Three-electron Systems Research Articles

Towards an accurate treatment of σ∗ ← σ transitions: Moving onto [formula omitted

Dimeric σ∗ radical anions are ubiquitous, and their formation, spectroscopy and outcome can often be elucidated by density functional theory. But for shorter interfragment distances, three-electron two-center systems can be reluctant to a single-determinant description, such as the hexanitrogen radical anion. For N6.-, we show that multireference configuration interactions calculations are required to recover its characteristic electronic excitation energy, while TDDFT fails even with modern exchange–correlation functionals. The effects of vibronic couplings on the absorption spectrum are delineated based on a full quantum mechanical dynamical treatment; this study opens the door towards an accurate description of the subtle solvatochromism of hemi-bonded systems.

Application of the time-dependent close-coupling approach to few-body atomic and molecular ionizing collisions

We review the recent progress made in applying the time-dependent close-coupling approach to ionizing collisions of electrons, photons, and ions with small atoms and molecules. The last twenty years have seen a proliferation of non-perturbative approaches applied to fundamental atomic and molecular scattering processes. Such processes form the building blocks of describing the dynamics of plasmas over a wide range of temperatures and densities, and also provide insight into the long-range Coulomb interactions between charged particles. Studies of the few-body Coulomb problem presented in electron, photon, or ion-impact ionization of small atoms and molecules, by direct solution of the time-dependent Schrodinger equation, are particularly useful because the complicated three-body boundary conditions of more than one continuum particle in a Coulomb potential are not required. With the continuing growth and increasing availability of high-performance computing resources, such methods can now be applied to a wide variety of scattering processes. The recent progress made using such a time-dependent approach is described in this colloquium. In this paper, we focus on the recent results obtained for one-, two-, and three-electron systems, thus building on a previous review of the time-dependent close-coupling method [M.S. Pindzola et al., J. Phys. B 40, R39 (2007)], which also described the application to multi-electron targets.

A variational approach to the Dirichlet boundary condition: application to confined H−, He and Li

The Rayleigh–Ritz variational approach is widely used to study confined systems with a Dirichlet boundary condition. To exploit this approach, we first derive a general and a simple expression for the energy functional which takes into account the confinement boundary condition that the wavefunction vanishes on a defined surface. This simple functional allows a straightforward variational optimization of any trial function including a cut-off factor fulfilling the Dirichlet boundary conditions. To illustrate the method, we propose a three-parameter correlated wavefunction for confined two- and three-electron systems located at the centre of an impenetrable spherical cavity. In addition, we also carry out a study on the effect of dimensionality of the confinement by considering a helium atom located on the axis of a cylinder which is confined along the transverse direction but free along the axis of the cylinder resulting in a two-dimensional confinement. An exponential cut-off function is used that ensures the satisfaction of the Dirichlet boundary condition and the cusp conditions related to the regular singularities of the Coulomb field. This function is different from the frequently employed linear and quadratic forms and is suitable for both small and large values of the cavity radius rc. We employ this correlated wavefunction to perform variational calculations of ground-state energies of H− ion, He, and Li atoms. We find that the electron–electron and electron–nucleus interactions are accurately described by a Handy–Jastrow factor. The results show that the electron–electron interaction is almost independent of the radius of confinement rc, whereas the electron–nucleus interaction is strongly dependent on the confinement.

Compact and accurate variational wave functions of three-electron atomic systems constructed from semi-exponential radial basis functions

The semi-exponential basis set of radial functions (A.M. Frolov, Physics Letters A {\bf 374}, 2361 (2010)) is used for variational computations of bound states in three-electron atomic systems. It appears that semi-exponential basis set has a substantially greater potential for accurate variational computations of bound states in three-electron atomic systems than it was originally anticipated. In particular, the 40-term Larson's wave function improved with the use of semi-exponential radial basis functions now produces the total energy \linebreak -7.47805413551 $a.u.$ for the ground $1^2S-$state in the ${}^{\infty}$Li atom (only one spin function $\chi_1 = \alpha \beta \alpha - \beta \alpha \alpha$ was used in these calculations). This variational energy is very close to the exact ground state energy of the ${}^{\infty}$Li atom and it substantially lower than the total energy obtained with the original Larson's 40-term wave function (-7.477944869 $a.u.$).

