Helium Atom

We report experimental results on correlated double ionization of magnesium (Mg) by near-infrared (0.8and 1.03-μm) circularly polarized laser fields. With 0.8 μm, we confirm the recollision interpretation of the observed "knee" structure of Mg [Gillen et al., Phys. Rev. A 64, 043413 (2001)], even though the experiments are performed in the multiphoton regime. Our experiments show that, even in the multiphoton regime, which is normally thought to be a purely quantum-mechanical territory, some ionization phenomena can still be understood classically.

The operator method is used to construct the solutions of the problem of the polaron in the strong coupling limit and of the helium atom on the basis of the Hartree-Fock equation. E 0 = -0.108 512 8052α 2 is obtained for the polaron ground-state energy. Energies for 2s and 3s states are also calculated. The other excited states are briefly discussed.

method, or the Gear version for stiff equations . These routines work with vector solutions. A precise method for solving systems of coupled ordinary differential equations of second order in one variable is presented. The In this paper we present an algebraic method of integrating method consists mostly of algebraic manipulations and is very effisystems of coupled differential equations with a regular singular cient on vector computers. The method is applied to the solution point at the origin. The method represents the solution on small of the three-body Schro ¨dinger equation. Besides giving, in contrast intervals of the independent variable, z, by matrix Taylor series; to variational methods, uniformly precise expectation values of opon the first interval, at z ϭ 0, a modified series in z is used. erators including the Hamiltonian, the method allows one to study the analytic structure of the wave function. Applications to the He The coefficients of the ODE are expanded in matrix Laurent atom, the muonic helium atom, and the Ȑdt molecular ion are preseries around z ϭ 0. The solution on each interval is obtained via sented. No extended precision intermediate calculations are rerecurrence relations. A small number of powers of z is required. quired.

Degenerate perturbation theory is used to study dipole susceptibilities of an excited helium atom in an external electric field. The dependence of the perturbed energy of levels in atoms on fine-structure effects and on the higher-order Stark effect is investigated. Numerical results have been obtained for the (1s3p) 3 P 0 and (1s3p) 3 P 2 states of helium. The magnitude of the electric field and the energy separation at the 0 Ϫ ϫ0 Ϫ anticrossing are calculated. Calculations of polarizabilities and hyperpolarizabilities are carried out using sums of oscillator strengths and, alternatively, with the excited electron Green function. An estimate is given based on the model potential method for the contribution of an infinite series over the bound states, including the integral over the continuum, for second-and higher-order matrix elements. The relativistic approach for evaluating reduced dipole matrix elements based on the relativistic no-pair Hamiltonian and including both the Coulomb and Breit interactions ͑configuration-interaction method͒ is analyzed. ͓S1050-2947͑99͒06108-9͔

Transition moments for the strongly allowed (ZIP, 1's) and for the (23P, 23S) transitions in the helium isoelectronic sequence are computed from tenth-order perturbation wave functions using the length, velocity, and acceleration forms of the requisite operator. In addition, more limited results for transitions from NP states are also considered. The internal consistency as well as the absolute values of the results are in satisfactory agreement with the results of C. L. Pekeris and coworkers based on totalvariational wave function calculations, and hence in agreement with experiment. Since the Z3P and 23S states are

We propose the generation of the recently discovered Langmuir Trojan states [1] in helium atom by the adiabatic sequence of electric, magnetic and electromagnetic fields turn-ons. First the Trojan wavepacket is generated from one electron originally in the circular state leaving the He+ ion core by the adiabatic rapid passage during Zener tunneling. Next, the plane of its circular motion is adiabatically shifted by the static Stark field turn on parallel to the magnetic field and than the second Trojan wavepacket from Helium He+ ion is generated but to much smaller off resonant orbit not much to influence the first one. Later the ``two color'' CP field is adiabatically adjusted by the frequency chirping to one frequency while turning off the symmetry breaking electric Stark field and turning on the magnetic field to stabilize the orbits at the same time. Numerical simulations using the time dependent Hartree method reducing the problem two Schrödinger equations in three dime...

