Finite Nuclear Mass Effects Research Articles

A program for accurate solutions of two-electron atoms

We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a two-electron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. The finite difference method is applied and subsequent extrapolations gives the continuous grid result. The program is written in C and is highly optimized. All computations are made in double precision. Despite this relatively low degree of floating point precision (48 digits are not uncommon), the accuracy in the results can reach about 10 significant figures. Both serial and parallel versions are provided. The parallel program is particularly suitable for shared memory machines such as the Sun Starcat series. The serial version is simple to compile and should run on any platform. Program summary Title of program: corr2el Catalogue identifier: ADUX Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUX Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computer for which the program is designed and others on which it has been tested: Computers: Sun Fire 15K StarCat, Sun Ultra SPARC III, PC Operating systems or monitors under which the program has been tested: Sun Solaris 9, Linux Programming language used: ANSI C Memory required to execute with typical data: 3 Mwords or more No. bits in a word: 32 No. processors used: arbitrary Has the code been vectorized or parallelized: parallelized Number of lines in distributed program, including test data, etc.:5885 Number of bytes in distributed program, including test data, etc.: 26 199 Nature of physical problem: The Schrödinger equation for two-electron atoms is solved using finite differences. Method of solution: An iterative eigenvalue-solver that requires only the action of the Hamiltonian on a trial function is applied. The two-electron wave function is expanded in a sum of partial waves. The finite difference method is then applied to approximate the derivatives of the pair functions. The total action of the Hamiltonian on the partial waves, including correlation effects, is computed using highly optimized routines. Restriction on the complexity of the problem: The Hamiltonian employed here does not take relativistic or finite nuclear mass effects into account. The amount of computing time may become unreasonable for excited states far above the ground state. The use of double precision puts a limit on the accuracy obtainable. Typical running time: This ranges from half a minute (to obtain 10 significant figures for the s-limit of the Helium ground state) to perhaps a day for advanced examples depending on the level of parallelization. Unusual features of the program: The implicitly restarted Arnoldi method used to obtain the eigenvalues is implemented by using the ARPACK/PARPACK program library [ http://www.netlib.org/arpack]. This package also depends on the standard numerical libraries BLAS and LAPACK [ http://www.netlib.org/lapack]. Good performance is obtained by using Sun's optimized performance libraries [ http://www.sun.com]. By using a 64-bit environment (Ultra SPARC III and Solaris 9), memory limitations are non-problematic. Shared memory is used in the parallel version. Fast communication between the nodes is made over shared memory using Sun's implementation of MPI.

Heteroisotopic molecular behavior: the valence-bond theory of the positronium hydride

We develop an adiabatic valence-bond theory of the positronium hydride, HPs, as a heteroisotopic diatomic molecule. Typical heteronuclear ionic behaviour comes out at bonding distances, yielded just by finite nuclear mass effects, but some interesting new features appears for short distances as well.

A new algorithm to handle finite nuclear mass effects in electronic calculations: the ISOTOPE program.

We report the development of a simple algorithm to modify quantum chemistry codes based on the LCAO procedure, to account for the isotope problem in electronic structure calculations. No extra computations are required compared to standard Born-Oppenheimer calculations. An upgrade of the Gamess package called ISOTOPE is presented, and its applicability is demonstrated in some examples.

On the status and perspectives of MCDF calculations and measurements of transition data in the Be isoelectronic sequence

Transition energies and rates between the and levels of ions in the Be isoelectronic sequence (nuclear charges Z = 7-28, 30, 36, 42) are calculated in the valence and core-valence limit using the multiconfiguration Dirac-Fock method. The Breit interaction and finite nuclear mass effects are taken into account in subsequent relativistic configuration-interaction calculations. Separately optimized orbital sets are used for the initial and final states in the transitions. The transition matrix elements are evaluated with an efficient transformation technique. The calculated energy separations and transition probabilities are compared with critically evaluated experimental values and with results from other recent calculations.

