States Of Two-electron Atoms Research Articles

Potential energy surfaces in atomic structure: The role of Coulomb correlation in the ground state of helium

For the $S$ states of two-electron atoms, we introduce an exact and unique factorization of the internal eigenfunction in terms of a marginal amplitude, which depends functionally on the electron-nucleus distances $r_1$ and $r_2$, and a conditional amplitude, which depends functionally on the interelectronic distance $r_{12}$ and parametrically on $r_1$ and $r_2$. Applying the variational principle, we derive pseudoeigenvalue equations for these two amplitudes, which cast the internal Schr\"odinger equation in a form akin to the Born-Oppenheimer separation of nuclear and electronic degrees of freedom in molecules. The marginal equation involves an effective radial Hamiltonian, which contains a nonadiabatic potential energy surface that takes into account all interparticle correlations in an averaged way, and whose unique eigenvalue is the internal energy. At each point $(r_1,r_2)$, such surface is, in turn, the unique eigenvalue in the conditional equation. Employing the ground state of He as prototype, we show that the nonadiabatic potential energy surface affords a molecularlike interpretation of the structure of the atom, and aids in the analysis of energetic and spatial aspects of the Coulomb correlation, in particular correlation-induced symmetry breaking and quantum phase transition.

Photoionization of two-electron atoms via highly doubly excited states: Numerical and semiclassical results

The quantum regime of highly doubly excited states in two-electron atoms has, so far, been largely inaccessible both to numerical methods as well as to experiments. Recent advances in semiclassical closed orbit theory in combination with a quantum mapping approach have shown a new way into this region of high dynamical complexity. In particular, new scaling laws near the double-ionization threshold as well as the dominant semiclassical contributions to the total photoionization cross section can be identified. We will present this new approach here in all its detail. It is based on representing the photoionization cross section in terms of quantum maps. These quantum maps or quantum propagators are used as a starting point for developing an efficient numerical method for calculating cross sections. Furthermore, by writing the quantum operators in semiclassical approximations, it is possible to interpret the quantum results in terms of classical triple collision orbits and to derive threshold laws near the three-particle breakup point. Semiclassical and numerical quantum results show excellent agreement for a model system, namely collinear helium.

Doubly excited [formula omitted] states of two-electron atoms

Precise energy eigenvalues of doubly excited 1 , 3 D o states originating from 2pnd ( n = 3 – 8 ) configuration of two-electron atoms ( Z = 3 – 7 ) have been calculated by using Ritz variational method in Hylleraas coordinates. Except for a few, the present non-relativistic energy values are the lowest yet obtained. The energy eigenvalues of a number of these states are being reported for the first time. Comparisons with existing experimental results are also made. The effective quantum numbers ( n ∗ ) for the states mentioned above have been calculated using the theory of quantum defect.

Three-dimensional statistical reduction of theN-body Schrödinger equation for electrons with pairwise Coulomb interactions

Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of $N$ electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the $N$-body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.

The state of an atomic-molecular system near the stability threshold

The state of an atomic-molecular system near its stability threshold with regard to the detachment of one of the particles is studied. The decay of the system upon a decrease in the charge of a binding particle, as well upon an increasasymmetry of masses of like-charged particles, is considered. A special variational principle that allows one to directly calculate the threshold state of the system without repeatedly calculating its energy for different values of masses and charges of particles is used. With the motion of all particles fully taken into account, the threshold states of two-electron atoms with different nuclear charges and of atomic-molecular systems corresponding to the attachment of a positron of variable mass to a neutral atom are calculated. On the basis of calculation of quantum-mechanical expectation values of the kinetic energy of particles and the potential energy of their interaction, the rearrangement of the wave function upon passage of the system through the decay threshold is examined. The threshold characteristics of a purely adiabatic system containing infinitely heavy particles are considered separately.

Coorbital Dynamics at Large Eccentricity and Inclination

We show that coorbital dynamics at large eccentricity and inclination exhibit hitherto unknown types of stable motion in the gravitational three-body problem. In the solar system, the secular stability is observed in the evolution of specific near-Earth asteroids. In particular, the slow diffusion of such objects through the Earth's coorbital region which leads to temporary capture suggests the existence of undiscovered retrograde moons of the Earth. In the Coulomb problem, the Kozai resonance is a primary candidate for stable low angular momentum states in two-electron atoms.

Large-dimension limit of higher-angular-momentum states of two-electron atoms

To apply the methods of dimensional scaling to higher-angular-momentum states, a formalism needs to be developed which factors the $D$-dimensional rotational degrees of freedom from the internal degrees of freedom. The rotational degrees of freedom multiply with increasing dimensionality, while the internal degrees of freedom remain finite in number. A suitable expansion which achieves this has been presented by the authors recently and is an $N$-electron $D$-dimensional generalization of the Schwartz expansion. A derivation of the coupled differential equations in the internal variables that result from the application of the Hamiltonian to this wave-function expansion for the atomic two-electron system has been presented by the authors in another recent paper. The coupled differential equations admit continuation in $D$ and clearly show the complete spectrum of exact interdimensional degeneracies of the two-electron system. However, to apply the methods of dimensional scaling to the two-electron system, the system of coupled differential equations have to be solved for large $D.$ This paper concerns itself with this issue.