The symmetric eikonal distorted wave method is extended to account for two-electron atom excitation by ion impact. In this formulation ͑SE2͒, the interaction between the projectile and each one of the two target electrons is taken into account in the initial and final wave functions by using the eikonal approximation. The active electron notion is not employed as both electrons are treated the same. When single configuration target wave functions are used, the SE2 transition matrix element can be expressed as a convolution of one-electron transition matrix elements. The theory is applied to the study of helium excitation by proton and highlycharged-ion impact. Experimental total and differential cross section data are well reproduced.

A quantum model of the Thomson helium atom is considered within the framework of stationary perturbation theory. It is shown that from a formal point of view this problem is similar to that of two electron states in a parabolic quantum dot. The ground state energy of the quantum Thomson helium atom is esti mated on the basis of Heisenberg's uncertainty principle. The ground state energies obtained in the first order of perturbation theory and qualitative estimate provide, respectively, upper and lower estimates of eigenvalues derived by numerically solving the problem for a quantum model. The conditions under which the Kohn the orem holds in this system, when the values of resonance absorption frequencies are independent of the Cou lomb interaction between electrons, are discussed.

The convergence behavior of symmetry-adapted perturbation theory (SAPT) expansions is investigated for an interacting system involving an excited, open-shell monomer. By performing large-order numerical calculations for the interaction of the lowest, 1s2s, triplet state of helium with the ground state of the hydrogen atom we show that the conventional polarization and symmetrized Rayleigh-Schrödinger expansions diverge in this case. This divergence is attributed to the continuum of intruder states appearing when the hydrogen electron is falling on the helium 1s orbital and the 2s electron is ejected from the interacting system. One of the dimer states resulting from the interaction becomes then a resonance, which presents a hard case to treat by a perturbation theory. We show that the SAPT expansions employing the strong symmetry-forcing procedure, such as the Eisenschitz-London- Hirschfelder-van der Avoird or the Amos-Musher theories, can cope with this situation and lead to conver...

A dramatic electric field dependence has been observed in the fluorescence yield spectrum of the doubly excited states in helium, where a rich phenomenology is encountered below the N 2 threshold. Fluorescence yields of certain states can be tuned to zero, while other dipole-forbidden states are significantly enhanced, for fields much weaker than 1 kV=cm. Using an R-matrix multichannel quantum defect theory, spherical-to-parabolic frame transformation method, we are able to reproduce the main features of the observed spectrum, and we discuss the qualitative behavior in terms of weak electric field mixing.

A dramatic electric field dependence has been observed in the fluorescence yield spectrum of the doubly excited states in helium, where a rich phenomenology is encountered below the N 2 threshold. Fluorescence yields of certain states can be tuned to zero, while other dipole-forbidden states are significantly enhanced, for fields much weaker than 1 kV=cm. Using an R-matrix multichannel quantum defect theory, spherical-to-parabolic frame transformation method, we are able to reproduce the main features of the observed spectrum, and we discuss the qualitative behavior in terms of weak electric field mixing.

We discuss the application of perturbation theory to a system of particles confined in a spherical box. A simple argument shows that the particles behave almost independently in sufficiently strong confinement. We choose the helium atom with a moving nucleus as a particular example and compare results of first order with those for the nucleus clamped at the center of the box. We provide a suitable explanation for some numerical results obtained recently by other authors.

We aim to show the importance of non-elastic excitation and deexcitation processes in He * (n) + He(1s 2 ) collisions with the principal quantum number n ≥ 3 for helium-rich white dwarf atmospheres. We compare the efficiencies of these processes with those of the known non-elastic electron-He * (n) atom processes in the atmospheres of some DB white dwarfs. We show that in significant parts of the considered atmospheres, which contain weakly ionized layers (the ionization degree 10 -3 ), the influence of the studied atom-Rydberg atom processes on excited helium atom populations is dominant or at least comparable to the influence of the concurrent electron-He * (n)-atom processes.