Energies and oscillator strengths for lithiumlike ions

The Hylleraas-type variational method is used to calculate the energies of the lithium ${1s}^{2}2s{}^{2}S$ and ${1s}^{2}2p{}^{2}P$ isoelectronic sequences up to $Z=20$. The oscillator strengths for the ${1s}^{2}2s{}^{2}\stackrel{\ensuremath{\rightarrow}}{S}{1s}^{2}2p{}^{2}P$ transitions are evaluated for $Z$ up to 20, including finite nuclear mass effects, and the corresponding lifetimes are determined. The $1/Z$ expansion method is used to extend the variational results to larger values of $Z$. Relativistic corrections to the dipole transition moment squared are estimated by comparison with the relativistic many-body perturbation theory results of Johnson et al. [At. Data Nucl. Data Tables 64, 279 (1996)]. The final lifetime for the ${1s}^{2}2p{}^{2}P$ state of lithium is in good agreement with recent high-precision measurements.

Excitation energies from density functional perturbation theory

We consider two perturbative schemes to calculate excitation energies, each employing the Kohn–Sham Hamiltonian as the unperturbed system. Using accurate exchange-correlation potentials generated from essentially exact densities and their exchange components determined by a recently proposed method, we evaluate energy differences between the ground state and excited states in first-order perturbation theory for the helium, ionized lithium and beryllium atoms. It was recently observed that the zeroth-order excitations energies, simply given by the difference of the Kohn–Sham eigenvalues, almost always lie between the singlet and triplet experimental excitations energies, corrected for relativistic and finite nuclear mass effects. The first-order corrections provide about a factor of two improvement in one of the perturbative schemes but not in the other. The excitation energies within perturbation theory are found to be more accurate than the excitations obtained within ΔSCF while, for a two-electron system, they coincide with the ones obtained in time-dependent density functional theory within the single-pole approximation using our accurate static exchange-correlation potential and the time-dependent optimized effective potential kernel. We find that the agreement between the experimental and the perturbative excitation energies deteriorates significantly if potentials from approximate functionals such as the local density approximation and the optimized effective potential method are employed instead of the true Kohn–Sham potential.

Methodology for superallowed Fermi beta-decay Part II. Reduction of data

This paper, Part II of the group, takes the Q EC-values and lifetimes from Part I and reduces them to a form suitable for further analysis in Part III. In reducing the Q EC-values to the phase space factors ƒ critical examination is made of: i) finite nuclear size and form effects; ii) screening by the orbital electrons; iii) finite nuclear mass effects; iv) convolution of the lepton and nucleonic wave functions and v) relativistic terms. In reducing the lifetimes there is a critical discussion of: i) branching ratios; ii) orbital electron capture and iii) the effect of atomic excitation in the beta-decay process.

A quantum Monte Carlo calculation of the ground state energy of the hydrogen molecule

We have calculated the ground state energy of the hydrogen molecule using the quantum Monte Carlo (QMC) method of solving the Schrödinger equation, without the use of the Born–Oppenheimer or any other adiabatic approximations. The wave function sampling was carried out in the full 12-dimensional configuration space of the four particles (two electrons and two protons). Two different methods were employed: the diffusion quantum Monte Carlo (DQMC) method and the Green’s function quantum Monte Carlo (GFQMC) method. This computation is very demanding because the configurations must be evolved on the time scale of the electronic motion, whereas the finite nuclear mass effects are resolved accurately only after equilibration on the much slower time scale of the nuclear motion. Thus, a very large number of iterations is required. The calculations were performed on the CM-2 Connection Machine computer, a massively parallel supercomputer. The enormous speedup afforded by the massive parallelism allowed us to complete the computation in a reasonable amount of time. The total energy from the DQMC calculations is −1.163 97±0.000 05 a.u. A more accurate result was obtained from the GFQMC calculations of −1.164 024±0.000 009 a.u. Expressed as a dissociation energy, the GFQMC result is 36 117.9±2.0 cm−1, including the corrections for relativistic and radiative effects. This result is in close agreement with accurate nonadiabatic-relativistic dissociation energies from variational calculations (corrected for radiative effects) in the range of 36 117.9–36 118.1 cm−1 and with the best experimentally determined dissociation energy of McCormack and Eyler 36 118.1±0.2 cm−1.

