Two-electron Contribution Research Articles

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration.

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

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

ABSTRACTThe 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.

Dressing Effects in the Attosecond Transient Absorption Spectra of Doubly-Excited States in Helium

SynopsisStrong-field manipulation of autoionizing states is a crucial aspect of electronic quantum control. Recent measurements of the attosecond transient absorption spectrum (ATAS) of helium dressed by a few-cycle visible pulse [C. Ott et al, Nature 516, 374 (2014)] provide evidence of the inversion of Fano profiles. With the support of accurate ab-initio calculations that reproduce the results of the latter experiment, here we investigate the new physics that arise from ATAS when the laser intensity is increased. In particular, we show that (i) previously unnoticed signatures of the dark 2p2 1S doubly excited state are observed in the experimental spectrum, (ii) inversion of Fano profiles is predicted to be periodic in the laser intensity, and (iii) the ac-Stark shift of the higher terms in the sp+2, n autoionizing series exceeds the ponderomotive energy, which is the result of a genuine two-electron contribution to the polarization of the excited atom.

Attractive electron–electron interactions within robust local fitting approximations

An analysis of Dunlap's robust fitting approach reveals that the resulting two-electron integral matrix is not manifestly positive semidefinite when local fitting domains or non-Coulomb fitting metrics are used. We present a highly local approximate method for evaluating four-center two-electron integrals based on the resolution-of-the-identity (RI) approximation and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair-atomic resolution-of-the-identity (PARI) approach, atomic-orbital (AO) products are expanded in auxiliary functions centered on the two atoms associated with each product. Numerical tests indicate that in 1% or less of all Hartree-Fock and Kohn-Sham calculations, the indefinite integral matrix causes nonconvergence in the self-consistent-field iterations. In these cases, the two-electron contribution to the total energy becomes negative, meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the standard Coulomb-metric RI method. The speed-up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple-zeta quality basis set. A positive semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky-decomposition techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calculations.

Local and nonlocal conductance enhancement due to Cooper pair splitting

Enhanced local conductance due to Andreev reflection is well known for high transparency Normal metal-Superconductor (NS) interface. For low transparency NS junctions, observation of two-electron tunneling contribution (enhanced Andreev reflection) to current was also reported previously. In our recent work [J Wei and V Chandrasekhar, Nat. Phys. 6, 494 (2010)], for a three-terminal Cooper pair splitter geometry, i.e., with two closely placed NS junctions sharing the same S terminal, we were able do a 2D scan of both local and nonlocal differential resistance, since for our ideal tunneling junctions there is little current redistribution (flow from one normal-metal lead to the other via the superconducting lead). In contrast to previous 1D nonlocal resistance measurements, 2D scans clearly show regime with pronounced contribution of the nonlocal processes to both local and nonlocal conductance enhancement. The enhanced local conductance and negative nonlocal resistance are consistent with enhanced Cooper pair splitting, and dynamical Coulomb blockade could be the origin of this enhancement.

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.

Electron-electron interaction in strong electromagnetic fields: the two-electron contribution to the ground-state energy in He-like uranium.

Radiative recombination transitions into the ground state of cooled bare and hydrogenlike uranium ions were measured at the storage ring ESR. By comparing the corresponding x-ray centroid energies, this technique allows for a direct measurement of the electron-electron contribution to the ionization potential in the heaviest He-like ions. For the two-electron contribution to the ionization potential of He-like uranium we obtain a value of 2248+/-9 eV. This represents the most accurate determination of two-electron effects in the domain of high-Z He-like ions, and the accuracy reaches already the size of the specific two-electron radiative QED corrections.

Semidirect parallel self-consistent field: the load balancing problem in the input/output intensive self-consistent field iterations

The full capacity of contemporary parallel computers can, in the context of iterative ab initio procedures like, for example, self-consistent field (SCF) and multiconfigurational SCF, only be utilized if the disk and input/output (I/O) capacity are fully exploited before the implementation turns to an integral direct strategy. In a recent report on parallel semidirect SCF http://www.tc.cornell.edu/er/media/1996/collabrate.html, http://www.fp.mcs.anl.gd/grand-challenges/chem/nondirect/index.html it was demonstrated that super-linear speedups are achievable for algorithms that exploit scalable parallel I/O. In the I/O-intensive SCF iterations of this implementation a static load balancing, however, was employed, dictated by the initial iteration in which integral evaluation dominates the central processing unit activity and thus determines the load balancing. In the present paper we present the first implementation in which load balancing is achieved throughout the whole SCF procedure, i.e. also in subsequent iterations. The improved scalability of our new algorithm is demonstrated in some test calculations, for example, for 63-node calculation a speedup of 104 was observed in the computation of the two-electron integral contribution to the Fock matrix.

X-ray transitions studied for decelerated bare and H-like uranium ions at the ESR electron cooler

Here we report on X-ray spectra induced by spontaneous capture of free electrons into decelerated bare- and hydrogen-like uranium ions which we measured recently at the cooler section of the ESR storage ring. The most intense lines observed in spectra can be attributed to direct transition of electrons into the K shell of the projectile ions and to characteristic L→K (Lyα) transitions. Radiative recombination lines into the K shell of bare and H-like uranium can be exploited for measuring the two-electron contribution to the ground state binding energy in helium-like uranium. The goal is to probe for high-Z ions bound-state QED corrections which are of the order of α2. Besides the dominant characteristic L→K transitions, the strongly reduced Bremsstrahlung (due to the low cooler voltage applied to the decelerated ions) allowed us to observe for the very first time RR transitions into the L shell as well as the balmer radiation located at the low-energy part of the spectra.