The Schrödinger equation is a linear partial differential equation that describes the temporal evolution of the state of a quantum mechanical system. One of the pillars of quantum mechanics, the Schrödinger equation has been used to describe a wide variety of phenomena such as wave-particle duality, atomic orbitals, quantum oscillations, and even some fluid mechanics problems. Despite its established importance in the development of quantum mechanics, a solution of the Schrödinger equation for an arbitrary distribution of quantum mechanical potential or for an arbitrary number of space dimensions remains unknown. This work seeks to rectify this theoretical roadblock by providing an analytical solution through a recasting of the well-known Cole-Hopf transformation which is used to linearize a broad class of PDEs of which the Schrödinger equation becomes a member after an earlier change of variables. These two successive transformations render the general wave function proportional to the solution of a forced heat equation, allowing an explicit analytical solution applicable to any system in any number of space dimensions. Two practical applications to deriving the explicit wave functions of multielectronic atoms and to the forced Burgers equation of fluid mechanics are discussed, enabling greater mathematical insight into quantum entanglement and nonlinear fluid systems.

An implementation of the Hartree-Fock Roothaan with six expansion terms of Gaussian Type Orbitals (GTO-6G) is described and used to study the Helium atom's ground state accurately. The objective of this research is to calculate the ground state energy of the Helium atom. The analysis was conducted using the numerical method using Matlab 2017b. By using the Hartree-Fock Roothaan method, the complexity of Hartree-Fock in integrating the differential equations of eigenvalues for each electron is repeatedly successfully overcome by expressing Hartree-Fock orbitals in the form of linear combinations, known as STO (Slater Type Orbitals) and GTO (Gaussian Type Orbital). The Hartree-Fock Roothaan approximation procedure begins by assigning an initial guess value to the elements of the density matrices and then constructing the first Fock matrix from these matrices. The calculation is performed using the various iterations in multiples of 50, show that the error value expressed in relative uncertainty is getting smaller at the 150 th iteration, which is around 1.84%. Compared with some literature, the relative uncertainty value is still within tolerance (below 5%). The contribution of theoretical implications in this study can be used as input for other researchers to review the Hartree-Fock Roothaan method and improve accuracy. This is an open-access article under the CC-BY-SA license.

The method derived recently for solving a multidimensional scattering problem is applied to a three-dimensional SchrSdinger equation. As compared with (,direct,, numerical methods of finite elements and finite differences, this approach gives sufficiently accurate upper and lower approximations to the helium-atom binding energy, which demonstrates its efficiency.

Employing two nondestructive methods, Fourier-transform infrared spectroscopy (FTlRS) and helium atom scattering (HAS), the adsorbate C~/NaCl(ODl) was investigated in the frequency range from < 10 cm-' to 5000 cm'. The 'internal' vibrations v2, +, v3 +2v2 and v3 +yl were studied using FTlRS. The v2 and v3 are split due to a correlation field between two molecules in each adsorbate unit cell. In addition the degeneracy of the v2 bending vibrations is lifted on the surface, yielding a quartet absorption. Outside the experimental resolution the combination vibrations ~3 + 2~2 and v3 + vl are not split, as expected from their low absorption cross section. The tempemture dependence of the v3 doublet is explaiued in terms of small changes of the adsorption geometry, possibly due to the excitation of adsorbate phonons ('external' modes, i.e. vibrations of the molecules in the adsorption potential). In HAS experiments these phonons were studied. Below 80 cni' a total of eight different modes was observed, some of which could be followed across the whole Brillouin zone. Above 80 cm-' several overtone/multiphonon excitations were also observed.

In order to investigate the effect of a medium with dissipation and dispersion and also the curvature of the physical space on the properties of the incident quantum states, we use the quantization of electromagnetic field based on phenomenological approach to obtain input-output relations between radiations on both sides of dielectric slab. By using these relations the fidelity, the Wigner function, and also the quantum correlation of the outgoing state through dielectric slab are obtained for a situation in which the rightward incident state is a nonlinear coherent state on a sphere and the leftward incident state is a vacuum state. Here, the incident states are considered monochromatic and the modeling of the medium is given by the Lorentz&#39; model. Accordingly, we study nonclassical properties of the output states such as the quantum entanglement. It will be observed that the nonclassical properties of the outgoing states depend strongly on the optical property of the medium a...

The gyromagnetic ratios of levels (the g-factors)-are one of the most important characteristics of atoms. There are no corresponding experimental data in the literature for npn`l configurations of carbon atom. That is why the theoretical study of the fine and the Zeeman structures for the determination of the g-factors is ongoing. All the calculations are effected in one configuration approximation, with the energy operator matrix, in which the maximum possible number of interactions is taken into account, including the magnetic, spin-orbit (own and other) and spin-spin interactions. The fine structures were studied in three approximations (LS, LK, jK) for the establishment of the character of the coupling in 2p5f C I configuration. During the study of Zeeman splitting (except the g-factors) features of it were determined: the fields of crossings and anticrossings of magnetic components. In all steps of calculations the numerical digitalization of corresponding energy operator matrices were effected, e.g. the results presented in the paper were obtain in the intermediate coupling approximation.