Semi-exponential basis for highly accurate computations of three-electron atomic systems

An advanced basis set of semi-exponential radial functions is proposed and applied to highly accurate computations of three-electron atomic systems. This set is found to be significantly more effective in highly accurate, bound state computations of three-electron atomic systems, than currently used set of Hylleraas radial functions.

Hylleraas-configuration-interaction study of the2 S2ground state of neutral lithium and the first five excitedS2states

High-precision Hylleraas-configuration-interaction (Hy-CI) method variational calculations are reported for the $2\text{ }^{2}S$ ground state of neutral lithium. The nonrelativistic energy is calculated to be $\ensuremath{-}7.478\text{ }060\text{ }323\text{ }451\text{ }9\text{ }\text{hartree}$, demonstrating that the Hy-CI technique is capable of sub-nanohartree accuracy for three-electron systems. A Hylleraas expansion without linked products of odd powers of ${r}_{ij}$ gives $\ensuremath{-}7.478\text{ }060\text{ }323\text{ }452\text{ }\text{hartree}$, showing the relative unimportance of such terms for lithium at the nanohartree level of accuracy. Hy-CI calculations are also reported for the $3\text{ }^{2}S$, $4\text{ }^{2}S$, $5\text{ }^{2}S$, $6\text{ }^{2}S$, and $7\text{ }^{2}S$ lithium excited states.

Fundamental vibrational transitions of theH3e H4e+andL7iH+ions calculated without assuming the Born-Oppenheimer approximation and with including leading relativistic corrections

Very accurate variational calculations of the fundamental pure vibrational transitions of the $^{3}\text{H}\text{e}\text{ }{^{4}\text{H}\text{e}}^{+}$ and ${^{7}\text{L}\text{iH}}^{+}$ ions are performed within the framework that does not assume the Born-Oppenheimer (BO) approximation. The non-BO wave functions expanded in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance are used to calculate the leading relativistic corrections. Up to $10\text{ }000$ Gaussian functions are used for each state. It is shown that the experimental $^{3}\text{H}\text{e}\text{ }{^{4}\text{H}\text{e}}^{+}$ fundamental transitions is reproduced within $0.06\text{ }{\text{cm}}^{\ensuremath{-}1}$ by the calculations. A similar precision is expected for the calculated, but still unmeasured, fundamental transition of ${^{7}\text{L}\text{iH}}^{+}$. Thus, three-electron diatomic systems are calculated with a similar accuracy as two-electron systems.

Compact variational wave functions for bound states in three-electron atomic systems

The variational procedure to construct compact and accurate wave functions for three-electron atoms and ions is developed. The procedure is based on the use of six-dimensional Gaussoids written in the relative four-body coordinates r 12, r 13, r 23, r 14, r 24, and r 34. The nonlinear parameters in each basis function have been carefully optimized. Using these variational wave functions, we have determined the energies and other bound state properties for the ground 12 S-states in a number of three-electron atoms and ions. The three-electron atomic systems considered in this work include the neutral Li atom and nine positively charged lithiumlike ions: Be+, B2+, C3+, ..., Na8+, and Mg9+. Our variational wave functions are used to determine the hyperfine structure splitting and field shifts for some lithium-like ions. The explicit formulas of the Q −1 expansion are derived for the total energies of these three-electron systems.

Semiclassical calculations of the quadruple excited four-electron systems

Highly excited atoms acquire very large dimensions and can be present only in a very rarified gas medium, such as the interstellar space. Multiply excited beryllium-like systems, when excited to large principal quantum numbers, have a radius of r ? 10 ?. We examine the semiclassical spectrum of quadruple highly excited four-electron atomic systems for the plane model of equivalent electrons. The energy of the system consists of rotational and vibrational modes within the almost circular orbit approximation, as used in a previous calculation for the triply excited three-electron systems. Here we present numerical results for the beryllium atom. The lifetimes of the semiclassical states are estimated via the corresponding Lyapunov exponents. The vibrational modes relative contribution to the energy levels rises with the degree of the Coulombic excitation. The relevance of the results is discussed both from the observational and heuristic point of view.