Spin–orbit coupling with model core potentials

A method has been proposed to compute one- and two-electron spin–orbit coupling using model core potentials generated with spin-free Hamiltonian without any need for separately producing either spin-dependent model core potentials or use of some fitted parameters, such as effective charges. The use of approximations (one-centre spin–orbit integrals and partial two-electron contribution) is verified to work within 2% accuracy compared to much costlier full results. The new method has been tested on atoms, hydrides, and homonuclear diatomic cations of P, As, and Sb. The accuracy has been found to be 3% or better within the tested set.

Application to argon ions of a new technique to measure the two-electron contribution to the ground state energy of helium-like ions

We have measured the two-electron contribution of the ground state energy of helium-like argon ions using an electron beam ion trap (EBIT). A two-dimensional map was measured showing the intensity of x-rays from the trap passing through a krypton-filled absorption cell. The independent axes of this map were electron beam energy and x-ray energy. From this map, we deduced the two-electron contribution of the ground state of helium-like argon. This experimentally determined value (312.4±9.5 eV) was found to be in good agreement with our calculated values (about 303.35 eV) and previous calculations of the same quantity. Based on these measurements, we have shown that a ten-day absorption spectroscopy run with a super-EBIT should be sufficient to provide a new benchmark value for the two-electron contribution to the ground state of helium-like krypton. Such a measurement would then constitute a test of quantum electrodynamics to second order.

Direct renormalization of two-electron self-energy contribution to energy of helium-like ions

A new practical approach to the calculation of two-electron self-energy corrections is presented for the energy of helium-like ions. The working formulas are derived within the modified adiabatic formalism. The direct renormalization procedure is used in terms of multiple commutator expansions. It is shown that the final expression does not contain ultraviolet and infrared divergences. This expression is used then to obtain numerical estimates for two-electron self-energy corrections in wide range of the nuclear charge Z.

Recent research using the Oxford electron beam ion trap

An overview is given of the current spectroscopic effort on the Oxford electron beam ion trap. Recent results from three different experiments are discussed: a precision measurement of the 1s2s3S1–1s2p3P2 transition in helium-like neon by VUV emission spectroscopy, a measurement of the two-electron contribution to the ground state energy of helium-like argon using absorption edge spectroscopy, and a study of the lifetime of the 1s22s2p3P2 level in beryllium-like argon.

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 238U 90+, based on currently available theory, is evaluated to be −261382.9(8) eV, without higher-order one-electron QED corrections.

Two-electron Lamb-shift calculations on heliumlike ions.

Complete second-order calculations of the two-electron contribution to the ground-state energy of heliumlike ions are presented and compared with recent experimental results. Calculations involve relativistic many-body effects and two-electron Lamb-shift (vacuum-polarization and self-energy) contributions as well as the recently presented nonradiative QED corrections. Agreement between experimental and theoretical results verifies the many-body part of the calculation. The accuracy of the results is not high enough to test the QED part of the calculations, but an improvement by half an order of magnitude would provide an experimental test of the QED effect for heavy ions beyond the first-order Lamb shift. {copyright} {ital 1996 The American Physical Society.}

MC SCF gradient optimization of the H 2CO→H 2 + CO transition structure

A compact algorithm is presented to obtain the two-electron integral derivative contribution to MC SCF gradients that avoids the storage of the integral derivatives or the back-transformed two-electron density matrix. The MC SCF gradients are applied to the optimization of the H 2CO→H 2 + CO transition structure at the 6–31G* basis-set level using a complete CI in a four-electron, four-orbital valence space.

Incoherent-scattering factors for atomic helium using a density-matrix approach

Both the one- and two-electron contributions to the differential cross section (DCS) for the Compton scattering of x rays from atomic helium have been computed at the Hartree-Fock level. An asymmetry in the DCS about $q=0$ is observed. The DCS has been used to obtain the incoherent-scattering factors for atomic helium and the results have been compared with the Waller-Hartree (WH) theory. It is found that the two-electron contribution is appreciable in the region of the small momentum transfer. The effect of the choice of the Breit-Dirac factor on the incoherent-scattering factors has also been studied.

Molecular symmetry. III. Second derivatives of electronic energy with respect to nuclear coordinates

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the analytic computation of the energy Hessian matrix. A ’’skeleton’’ Hessian matrix is calculated from the unique blocks of electron repulsion integrals. The true Hessian matrix is generated by projecting the symmetric component out of the skeleton Hessian. The analysis is valid for many wave functions, including closed- or open-shell restricted and unrestricted Hartree–Fock wave functions, multiconfiguration Hartree–Fock wave functions, and configuration interaction wave functions. We also extend the use of translational invariance previously used for energy gradient calculations. To illustrate the method, we compare the computer time required for the two-electron contribution to the Hessian matrix of eclipsed ethane, using Pople’s 6-31G basis set and D3h symmetry and various subgroups of D3h. Computational times are roughly inversely proportional to the order of the point group.

Separation of 1La Excitation Energies of Conjugated Hydrocarbons into Orbital Energy Difference and Two-Electron Term Contributions

Working within the framework of the self-consistent-field approximation, an estimate is given for the value of experimental counterparts of the orbital energy difference and the two-electron term contribution to the 1→−1 excitation energy for eight conjugated hydrocarbons. The two-electron term is almost constant for the benzenoid hydrocarbons examined but markedly different in the case of azulene. The orbital energy difference is responsible for the variation of the excitation energy in the benzenoid series, and, in all cases examined, it is simply related to the topology of the molecule. Semiempirical limited configuration-interaction calculations in the π-electron approximation (Pariser—Parr—Pople) are in qualitative agreement with these findings. The well-known facts that simple Hückel calculations lead to a good correlation with the excitation energies of the benzenoid hydrocarbons if the regression line is not forced to go through the origin and that azulene is far off the regression line are shown to be in accordance with the findings.