Quantal calculations for scattering of ground-state antihydrogen by metastable ͑n =2S͒ helium atoms have been performed using the nonadiabatic, atomic orbital expansion technique at thermal energies. The zeroenergy elastic cross sections of the present systems are much greater than the corresponding value for the ground-state helium target. The low-energy elastic cross section for the singlet metastable helium ͓He͑2 1 S͔͒ target is higher than the corresponding value when the target is in the metastable triplet state ͓He͑2 3 S͔͒.

Quantal calculations for scattering of ground-state antihydrogen by metastable ͑n =2S͒ helium atoms have been performed using the nonadiabatic, atomic orbital expansion technique at thermal energies. The zeroenergy elastic cross sections of the present systems are much greater than the corresponding value for the ground-state helium target. The low-energy elastic cross section for the singlet metastable helium ͓He͑2 1 S͔͒ target is higher than the corresponding value when the target is in the metastable triplet state ͓He͑2 3 S͔͒.

Scattering of orthopositronium off helium target has been investigated using close-coupling method in the energy range 0-110 eV. Two basis sets, ͑a͒ Ps(1s)ϩHe(1s 2 , 1s2 1 s, 1s2 1 p) and ͑b͒ Ps(1s,2p) ϩHe(1s 2 , 1s2 1 s, 1s2 1 p), have been employed to find the scattering parameters. Low-order phase shifts, scattering length, and elastic and excitation cross section up to nϭ2 are reported using close-coupling approximation. Total cross section is also given by adding other partial cross section and compared with available measured data and existing theoretical predictions. Our total cross section at zero energy is very close to the theoretical prediction of Drachman and Houston ͓J. Phys B 3, 1657 ͑1970͔͒ and measured data of Canter et al. ͓Phys. Rev. A 12, 375 ͑1975͔͒. Present total cross section is in qualitative agreement with measured data of the University College London group in the energy range considered. In particular, present predictions are in good agreement with the UCL group in the energy range 20-30 eV. It has been found that elastic cross section is dramatically reduced, at zero or near zero energies, with the inclusion of target excitation in the expansion scheme. Moreover, ionization cross section of the Ps atom is found to be a major contributor to the total cross section above the ionization threshold.

We aim to show the importance of non-elastic excitation and deexcitation processes in He * (n) + He(1s 2) collisions with the principal quantum number n ≥ 3 for helium-rich white dwarf atmospheres. We compare the efficiencies of these processes with those of the known non-elastic electron-He * (n) atom processes in the atmospheres of some DB white dwarfs. We show that in significant parts of the considered atmospheres, which contain weakly ionized layers (the ionization degree 10 −3), the influence of the studied atom-Rydberg atom processes on excited helium atom populations is dominant or at least comparable to the influence of the concurrent electron-He * (n)-atom processes.

A helium-helium interatomic potential energy curve was determined from quantummechanical ab initio calculations. Very large atom-centred basis sets including a newly developed d-aug-cc-pV8Z basis set supplemented with bond functions and ab initio methods up to Full CI were applied. The aug-cc-pV7Z basis set of Gdanitz (J. Chem. Phys., 113, 5145 (2000)) was modified to be more consistent with the aug-cc-pV5Z and aug-cc-pV6Z basis sets. The diagonal Born-Oppenheimer corrections as well as corrections for relativistic effects were also calculated. A new analytical representation of the interatomic potential energy was fitted to the ab initio calculated values. In a following paper this potential model will be used in the framework of the quantum-statistical mechanics and of the corresponding kinetic theory to calculate the most important thermophysical properties of helium governed by two-body and three-body interactions.

A many-body procedure applied earlier to helium is utilized here to study the response of a neon atom in its ground state to a time-dependent electric field. The theory of the method, omitted in the helium paper for reasons of brevity, is included here. Our results for the frequency-dependent polarizabilities n (co) lead to refractive indices n(e), in excellent agreement with experiment. As a by-productof the calculation, we obtain [from the poles of n(~)] excitation energies for 2p-ns and 2p nd transitions in good agreement with experiment.

