Two-electron self-energy contribution to the ground-state energy of helium-like ions

The two-electron self-energy contribution to the ground-state energy of helium-like ions is calculated both for a point nucleus and an extended nucleus in a wide interval of Z. All the two-electron contributions are compiled to obtain most accurate values for the two-electron part of the ground-state energy of helium-like ions in the range Z = 20–100. The theoretical value of the ground-state energy of ²³⁸U⁹⁰⁺, based on currently available theory, is evaluated to be −261382.9(8) eV, without higher-order one-electron QED corrections.     

Ground-state energy of uranium diatomic quasimolecules with one and two electrons

Ground-state energies of the one- and two-electron uranium dimers are calculated for internuclear distances in the range D=40–1,000 fm and compared with the previous calculations. The generalization of the dual-kinetic-balance approach for axially symmetric systems is employed to solve the two-center Dirac equation without the partial-wave expansion for the potential of two nuclei. The one-electron one-loop QED contributions (self-energy and vacuum polarization) to the ground-state energy are evaluated using the monopole approximation for the two-center potential. Interelectronic interaction of the first and second order is taken into account for the two-electron quasimolecule. Within the QED approach, one-photon-exchange contribution is calculated in the two-center potential, whereas the two-photon-exchange contribution is treated in the monopole approximation.     

Construction of the two-electron contribution to the Fock matrix by numerical integration

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10^?12 E _h using the numerical approach.     

Horizontal rolls in convective flow above a partially heated surface

Considering the nonlinearity arising from the interaction between electrons and lattice vibrations, an effective electronic model with a self-interaction cubic term is employed to study the interplay between electron-electron and electron-phonon interactions. Based on numerical solutions of the time-dependent nonlinear Schroedinger equation for an initially localized two-electron singlet state, we show that the magnitude of the electron-phonon coupling χ necessary to promote the self-trapping of the electronic wave packet decreases as a function of the electron-electron interaction U. We show that such dependence is directly linked to the narrowing of the band of bounded two-electron states as U increases. We obtain the transition line in the χ × U parameter space separating the phases of self-trapped and delocalized electronic wave packets. The present results indicates that nonlinear contributions plays a relevant role in the electronic wave packet dynamics, particularly in the regime of strongly correlated electrons.     

Spin-orbit matrix elements for internally contracted multireference configuration interaction wavefunctions

An efficient method for the calculation of Breit-Pauli spin-orbit matrix elements for internally contracted multireference configuration interaction wavefunctions is presented. Instead of taking all two-electron contributions of the wavefunction explicitly into account, the most important two-electron contributions of the spin-orbit operator are incorporated by means of an effective one-electron Fock operator. As a further refinement, explicit two-electron contributions can be reinstated for the dominant all-internal parts of the wavefunctions.     

A one-electron variant of direct perturbation theory for the treatment of scalar-relativistic effects

ABSTRACT

The different importance of scalar-relativistic two-electron contributions in second-order direct perturbation theory (DPT2) and the spin-free one-electron variant of exact two-component theory (SFX2C-1e) is analysed. The apparent discrepancy is traced back to the fact that SFX2C-1e is not ignoring the total DPT2 two-electron contribution rather just a so-called commutator term which originates from a rewrite of the small-component density (matrix) in terms of the related, though different kinetic-energy density (matrix). This commutator term is shown to be significantly smaller (10% and less) and to have, unlike the total DPT2 two-electron contribution, a negative sign. Based on our findings, we propose a one-electron variant of direct perturbation theory, referred to as DPT-1e, and report on its implementation for the computation of energies and first-order properties at the second-order level, i.e. DPT2-1e. Numerical results are presented for the hydrogen halide series HX, X=F, Cl, Br, I, and At, as well CuF and CuCl in order to investigate its performance in comparison to DPT2 and SFX2C-1e.     

