introduces strong correlation effects (radial and/or angular) with a non-local character. These correlation effects are analyzed in terms of inflation and depletion zones, which are easily identifiable, thanks to the nodes of the orbitals composing the wave function. Also, we show that the correlation effects induced by the antisymmetrization occur also for anti-parallel spins since all Ms components of a given spin state have the same N-body densities. Finally, we illustrate that these correlation effects occur also for the singlet states, but they have strictly opposite impacts: the inflation zones in the triplet become depletion zones in the singlet and vice versa.

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.

D-wave electron-H,−He+, and−Li2+elastic scattering and photoabsorption inPstates of two-electron systems

In previous papers [A. K. Bhatia, Phys. Rev. A 85, 052708 (2012); A. K. Bhatia, Phys. Rev. A 86, 032709 (2012); A. K. Bhatia, Phys. Rev. A 87, 042705 (2013)] electron-H, $\ensuremath{-}{\mathrm{He}}^{+}$, and $\ensuremath{-}{\mathrm{Li}}^{2+}$ $P$-wave scattering phase shifts were calculated using the variational polarized orbital theory. This method is now extended to the singlet and triplet $D$-wave scattering in the elastic region. The long-range correlations are included in the Schr\"odinger equation by using the method of polarized orbitals variationally. Phase shifts are compared to those obtained by other methods. The present calculation provides results which are rigorous lower bonds to the exact phase shifts. Using the presently calculated $D$-wave and previously calculated $S$-wave continuum functions, photoionization of singlet and triplet $P$ states of He and ${\mathrm{Li}}^{+}$ are also calculated, along with the radiative recombination rate coefficients at various electron temperatures.

Modified atomic orbital theory applied to the calculation of high-lying <sub>2</sub>(K,T)<sub>n</sub><sup>±1,3</sup>P° rydberg series of he-like ions

The 2(1,0)n-1P°, 2(1,0)n +3P°, 2(0,1)n+1P°, and 2(0,1)n-3P° intershell Rydberg series of the helium-like ions are investigated in the framework of the modified Atomic Orbital Theory (MAOT). High-lying energy resonances of He and excitation energy of the he- lium-like Li+ up to n = 10 are tabulated. In addition, total energy positions for low-lying states (n1,3P° autoionizing states of two-electron systems.

Configuration-interaction approach for high-lying singly and doubly excited states of two-electron systems

We introduce a configuration-interaction expansion in terms of Sturmian functions, which provides very accurate nonrelativistic energies for the high-lying singly and doubly excited states of a two-electron system. The approach, which requires a substantially smaller basis size by contrast to the standard configuration interaction methods, provides accurate results even for large total angular momenta. Key features are the use of different numbers of Sturmian functions for each electron, the inclusion of many different pairs of nonlinear parameters in the expansion, as well as complex scaling. The accuracy and convergence of the method, applied to helium, increases with the degree of excitation of one of the two electrons.

Double photoionization of excited 1S and 3S states of the helium isoelectronic sequence.

We present results for the photoionization of ground and excited S states of two-electron systems by x-ray photons. The method uses the asymptotic formulation of Dalgarno and Stewart [Proc. Phys. Soc. London 76, 49 (1960)] with highly correlated Frankowski-Pekeris--type wave functions for N $^{1}\mathrm{S}$ and N $^{3}\mathrm{S}$ initial states. The calculations show a smaller ratio of double-to-single photoionization R for triplet states due to the spatial antisymmetry of the wave function near the nucleus. We find an asymmetric population of photoion satellite states and provide expressions for the asymptotic single and total photoionization, and R as a function of nuclear charge. We also make predictions for the differential double photoionization cross sections in the extreme energy sharing case when one electron is ejected with zero energy.

Electron-pair wave packets.

We present a computational scheme to propagate wave packets describing S states of two-electron systems. Body-fixed polar coordinates (${\mathit{r}}_{1,}$${\mathit{r}}_{2}$,\ensuremath{\theta}) are used to represent the time-dependent wave functions on a grid in three dimensions. We handle the Coulomb singularities by transformation of the wave function and a modified finite-difference formula to evaluate the kinetic energy. We use a finite-difference split-operator scheme for the time propagation and, as a first application, we compute the low-energy spectrum of helium and ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$.

Z dependences of the energy levels and autoionization rates for 2l3l' states of two-electron systems

Three different calculational methods (AUTOLSJ, MZ, MCDF) were used to obtain energy levels and autoionization rates for doubly excited states 2l3l' of two-electron systems (10<or=Z<or=42). When all the transpositions of the levels are done according to the Wigner-von Neuman theorem (noncrossing of the levels with the same total angular momentum J and parity of the system along the Z isoelectronic sequence), it becomes possible to compare energy levels and autoionization rates obtained by the three methods. The Z dependences of the quantities are investigated for each calculational method for 10<or=Z<or=42. The final comparison reveals rather good agreement between the results obtained by the three methods.

Dipole transitions from excited S states of two-electron systems in time-dependent Hartree-Fock theory

Dipole transitions from excited S states of two-electron systems in time-dependent Hartree-Fock theory

Widths and configuration mixings of two-electron systems below the N=2 threshold

Width calculations have been carried out for all doubly excited states of two-electron systems with orbital angular momentum L<or=3, and Z=1,2,3,4,5 below the N=2 threshold, using a procedure which combines the truncated diagonalisation method with the open-channel close-coupling approximation. The properties of these states are discussed with regard to series, width, configuration mixings and quantum defects. Comparison is made with other calculations and with experiment.

Autoionising states of three-electron atomic systems

The excitation energies and the probabilities for the radiative and radiationless decay of the autoionising states belonging to the 2s22p, 2s2p2 and 2p3 configurations have been calculated for three-electron atomic systems on the basis of perturbation theory with respect to inter-electron interaction in a broad range of values of atomic number Z. Using the results of the diagonalisation calculations of the autoionising states of two-electron systems, the authors have obtained estimates of corrections to the autoionisation widths calculated by perturbation theory. The calculations of the radiative transition probabilities for Fe and Ni ions are in good agreement with the 1976 results of Nussbaumer.