A helium-helium interatomic potential energy curve was determined from quantummechanical ab initio calculations. Very large atom-centred basis sets including a newly developed d-aug-cc-pV8Z basis set supplemented with bond functions and ab initio methods up to Full CI were applied. The aug-cc-pV7Z basis set of Gdanitz (J. Chem. Phys., 113, 5145 (2000)) was modified to be more consistent with the aug-cc-pV5Z and aug-cc-pV6Z basis sets. The diagonal Born-Oppenheimer corrections as well as corrections for relativistic effects were also calculated. A new analytical representation of the interatomic potential energy was fitted to the ab initio calculated values. In a following paper this potential model will be used in the framework of the quantum-statistical mechanics and of the corresponding kinetic theory to calculate the most important thermophysical properties of helium governed by two-body and three-body interactions.

With the help of the boost operator we can model the interaction between a weakly interacting particle(WIMP) of dark matter(DAMA) and an atomic nuclei. Via this &quot;kick&quot; we calculate the total electronic excitation cross section of the helium atom. The bound spectrum of He is calculated through a diagonalization process with a configuration interaction (CI) wavefunction built up from Slater orbitals. All together 19 singly- and doubly-excited atomic sates were taken with total angular momenta of L=0,1 and 2. Our calculation may give a rude estimation about the magnitude of the total excitation cross section which could be measured in later scintillator experiments. The upper limit of the excitation cross section is $9.7\cdot 10^{-8}$ barn.

In the presence of an environment of mobile charges, the bound-state Schrödinger Hamiltonian for an embedded He atom differs from its vacuum form. The central problem of incorporating screening in the nucleus-bound-electron and bound-electron-bound-electron terms of this Hamiltonian is investigated here for the He ground-state in a comparative manner by using two models, and the same product form of 1s-type parametric hydrogenic functions to perform exploratory variational calculations. Both models employ induced charge densities in the corresponding Poisson's equations with a fixed point-like nucleus, but the underlying charge-density response of the host system is generated by differently chosen perturbations. These are the point-charge nucleus and the nucleusbound-electron charge distribution as external perturbations. The repulsive bound-electron-boundelectron interaction in the Hamiltonian is modeled by a parametric Yukawa-type potential. Using the consistent variational results for the binding energies and wave functions, the charge-state dependent stopping power of a metallic target for slowly moving He is briefly discussed.

The effect of Debye plasma on the 1s2s 2 2 S resonance states in the scattering of electron from helium atom has been investigated within the framework of the stabilization method. The interactions among the charged particles in Debye plasma have been modelled by Debye-Huckel potential. The 1s2s excited state of the helium atom has been treated as consisting of a He + ionic core plus an electron moving around. The interaction between the core and the electron has then been modelled by a model potential. It has been found that the background plasma environment significantly affects the resonance states. To the best of our knowledge, such an investigation of 1s2s 2 2 S resonance states of the electron-helium system embedded in Debye plasma environment is the first reported in the literature.

Coherent control calculations are presented for helium. With the help of a genetic algorithm (GA) phase-modulated extreme ultra violet (XUV) laser pulses were controlled to maximize or minimize the non-resonant two-photon 1s1s → 1s3s excitation. Linearly polarized laser pulses were chosen at the frequency of 0.4 a.u. with a duration of about 15 fs and 10 14 Wcm −2 peak intensity. We verified the theory that an antisymmetric spectral phase around the two-photon resonance frequency maximizes the transition probability. Dark pulses which do not excite the system at all have however symmetric spectral phases around the two-photon resonance frequency.

Coherent control calculations are presented for helium. With the help of a genetic algorithm (GA) phase-modulated extreme ultra violet (XUV) laser pulses were controlled to maximize or minimize the non-resonant two-photon 1s1s → 1s3s excitation. Linearly polarized laser pulses were chosen at the frequency of 0.4 a.u. with a duration of about 15 fs and 10 14 Wcm −2 peak intensity. We verified the theory that an antisymmetric spectral phase around the two-photon resonance frequency maximizes the transition probability. Dark pulses which do not excite the system at all have however symmetric spectral phases around the two-photon resonance frequency.