Quadruply excited beryllium-like atoms – a semiclassical model

The semiclassical spectrum of quadruply highly excited four-electron atomic systems has been calculated for the plane model of equivalent electrons. The energy of the system consists of rotational and vibrational modes within the circular skeleton orbit approximation, as used in a previous calculation for the triply excited three-electron systems. The full dynamical analysis is carried out within the Hamiltonian theory, accounting for the inertial effects and the complete coupling between different degrees of freedom. Here we present numerical results for energy spectrum of the beryllium atom. The lifetimes of the semiclassical states are estimated via the corresponding Lyapunov exponents. The vibrational modes relative contribution to the energy levels rises with the degree of the Coulombic excitation.

Double autoionization of hollow-atom states

A time-dependent close-coupling method for three-electron atomic systems is formulated to calculate the double autoionization of hollow-atom states. Initial excited states are obtained by relaxation of the Schr\"odinger equation in imaginary time, while autoionization rates are obtained by propagation in real time. A 12-coupled-channels nonperturbative calculation on a three-dimensional radial lattice yields a double-autoionization rate for the $\mathrm{Li}(2{s}^{2}2p)\ensuremath{\rightarrow}{\mathrm{Li}}^{2+}(1s)+2{e}^{\ensuremath{-}}$ transition that that is somewhat smaller than earlier many-body perturbation theory calculations and in reasonable agreement with rates extracted from resonance profiles found in more recent $\ensuremath{\gamma}+\mathrm{Li}$ experiments.

Density functional studies on the hollow resonances in the Li-isoelectronic sequence (Z = 4–10)

Following from our work on triply excited hollow resonances in three-electron atomic systems, a density functional theory (DFT)-based formalism is employed to investigate similar resonances in the Li-isoelectronic series (Z = 4–10). A combination of the work-function-based local nonvariational exchange potential and the popular gradient plus Laplacian-included Lee–Yang–Parr correlation energy functional is used. The generalized pseudospectral method provides nonuniform and optimal spatial discretization of the radial Kohn–Sham equation. First, all the eight n = 2 intrashell states of B2+, N4+ and F6+ are presented, which are relatively less studied in the literature compared to the remaining four members. Then calculations are performed for the eight 2l2l′nl″ (3 ⩽ n ⩽ 6) hollow resonance series, namely 2s2ns 2Se, 2s2np 2Po, 2s2nd 2De, 2s2pns 4Po, 2s2pnp 4De, 2p2ns 4Pe, 2p2np 4Do and 2p2ns 2De, of all the seven positive ions. Next, as an illustration, higher resonance positions of the 2s2ns 2Se series are calculated for all the ions with a maximum of n = 25. The calculated excitation energies are in excellent agreement with the available literature data (for the n = 2 intrashell states the deviation is within 0.125% and excepting only one case, the same for the resonance series is well below 0.5%). With an increase in Z, the deviations tend to decrease. Radial densities are also presented for some of the selected states. The only result available in the literature for the lower resonances (corresponding to a maximum of n = 17) have been reported very recently. The n > 16 (17 for F6+) resonances are examined here for the first time. This gives a promising viable and general DFT scheme for the accurate calculation of these and other hollow resonances in many-electron atoms.

Three-Electron Systems in Quantum Nanostructures: Ground State Transition from a Spin-Doublet State to a Spin-Quartet State

The ground state and the lowest excited state of three electrons confined in quantum structures (quantum dots) are calculated. In general, the ground state is a spin-doublet one in a small dot, level crossing occurs as the dot size increases, and the ground state is a spin-quartet state in a large dot. It is found that the spin-quartet ground state is more likely to be realized in a quantum ring (QR) than in a harmonic confinement. In a narrow QR, two-electron exchanges are unlikely to occur because of the large Coulomb barrier while three-electron exchanges are little suppressed, leading to the fully spin-polarized (spin-quartet) ground state. The present calculation conforms to Herring's theorem that the ground state of three spin-1/2 fermions on a one-dimensional ring is a fully spin-polarized one if the repulsive interaction between fermions is sufficiently strong. The extensions of the present result to more complicated cases are also discussed.

Highly accurate evaluation of atomic three-electron integrals of lowest orders.

