The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a 1/r singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers n + 1 of mesh points, where n is the principal quantum number. Numerically exact mean values of powers −2 to 3 of the radial coordinate r can also be obtained with n + 2 mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 or fewer mesh points.

Relativistic dipolar to hexadecapolar polarizabilities of the ground state and some excited states of hydrogenic atoms are calculated by using numerically exact energies and wave functions obtained from the Dirac equation with the Lagrange-mesh method. This approach is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. The partial polarizabilities conserving the absolute value of the quantum number κ are also numerically exact with small numbers of mesh points. The ones where |κ| changes are very accurate when using three different meshes for the initial and final wave functions and for the calculation of matrix elements. The polarizabilities of the n = 2 excited states of hydrogenic atoms are also studied with a separate treatment of the final states that are degenerate at the nonrelativistic approximation. The method provides high accuracies for polarizabilities of a particle in a Yukawa potential and is applied to a hydrogen atom embedded in a Debye plasma.

Collinear laser spectroscopy was performed on Zn (Z=30) isotopes at ISOLDE, CERN. The study of hyperfine spectra of nuclei across the Zn isotopic chain, N=33–49, allowed the measurement of nuclear spins for the ground and isomeric states in odd-A neutron-rich nuclei up to N=50. Exactly one long-lived (>10 ms) isomeric state has been established in each ^69–79Zn isotope. The nuclear magnetic dipole moments and spectroscopic quadrupole moments are well reproduced by large-scale shell–model calculations in the f₅pg₉ and fpg₉d₅ model spaces, thus establishing the dominant term in their wave function. The magnetic moment of the intruder I^π=1/2⁺ isomer in ⁷⁹Zn is reproduced only if the νs_1/2 orbital is added to the valence space, as realized in the recently developed PFSDG-U interaction. The spin and moments of the low-lying isomeric state in ⁷³Zn suggest a strong onset of deformation at N=43, while the progression towards ⁷⁹Zn points to the stability of the =28 and =50 shell gaps, supporting the magicity of ⁷⁸Ni

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Relativistic two-photon decay rates of the 2s 1/2 and 2p 1/2 states towards the 1s 1/2 ground state of hydrogenic atoms are calculated by using numerically exact energies and wave functions obtained from the Dirac equation with the Lagrange-mesh method. This approach is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. Highly accurate values are obtained by a simple calculation involving different meshes for the initial, final, and intermediate wave functions and for the calculation of matrix elements. The accuracy of the results with a Coulomb potential is improved by several orders of magnitude in comparison with benchmark values from the literature. The general requirement of gauge invariance is also successfully tested, down to rounding errors. The method provides high accuracies for two-photon decay rates of a particle in other potentials and is applied to a hydrogen atom embedded in a Debye plasma simulated by a Yukawa potential.

The present work reports results from systematic multiconfiguration Dirac-Hartree-Fock calculations of electronic isotope shift factors for a set of transitions between low-lying levels of neutral aluminium. These electronic quantities together with observed isotope shifts between different pairs of isotopes provide the changes in mean-square charge radii of the atomic nuclei. Two computational approaches are adopted for the estimation of the mass-and field-shift factors. Within these approaches, different models for electron correlation are explored in a systematic way to determine a reliable computational strategy and to estimate theoretical error bars of the isotope shift factors.

The relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) and the non-relativistic multiconfiguration Hartree-Fock (MCHF) methods have been employed to calculate the magnetic dipole and electric quadrupole hyperfine structure constants of zinc. The calculated electric field gradients for the 4s4p 3 P o 1 and 4s4p 3 P o 2 states, together with experimental values of the electric quadrupole hyperfine structure constants, made it possible to extract a nuclear electric quadrupole moment Q( 67 Zn) = 0.122(10) b. The error bar has been evaluated in a quasi-statistical approach -the calculations had been carried out with eleven different methods, and then the error bar has been estimated from the differences between the results obtained with those methods.

Relativistic atomic structure calculations are carried out in alkaline-earth-metal ions using a semiempirical-core-potential approach. The systems are partitioned into frozen-core electrons and an active valence electron. The core orbitals are defined by a Dirac-Hartree-Fock calculation using the grasp2k package. The valence electron is described by a Dirac-like Hamiltonian involving a core-polarization potential to simulate the core-valence electron correlation. The associated equation is solved with the Lagrange-mesh method, which is an approximate variational approach having the form of a mesh calculation because of the use of a Gauss quadrature to calculate matrix elements. Properties involving the low-lying metastable 2 D 3/2,5/2 states of Ca + , Sr + , and Ba + are studied, such as polarizabilities, one-and two-photon decay rates, and lifetimes. Good agreement is found with other theory and observation, which is promising for further applications in alkali-like systems.