The variational quantum Monte Carlo method was applied to investigate the ground states of the helium atom and helium like ions with atomic number from 1 to 10 and the first four excited states of the helium atom. Furthermore, the investigation of the ground state of helium, Li + , and Be 2+ in a confined impenetrable spherical box. Moreover, the calculation of the ground state of the helium atom in a strong magnetic field using four simple trial wave functions. The trial wave functions consist of usual orbital hydrogen wave functions multiplied by correlation function. Using four different correlation wave functions, we describe the interaction of the two electrons with each other and having a small number of variational parameters.

In paper (Flad and Harutyunyan in Discrete Contin Dyn Syst 420-429, 2011) is shown that the Hamiltonian of the helium atom in the Born-Oppenheimer approximation, in the case if two particles coincide, is an edge-degenerate operator, which is elliptic in the corresponding edge calculus. The aim of this paper is an analogous investigation in the case if all three particles coincide. More precisely, we show that the Hamiltonian in the mentioned case is a corner-degenerate operator, which is elliptic as an operator in the corner analysis.

Helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and represents a workbench for any many-body methodology. Here, we check the ab initio many-body GW approximation and Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from Hartree-Fock, we show that GW and BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving time-dependent Hartree-Fock. These findings suggest that the accuracy of BSE and GW approximations is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent density-functional theory.

High-order harmonic generation (HHG), attosecond pulse train (APT), and non-sequential double ionization (NSDI) in the He atom under high intense femtosecond laser pulses are calculated by time-dependent Schrodinger equation (TDSE) in one dimension (1D). By considering the mutual electron-electron and electron-nuclei interactions along with calculating the He atom ground state wave function by imaginary time propagation (ITP) method, besides calculating probability density of electrons, dipole acceleration, HHG, and APT, we could generate the well-known "knee structure" in the probability of the He atom ionization against the intensity in an ionization 2 boundary condition model. The results are in good agreement with the experimental data reported by Walker et al. [B. Walker et al. Phys. Rev. Lett. 73, 1227 (1994)].

The long-distance entanglement distribution is an important issue in quantum information processing. Our aim in this paper is to design a new protocol of quantum repeater by superconducting qubits to distribute entanglement between two superconducting qubits which are far from each other. So, we consider four pairs of entangled qubits ) 1 , ( + n n where 7 , 5 , 3 , 1 = n . Entanglement between each adjacent separable qubits (1,4) and (5,8) is created by performing interactions in the presence of an external magnetic field between the separable superconducting qubits (2,3) and (6,7) and operating proper measurement. Finally, the entanglement is distributed between two target qubits (1,8) by performing interaction between qubits (4,5) and operating proper quantum measurements. The entanglement is considered via concurrence and the success probability of the distributed entangled states is calculated. The behavior of the above two-mentioned parameters is periodic, however, they posses...

An efficient method of solving the three-body Schroedinger equation is presented. The wave function is decomposed into the product of a correlation factor describing the singularity and clustering structure, and a smooth factor expanded in hyperspherical harmonics. The application to the Helium atom yields a ground state energy of 2.9037244 (2.9033052) au for infinite (finite) nuclear mass. The convergence pattern shows that the accuracy of these values is better than a few parts in 14/1

The Schrodinger equation is solved directly for the ground state and excited 2 S state of the helium atom by using a rapidly convergent hyperspherical method which involves no adjustable parameters. The double and triple coalescence points are taken into account analytically. The center-of