Calculations of three-electron atomic systems in Hylleraas coordinates require integrals involving all the interparticle distances r(ij), which have usually been evaluated by introducing series expansions. For integrals with the smallest powers of r(ij) these expansions do not converge at a satisfactory rate, leading some investigators to introduce convergence-acceleration procedures. This paper recommends the alternative of evaluating these integrals in closed form and presents stable explicit formulas for so doing. Some of the formulas are more compact versions of those in the literature; others have not been previously reported. It is also shown that finite-difference methods can be used with advantage to obtain additional low-order integrals. Sample integral values have been provided for test purposes.

Triply excited three-electron systems--semiclassical model

We calculate rovibronic intrashell spectra of the triply highly excited atomic hydrogen dianion, helium anion and lithium atom, within a simple semiclassical model. Zero-order electronic energy levels and half-lives are calculated for a number of principal quantum numbers and approximate thresholds for the appearance of vibronic modes are estimated. Since no quantum-mechanical and experimental data are available for the highly excited levels (N > 5), where the semiclassical models apply, no comparison with other results are possible at present. The problem of comparing semiclassical and quantum-mechanical calculations for moderately large quantum numbers, which seem attainable by the present day experimental technique, has been discussed.

Energy levels and classifications of triply excited states of N 4+, O 5+, F 6+, and Ne 7+

We report and tabulate the energies, classifications, effective quantum numbers, and configuration mixings of the triply excited 2,4S e,o, 2,4P e,o, 2,4D e,o, and 2,4F e,o states of three-electron systems from Z=7–10, namely, N 4+, O 5+, F 6+, and Ne 7+. For all states, no electron is in the 1s state. Our results are based on calculations using the truncated diagonalization method (TDM) with hydrogenic basis functions. Where available, we compare our results with other calculations.

Hyperspherical angular adiabatic separation for three-electron atomic systems

The hypothesis of treating three-electron systems as a two-electron core plus a bound electron in the hyperspherical adiabatic approach introduces a second adiabatic separation into hyperangular equations. Differently from the main radial-angular hyperspherical adiabatic separation, the resulting couplings are necessary to guarantee permutational and rotational invariance of the Hamiltonian. Thus, any kind of approximation, disregarding such couplings, represents a loss of symmetry. This paper explores the consequences of such approximations in the potential curve calculations for the lithium atom, showing that this symmetry breaking is quite smooth and can be recovered within a good precision adding few couplings to the system of angular differential equations.

Symmetric rotor of lithiumlike hollow atoms

We present an explicit analytical construction of highly correlated triply excited hollow states in Coulombic three-electron systems. The construction is based on the simple idea that the three electrons will orient themselves in maximally polarized Stark states along three different axes with 120\ifmmode^\circ\else\textdegree\fi{} between them to minimize their mutual repulsion. This ansatz for the three-electron state, supplemented with an external rotational average and proper antisymmetrization, produces configuration-interaction mixing coefficients in good agreement with data obtained from ab initio calculations. Our ansatz leads to a simple and fairly accurate Z-scaling relation for energy splittings and dipole matrix elements along the lithium isoelectronic sequence. Despite our prediction of isomorphic correlation patterns and corresponding quasigood quantum numbers for L series of states, there is no numerical evidence that energies are well represented by rigid-rotor rotational spectra. We explain this by the explicit failure in multiply excited atoms of the nuclear and molecular physics theories of rotational structure.

A posteriori corrections to systematic failures of standard density functionals: The dissociation of two-center three-electron systems

The method we proposed recently [J. Chem. Phys., 114, 1447 (2000)] to a posteriori correct the unphysical dissociation behavior of radical homonuclear diatomic cations obtained in density functional theory calculations has been enlarged to nonsymmetric three-center two electrons systems. This approach, which is derived from Slater’s transition state technique, allows to remove most of the self-interaction energy error contained in the current exchange functionals. It has been shown that this is the main contribution to the overestimation of the bonding energy of systems with delocalized charges. Although approximate, the method yields a better agreement with experimental bonding energies than more sophisticated methods.

Correlated electrons in lithiumlike hollow atoms.

We present a scheme for the explicit construction of highly correlated triply excited hollow states in Coulombic three-electron systems. Our analytical ansatz for the three-electron state is physically well justified and it produces configuration mixing coefficients in good agreement with data obtained from ab initio calculations.