Bound state solutions of the two-electron Dirac-Coulomb equation

We present a variational method for solving the two-electron Dirac-Coulomb equation. When the expectation value of the Dirac-Coulomb Hamiltonian is made stationary for all possible variations of the different components of a well-behaved trial function one obtains solutions representative of the physical bound state wave functions. The ground state wave function is derived from the application of a minimax principle. Since the trial function remains well-behaved, the method remains safe from the twin demons of variational collapse and continuum dissolution. The ground state wave function thus derived can be interpreted as a linear combination of different configurations. In particular, the admixing of intermediate states having one (two) electron(s) deexcited to a negative-energy orbital (orbitals) contributes a second-order level shiftE ₀₋ ⁽²⁾ which can be identified with the second-order shift due to the Pauli blocking of the production of one (or two) virtual electron-positron pair(s). Thus the minimax solution corresponds to the renormalized ground state in quantum electrodynamics, with deexcitations to negative-energy orbitals taking the place of the avoidance of virtual pairs. If one extends the relativistic configuration interaction (RCI) treatment by additionally including negative-energy and mixed-energyeigenvectors of the Dirac-Hartree-Fock hamiltonian matrix in the two-electron basis, the calculated energy will be shifted from the conventional RCI value by an amount that is much smaller thanE ₀₋ ⁽²⁾ . For two-electron atoms, we have derived expressions for the all-spinor limit (δE) and thes-spinor limit (δE _s) of this shift in leading orders. The all-spinor limit (δE) is of orderα ⁴ Z ^{4 1/3} whereas thes-spinor limit (δE _s) is of orderα ⁴ Z ^{3 2/3}. leading components are related to the 1-pair component ofE ₀₋ ⁽²⁾ in a simple way, and the relationships offer the possibility of computing energy due to virtual pairs. Numerical results are discussed.     

Continuation of the Schrödinger Equation for Higher Angular-Momentum States to D Dimensions and Interdimensional Degeneracies

The application of the techniques of dimensional scaling, and in particular the 1/D expansion, to higher angular-momentum states of multi-electron atoms requires the generalized Euler angles, which multiply with increasing D to be “factored out” of the wave function. The factorization must be performed in a way that produces from the Schr?dinger equation a tractable set of differential equations which admit continuation in the dimension D. In two recent works the authors have achieved the necessary factorization of the wave function by generalizing the Schwartz expansion to N electrons in D dimensions. The present paper applies the N-electron D-dimensional Schwartz expansion to the two-electron problem in D dimensions. The resulting set of coupled differential equations in the internal variables admit continuation in D, enabling the methods of dimensional scaling to be applied to higher-angular-momentum states. In addition, the coupled differential equations clearly show the complete spectrum of exact interdimensional degeneracies of the two-electron system. Received November 6, 1995; accepted in final form February 10, 1996     

Molecular relativistic electric field gradient calculations suggest revision of the value of the nuclear electric quadrupole moment of 127I

Relativistic ab initio methods are used to compute the electric field gradient at the iodine nucleus in nine different closed-shell diatomic molecules. Combining these theoretical electric field gradients with experimental nuclear quadrupole coupling constants gives a consistent value of the nuclear quadrupole moment of ¹²⁷I of—696(12)millibarn. We argue that this value is more precise than the current standard value of the nuclear quadrupole moment of ¹²⁷I and recommend adjusting the reference value accordingly. The precision of this determination is still determined by technical limitations in the theoretical work, in particular the neglect of the two-electron Gaunt interaction in the Hamiltonian and correlation contributions beyond those described at the CCSD(T) level of theory, but the errors are reduced relative to the theoretical work that underlies the current standard value of this nuclear quadrupole moment. As a secondary study we also considered the calculation of the small electric field gradient at the gold nucleus in the AuI molecule and conclude that this computation remains a challenge for theoreticians.     

Self-trapping of interacting electrons in crystalline nonlinear chains

Considering the nonlinearity arising from the interaction between electrons and latticevibrations, an effective electronic model with a self-interaction cubic term is employedto study the interplay between electron-electron and electron-phonon interactions. Basedon numerical solutions of the time-dependent nonlinear Schroedinger equation for aninitially localized two-electron singlet state, we show that the magnitude of theelectron-phonon coupling χ necessary to promote the self-trapping of theelectronic wave packet decreases as a function of the electron-electron interactionU. We show that such dependence is directly linked to the narrowing ofthe band of bounded two-electron states as U increases. We obtain thetransition line in the χ × U parameter space separatingthe phases of self-trapped and delocalized electronic wave packets. The present resultsindicates that nonlinear contributions plays a relevant role in the electronic wave packetdynamics, particularly in the regime of strongly correlated electrons.     

Tests of quantum electrodynamics using ion traps

Experiments in ion traps on the g factors for the free and the bound electron in low-Z, hydrogen-like ions have provided the most accurate tests of quantum-electrodynamics calculations. Moreover they have been used to determine new and precise values for fundamental constants. Extensions to more stringent tests using ions of higher values of the nuclear charge Z are on the way. Also other QED tests such as Lamb shifts or hyperfine structures in H-like ions using traps will be feasible in the near future. The tests in bound systems, however, will be limited by nuclear structure effects which are difficult to calculate. Assuming the QED calculations as correct, the experimental results may be used to determine nuclear contributions and thus support nuclear models. Contribution presented at the TCP06, Vancouver Island, 2006.     

An expansion method for calculating molecular properties IV.