Relativistic and finite-size corrections are calculated by using an accurate direct solution of the Schrodinger equation [the correlation function hyperspherical harmonic (CFHH) method] for the ground state of the helium atom. In the CFHH method the solution is a product of a correlation function and of a smooth factor expanded into hyperspherical harmonic functions. Given the proper correlation function, chosen from physical considerations, the method generates wave functions, accurate in the whole range of interparticle distances, which leads in turn to precise estimates of the relativistic operators. Our nonvariational results are compared with those obtained in recent variational calculations. PACS number(s): 31.30.Jv Recently [1,2] the two-electron helium Lamb shifts were measured very precisely for different states. Their values have shown very large deviations from the theoretical predictions obtained from sophisticated variational calculations by Drake and co-workers [3-7], which are correlated with the size of two-electron contributions. Determination of the Lamb shifts from experimental data is made by taking the difference between the theoretical energy, in which relativistic (non-QED) corrections were calculated perturbatively with the most accurate variational wave functions [3-7], and the measured wave number. An especially poor agreement of about twenty experimental uncertainties between theory and experiment, entirely due to the 2 'S Lamb shift, has been seen in the measurement of the 2'S-3'P transition [2]. The discrepancy is, however, reported to have been decreased by more precise calculation of the Bethe logarithm [8] and of quantum-electrodynamic corrections of order O(mc a 1na) to the electron-electron interaction that were evaluated for several states in Ref. [9]. Theoretically, variational wave functions are accurate only in the region where the probability density is high [10]. We still do not know, therefore, the correct analytical structure of three-body wave functions, since inclusion or omission of logarithmic terms, or negative powers of interparticle distances, has negligible effect on the value of the variational energy [11]. A variational function coincides with the precise one only on the average, and could wildly or even infinitely deviate from it locally [12]. The local discrepancies could lead to wrong estimates of expectation values of different operators that have significant contributions from the regions of the configuration space~here the deviations occur. This could result, in principle, in an error in the estimates of relativistic, finite size, and QED effects. e e e+ (also denoted Ps) consisting of particles of equal masses [18,24]. It was shown [15-24] that the direct bound state solution of the Schrodinger equation obtained by the CFHH method yields precision compara

Relativistic and QED corrections are calculated by using a direct solution of the Schrodinger equation for the 2 S excited state of the helium atom obtained with the correlation-function hypersphericalharmonic method. Our extremely accurate nonvariational results for relativistic, QED, and finite-size corrections coincide exactly (up to 0.00003 cm) with the values obtained in precision variational calculations of Drake [Nucl. Instrum. Methods Phys. Res. B 5, 2207 (1988)] and Baker, Hill, and Morgan [in Relativistic, Quantum Electrodynamic and Weak Interaction sects in Atoms, edited by Walter Johnson, Peter Mohr, aud Joseph Sucher, AIP Conf. Proc. No. 189 (AIP, New York, 1989), p. 123] for both infinite and finite nuclear masses. This confirms that a discrepancy of 0.0033 cm between theory and experiment is not a result of an inaccuracy of variational wave functions, but is rooted in our inadequate knowledge of the QED operators. A better understanding of the different QED contributions to the operators (such as, for example, a more precise estimate of the Bethe logarithm) is therefore needed to explain the discrepancy.

We have recently formulated an expansion of the N electron wavefunction in an appropriate set of harmonics on the 3 N-dimensional hypersphere. Angular correlation appears in the usual way, While radial correlation appears as a "generalized angular" correlation. Calculations on aS helium have been performed to explore the convergence of this expansion. Energies for various angular approximations have been compared with Bunge's angular limits and show a fractional error <3.5 x 10-*. A theoretical contraction procedure is shown to usefully reduce basis size without forfeiting accuracy

We put to the test an effective three-dimensional electrostatic potential, obtained effectively by considering an electrostatic source inside a (5+p)-dimensional braneworld scenario with p compact and one infinite spacial extra dimensions in the RS II-p model, for p = 1 and p = 2. This potential is regular at the source and matches the standard Coulomb potential outside a neighborhood. We use variational and perturbative approximation methods to calculate corrections to the ground energy of the Helium atom modified by this potential, by making use of a 6 and 39-parameter trial wave function of Hylleraas type for the ground state. These corrections to the ground-state energy are compared with experimental data for Helium atom in order to set bounds for the extra dimensions length scale. We find that these bounds are less restrictive than the ones obtained by Morales et. al. through a calculation using the Lamb shift in Hydrogen.

Persistent efforts in both theory and experiment have yielded increasingly precise understanding of the helium atom. Because of its simplicity, the helium atom has long been a testing ground for relativistic and quantum electrodynamic effects in few-body atomic systems theoretically and experimentally. Comparison between theory and experiment of the helium spectroscopy in 1s2p 3 PJ can potentially extract a very precise value of the fine structure constant α. The helium atom can also be used to explore exotic nuclear structures. In this paper, we provide a brief review of the recent advances in precision calculations and measurements of the helium atom.