Muon reactivation in muon-catalyzed D-T fusion

We comprehensively reanalyze and search for the density dependence of the effective muon alpha sticking fraction ω s ff observed experimentally in muon catalyzed deuterium-tritium fusion. In our work particular emphasis has been put on the density dependent dense hydrogen stopping power. The main technical details and improvements in this work are: The (αμ) + 2s and 2p states are treated independently and are assigned individual reaction rates. The essential muonic excitation rates have been recalculated taking into account finite nuclear mass effects. The stopping power for a charged projectile in liquid heavy hydrogen is modified to account for dynamic screening effects and a density dependent effective ionization potential. It is shown that the medium dependent stopping power for the (αμ) + ion is the crucial factor controlling the density dependence of the effective sticking fraction. It is also pointed out that the muonic helium K α X-ray yield and the sticking fraction at high density can not be simultaneously brought into agreement with the experimental results without invoking novel mechanisms suppressing Stark mixing in the (Heμ) L-shell.

A simple determination of the nuclear mass of deuterium

Many A level syllabuses in chemistry and physics include sections on the Bohr theory of the hydrogen atom as an introduction to atomic structure and interference in thin films, as demonstrated in the Fabry-Perot interferometer. The experiment described in this article clearly demonstrates the effect of finite nuclear mass in modifying the simple Bohr model of single electron atoms and the capability of a February-Perot interferometer to resolve spectral lines differing in wavelength by 10 nm. The experiment therefore forms a useful bridge between the subject areas of atomic physics and optics and illustrates how atomic parameters may be derived from simple macroscopic measurements.

Finite Nuclear Mass and Relativistic Effects on the Rotational Constant of Molecular Hydrogen

Corrections to the rotational constant of molecular hydrogen associated with finite nuclear mass and relativistic effects are calculated. They are significantly smaller in magnitude than are the corresponding corrections previously employed. After these refinements are made, the values 60.8566±0.004, 45.6548±0.003, and 30.4527±0.002 cm−1 are obtained for the Dunham Be values of H2, HD, and D2. The present theoretical results show reasonably good agreement with Stoicheff's measurements on H2, HD, and D2, and with recent measurements on H2 by Fink, Wiggins, and Rank, while showing less good agreement with the measurements of Herzberg and Howe. The present agreement represents decided improvement over the results of previous analysis.

Electrodynamic Displacement of Atomic Energy Levels. II. Lamb Shift

The vacuum fluctuations of the photon and pair fields modify the interaction of an electron with an electromagnetic field. The effects on the energy levels are conveniently described in terms of the mass operator and the vacuum polarization potential. An operator calculus for handling the mass operator is used to separate the contribution of low energy quanta, for which the external electromagnetic field may not be treated as weak. The remainder is given as a power series in the field. The additional displacement of order $Z\ensuremath{\alpha}$ with respect to the published values for the Lamb shift is calculated by specializing the external field to the Coulomb field of the nucleus and by taking the matrix element of the operators in an $S$-state of a hydrogen-like atom. It is found that the $\mathrm{nS}$-level is raised by $\frac{{Z}^{5}{\ensuremath{\alpha}}^{4}}{{n}^{3}}\mathrm{Ry}\left(1+\frac{11}{128}\ensuremath{-}\frac{1}{2}\mathrm{ln}2+\frac{5}{192}\right).$ The theoretical value for the Lamb shift of an atom with infinitely heavy nucleus becomes 1058.42 Mc/sec. Effects of finite nuclear mass and size change this value to 1057.8 Mc/sec and 1058.9 Mc/sec for hydrogen and deuterium, respectively.