Long‐Range Casimir Forces: Theory and Recent Experiments on Atomic Systems

An Overview of LongRange Casimir Interactions L. Spruch. Experimental Studies of High-L Rydberg States in Helium S.R. Lundeen. High Precision Calculations for the Rydberg States of Helium G.W.F. Drake. High Rydberg States of Two-Electron Atoms in Perturbation Theory R.J. Drachman. Long-Range Electromagnetic Forces in Quantum Theory J. Sucher, G. Feinberg. Index.

Rates of convergence of the partial-wave expansions of atomic correlation energies

The coefficients of the leading terms of the partial-wave expansion of atomic correlation energies in powers of (l+1/2)−1 are derived for the second- and third-order perturbed energies in the 1/Z expansion for all possible states of two-electron atoms, and for second-order Mo/ller–Plesset (many-body perturbation) theory for arbitrary n-electron atoms. The expressions for these coefficients given in Table I involve simple integrals over the zeroth-order wave functions (for the third order energies first-order wave functions are also involved). The leading term of E(2) goes as (l+1/2)−4 for natural parity singlet states, as (l+1/2)−6 for triplet states, and as (l+1/2)−8 for unnatural parity singlet states. There are no odd powers of (l+1/2)−1 present in E(2), and the coefficient of the (l+1/2)−6 term for natural parity singlet states of two-electron systems in the 1/Z expansion is generally −5/4 times the coefficient of the (l+1/2)−4 term. In E(3) there are terms that go as odd powers of (l+1/2)−1; the leading term is expressible in terms of the zeroth-order wave function and goes as (l+1/2)−5 for natural parity singlet states, as (l+1/2)−7 for triplet states, and as (l+1/2)−9 for unnatural parity singlet states. Numerical values of the coefficients for E(2) in two-electron atoms are given in Table II. The relation of our results to those obtained by Hill for variational calculations on the He ground state is discussed and generalizations of Hill’s formulas are conjectured. The unexpected results for unnatural parity singlet states are traced back to the behavior of their wave functions at the point of coalescence of two electrons. In terms of the relative coordinate of the two electrons they represent d waves, in contrast to s waves for natural parity singlet states and p waves for triplet states of either parity.

Lowest-order contribution to the width and shift of autoionizing states of two-electron atoms.

A combined complex rotation and Feshbach-projection method is implemented within Z-dependent perturbation theory to obtain the lowest-order contribution to the width and shift of autoionizing states of two-electron atoms. This approach relies on the fact that in the complex rotation method, it is the open-channel part of the wave function that produces the imaginary component of the energy. Calculation of this part of the wave function involves the solution of a simple one-electron differential equation which, in lowest order, can be obtained to any desired level of accuracy. Use of the complex rotation method here simultaneously yields the lowest-order contributions to both the width and shift. It also simplifies the calculation by making this wave function square-integrable while simultaneously rotating the interacting continuum away from the real energy axis, thereby eliminating the degeneracy in zeroth order between the doubly excited states and the adjacent continuum. In the present paper, the lowest-order contributions to the width and shift for the 50 lowest singlet and triplet autoionizing P states below the n=2 threshold are calculated. These results represent the limiting values of the width and shift for high Z in the nonrelativistic approximation. The dependence of these results on the level of excitation N is studied. Comparisons to total values for the isoelectronic series are made, indicating that even for relatively low values of Z, these lowest-order results can yield useful estimates of the total widths and shifts. This approach is particularly useful for states with extremely narrow widths, as these are difficult to calculate accurately with other methods. Hence, the lowest-order estimates obtained here give useful information for such states, even for small values of Z.

Rydberg analysis of doubly excited states.

Numerically available results on energy levels of doubly excited states of two-electron atoms with Z = 1, 2, and 7 are analyzed on the basis of a single Rydberg expression for the pair. Pair quantum defects and other parameters are studied as functions of Z and pair quantum numbers.

Faddeev calculation of energy states in two-electron atoms

The Faddeev equations are used to calculate the ground state as well as a few ortho- and para-excited states of e−e+e−, H−, He, Li+ and Be++ atoms. The Coulomb potential is represented in a separable form using the appropriate Coulombic Sturmians for both the attractive and repulsive interactions. Up to 36 Sturmians are employed in each interaction leading to the solution of a maximum of 56 coupled one-variable integral equations. Results are compared with data or predictions from variational calculations.

Hylleraas–Pekeris treatment of D-dimensional two-electron atoms

The algorithm of Pekeris for S states of two-electron atoms is generalized to spaces of arbitrary dimensionality. Numerical calculations are reported for the ground state (1 1S) and first two excited states (2 3S and 2 1S) for a wide range of dimensions, 1<D<∞, and nuclear charge, 1≤Z≤6. The accuracy is typically better than one part in 108. The energy eigenvalues may be continued to arbitrary real values of the parameter δ=1/D. Real atoms, with D=3, connect smoothly with simple, exactly known limits at D→1 and D→∞. Analysis of the data permits several further terms in the 1/D expansion for the ground state energy to be determined, up to order D−12. This indicates that the expansion does not converge for D=3 but terms of third to sixth order do conform approximately to a geometric series form, as previously postulated in order to carry out dimensional interpolation. The excited state data exemplify near continuum motion at D→1 and quasivibrational asymmetric and symmetric stretching modes of electron motion as D→∞.