The energies of the 2pσ and 2sσ states of H₂ ⁺ and HeH²⁺ and those of the a ³, A ¹, b ³, B ¹Σ⁺ states of HeH⁺ are calculated up to second order in the heavy nuclear charge. For the one-electron systems the first-order equation is solved analytically employing a special separation technique for degeneracies. The consideration of screening enables a recently proposed interchange result for the degenerate case to be tested. The results for the 2sσ states are good while those for the 2pσ states are only good at small internuclear distances.

For the two-electron systems the method employed is a single perturbation approach for two perturbations in the presence of degeneracy. The results for HeH⁺ appear to be good only at small internuclear distances. The corresponding results for the excited states of H₂ are omitted since an interesting problem arises at first order.     

Majorana Solutions to the Two-Electron Problem

The two-electron atom is the simplest nontrivial quantum system not amenable to exact solutions. Today, its relevance in the development of quantum mechanics and its pedagogical value within the realm of atomic physics are widely recognized. In this work, an historical review of the known different methods and results devised to study such a problem is presented, with an emphasis to the calculations of the ground state energy of helium. Then we discuss several, related, unpublished results obtained around the same years by Ettore Majorana, which remained unknown till recent times. Among them a general variant of the variational method appears to be particularly interesting, even for current research in atomic and nuclear physics: it takes directly into account, already in the trial wavefunction, the action of the full Hamiltonian operator of a given quantum system. Further relevant contributions, specialized to the two-electron problem, include the introduction of the remarkable concept of an effective nuclear charge different for the two electrons (thus generalizing previous known results) and an application of the perturbative method, where the atomic number Z was treated effectively as a continuous variable. Finally a survey of results, relevant mainly for pedagogical reasons, is given; in particular we focus on simple broad range estimates of the helium ionization potential, obtained by suitable choices for the wavefunction, as well as on a simple alternative to Hylleraas’ method, which led Majorana to first order calculations comparable in accuracy with well-known order 11 results derived, in turn, by Hylleraas.     

Theoretical study of the fine-structure coupling constants in the 2p 3Π u state of H2

The one and two-electron fine-structure constants for the 2p ³Π _u state of the H₂ molecule have been calculated using all-integral, ab initio methods for a variety of molecular wavefunctions. The results have been averaged over the first three vibrational states and are compared with previous calculations and with experiment.     

On highly accurate calculations of the excited n<Superscript>1</Superscript>S(L = 0)-states in helium atoms

The total energies and various bound state properties of the excited 2¹S(L = 0)-states in two-electron helium atoms, including the ^￥He^{\infty}\rm He, ⁴He^4\rm He and ³He^3\rm He atoms, are determined to very high numerical accuracy. The convergence of the results obtained for some electron-nuclear and electron-electron expectation values and, in particular, for the electron-nuclear and electron-electron cusp values, is discussed. The field component of the isotope shift and lowest order QED correction are estimated for the 2¹S(L = 0)-states in the ⁴He and ³He atoms. We also apply our highly accurate methods to numerical computations of the excited n ¹S-states (for n = 3 and 4) in two-electron atomic systems.     

Polarization analysis in soft X‐ray diffraction to study magnetic and orbital ordering

An experimental approach to the analysis of charge, magnetic and orbital ordering in 3d transition‐metal oxides is presented. The technique combines two important components: azimuthal rotations around the Bragg wavevector and polarization analysis of the Bragg intensities in the range 500–900 eV. The polarization analysis is performed using graded multilayers, which are translated and rotated in the vacuum chamber. It is shown why these two components are important to determine the origin of the Bragg scattered signals and how they allow us to separate the different contributions. Examples are given for the oxygen K and the Mn, Co, Ni and Cu L_2,3‐edges, and the advantages and drawbacks of this experimental technique are discussed.     

A simple model potential description of the alkaline earth isoelectronic sequences

We have developed a simple model potential with a hard core and the correct large-r Coulombic behaviour, to describe the interaction of an electron with a closed shell. One has an exact, analytic ground state wave function for this potential. This potential is used to develop two-electron perturbed and unperturbed wave functions, with the correct asymptotic behaviour and cusp conditions. These wave functions allow us to obtain accurate values for the two-electron energies, polarisabilities, hyperpolarisabilities, and dispersion coefficients of alkaline earth sequences. Many of these results are the only ones available in the literature. Received 29 July 1999 and Received in final form 16 November 1999     

On the group function model for molecular electronic structure

A group function model for the electronic structure of molecules is developed in which the strong orthogonality condition is not necessarily invoked. The group functions are required to satisfy orthogonality constraints which are less restrictive than the strong orthogonality condition. The particular condition which two group functions obey reflects the extent to which the electrons they describe are localized in well-separated regions of space and the degree of interaction between them. This will be called the interacting group function model. General expressions for the expectation values of one and two-electron spin-independent operators are derived using the spin-coupled formalism. A graphical technique is found to be useful when the molecule is divided into several subsystems.