An Excition Approach to the Excited States of Two Electron Atoms. I. Formalism and Interpretation

The exciton model is formally applied to a description of the excited states of two electron atoms with the explicit inclusion of exchange. The model leads to a conceptually simple framework for the discussion of the electronic properties of the archetypical atomic electron pair.

Asymptotic calculation of the Rydberg series quantum defects for the doubly excited states of two-electron atoms

The quantum defects are studied for the doubly excited states of two-electron atom with one electron excited much more than the other. The dipole approximation for the electron-electron interaction allows the Rydberg series of the doubly excited states to be constructed with account for the electronic correlations. The latter are particularly important due to the high degeneracy of the atomic states when the electron-electron interaction is switched off. Most attention is paid to the semiclassical region of high total orbital momentum of the atom where closed analytical expressions are obtained for the quantum defect in the dipole approximation as well as for the Coulomb and exchange correction accounting for the residual electron-electron interaction in the first order of perturbation theory. The quantum defects for the Rydberg series of the helium atom and helium-like ions are calculated in the framework of asymptotic theory and compared with the data from energy level numerical calculations for the D and F series of He, Li+, Be2+, B3+.

On exact analytical solutions for the few-particle Schrodinger equation. IV. The asymptotic form and normalisability of the wavefunction

For pt.III see ibid., vol.16, nol.18, p.4237-53 (1983). Formal series solutions for the Schrodinger equations of few-particle systems contain infinitely many associated with the normalisability, i.e. square-integrability of the wavefunction. The energy is one of these parameters. A method for determining the parameters by examining the asymptotic behaviour of the series has been developed. No integration, matrix inversions or trial and error procedures are involved. The method is directly applicable to the 1S states of two-electron atoms. If the wavefunction is expressed as a multipole expansion in spherical polar coordinates the radial functions have asymptotic properties which give rise to relations between the parameters. Values assigned to the parameters not specified by these relations (one per multipole) ensure that the wavefunction is normalisable.

Investigation of the Stark effect in krypton autoionizing Rydberg series with the use of two-step laser excitation

The first detailed observation of the perturbations by an electric field of the Rydberg autoionizing resonances of a rare-gas atom is reported. The Stark spectroscopy of autoionizing Rydberg states converging to the second ionization limit $4{p}^{5}^{2}P_{\frac{1}{2}}^{o}$ in Kr I is performed in an atomic beam with the use of a two-step laser excitation from the $4{p}^{5}5s{[\frac{3}{2}]}_{2}^{0}$ metastable state. In addition to the verification of the ${({n}^{*})}^{\ensuremath{-}3}$ dependence of the zero-field width of two autoionizing series and of the linear dependence of the Stark-manifold splittings with the electric field, we have observed an ${(n)}^{0.7}$ law for the dependence of the Stark-manifold splitting along the series, an important narrowing and dissymmetrization of the $n{d}^{\ensuremath{'}}$ resonances, and a nonlinear-field behavior of Stark-manifold components in the vicinity of the first anticrossing. Discussions are carried out with the use of the recent work on Rydberg-Stark spectroscopy and on autoionizing states in two-electron atoms.

S,P, andDstates of two-electron ions viaZ-dependent perturbation theory

A perturbation study of singly excited $S$, $P$, and $D$ states of two-electron atoms has been made. Perturbation expansion coefficients for the nonrelativistic energies are computed and estimates of the relativistic corrections to low orders in $Z$ are also obtained. A simple one-electron Lamb-shift correction is then combined with these results to yield theoretical transition energies in excellent agreement with experimental data for HeI through FeXXV.

Parity violation effects in decays of autoionisable states of two-electron atoms and ions

Parity non-conservation effects in the decays of autoionisable states of two-electron atoms and ions arising in the presence of neutral weak currents are considered. The degree of the photon circular polarisation and electron emission asymmetry P is calculated. The value of P is of order 10-6 for photon emission and 10-8 for electron emission in the Weinberg model (1972).

The symmetry of the electron-electron interaction operator in the dipole approximation

The operator for the interaction of a charged quantum particle with an excited hydrogen-like atom or ion is considered in the dipole approximation. A new additional integral of motion and related symmetry group are found. A broader approximate symmetry group and expressions for the eigenvalues and eigenfunctions are proposed in the semiclassical limit of large total orbital momentum. The qualitative peculiarities of the spectrum are discussed and limiting cases are considered. A comparison with the results of other authors and with analytical and numerical data is made. The results are applicable to the doubly excited states of two-electron atoms, scattering of charged particles on hydrogen-like atoms or ions, and spectral line broadening etc.