I P Grant

GRASP: A general-purpose relativistic atomic structure program

Abstract The Oxford MCP/MCDF and MCBP/BENA packages have been rewritten in FORTRAN 77 and combined in the new code, GRASP. This is more versatile than its predecessors, contains more stable and accurate numerical procedures and a simplified but more flexible interface. Array dimensions and installation-dependent parameters may be set by the user. All known errors in previous versions have been eliminated. A comprehensive users manual is now provided as supplementary documentation.

The grasp2K relativistic atomic structure package

This paper describes grasp2K, a general-purpose relativistic atomic structure package. It is a modification and extension of the GRASP92 package by [F.A. Parpia, C. Froese Fischer, I.P. Grant, Comput. Phys. Comm. 94 (1996) 249]. For the sake of continuity, two versions are included. Version 1 retains the GRASP92 formats for wave functions and expansion coefficients, but no longer requires preprocessing and more default options have been introduced. Modifications have eliminated some errors, improved the stability, and simplified interactive use. The transition code has been extended to cases where the initial and final states have different orbital sets. Several utility programs have been added. Whereas Version 1 constructs a single interaction matrix for all the Js and parities, Version 2 treats each J and parity as a separate matrix. This block structure results in a reduction of memory use and considerably shorter eigenvectors. Additional tools have been developed for this format. The CPU intensive parts of Version 2 have been parallelized using MPI. The package includes a “make” facility that relies on environment variables. These make it easier to port the application to different platforms. The present version supports the 32-bit Linux and ibmSP environments where the former is compatible with many Unix systems. Descriptions of the features and the program/data flow of the package will be given in some detail in this report.

Gauge invariance and relativistic radiative transitions

The gauge invariance of the matrix elements of electromagnetic interaction is a property that is usually taken for granted. The author describes a unified derivation in which the dependence of the matrix elements on the gauge in which the electromagnetic potentials are written is exhibited explicitly. The necessary and sufficient condition for the transition matrices for all multipoles to be gauge invariant is that the transition matrix for longitudinal photons should vanish identically. This imposes conditions on the initial and final wavefunctions which are automatically satisfied for a single-particle model but may not necessarily hold in, say, the Hartree-Fock model. For electric dipole transitions, it is shown that the Coulomb gauge leads to the dipole velocity form of the matrix element in the nonrelativistic limit and that a different choice is needed to give the dipole length form. Arguments are advanced suggesting that the dipole velocity form should be given a privileged position in approximate calculations of atomic and molecular transition probabilities.

Foundations of the relativistic theory of atomic and molecular structure

Publisher Summary This chapter discusses the basics of the relativistic theory of atomic and molecular structure. The dominant role of the atomic nucleus in the isolated atom means that the directional dependence of the wave function can be handled algebraically, so that only the dependence on the radial coordinate need can be treated by numerical methods. An understanding of the analytic behavior of Eigen solutions of the Dirac operator is fundamental to the construction of a rigorous description of the relativistic quantum mechanics of atoms and molecules. One-particle models provide a remarkably good first approximation for many processes in atoms and it is no coincidence that most many-body theories are built from one-body wave functions. In quantum electrodynamics textbooks, plane wave solutions of Diracs equation for non-interacting free electrons are the commonest building blocks. Because the theory only conserves total charge, but not the individual numbers of electrons and positrons, excitations producing electron-positron pairs will appear only as intermediate states in perturbation expansions in low energy processes. At higher energies, real pairs may be produced.

Dirac–Fock calculations of X-ray scattering factors and contributions to the mean inner potential for electron scattering

Tables of X-ray scattering factors for naturally occurring elements and ions have been calculated using a multiconfiguration Dirac–Fock code. The complete data set takes up about 250 kbytes of space. For accurate values, it is preferable to interpolate directly. For consistency with previous work, parametric fits are also presented with error estimates and a range of validity. The limiting values of the electron scattering factor as sinθ/λ tends to zero, which are needed to calculate the mean inner potential for electron scattering, are also tabulated. The results show that the mean inner potential can be very sensitive to ionization state.

Matrix representation of operator products

A systematic study of finite dimensional matrix approximations to products of quantum mechanical operators is made. It is shown that the matrix representation of the product of two operators is equal to the product of the matrix representation of each of the operators if the domains of the matrices are suitably chosen. The importance of these results to relativistic molecular structure calculations and to perturbation theory studies is emphasised.

The Dirac equation in the algebraic approximation

The construction of variational approximations to the solutions of the Dirac equation is discussed. The importance of imposing the physical boundary conditions for both the point- and finite-nuclear models on the set of trial functions is emphasised. The low-frequency Breit interaction is shown to cause no variational failure when included in the iterative solutions of the Dirac-Hartree-Fock equations. The practical advantages of calculations which variationally include the Breit interaction over conventional treatments which employ only the Coulomb interaction are discussed. Manybody perturbation theory calculations of the radial correlation energy of helium- and beryllium-like systems are reported

Ab initio relativistic quantum chemistry: four-components good, two-components bad!*

In view of the debate which resulted from the introductory lecture at this meeting, this article has been submitted to address the issues raised concerning the validity and implementation of relativistic theories of many-electron systems. We present the formulation and construction of BERTHA, our relativistic molecular structure program, and illustrate features of relativistic electronic structure theory with examples. These include magnetic and hyperfine interactions in small molecules, the use of spinor basis functions which include a dependence on a magnetic field strength, NMR shielding constants, P-odd interactions in chiral molecules, and computational details of a relativistic ab initio treatment of germanocene.

Application of relativistic theories and quantum electrodynamics to chemical problems

The BERTHA program embodies a new formulation of relativistic molecular structure theory within the framework of relativistic quantum electrodynamics (QED). This leads to a simple and transparent formulation of Dirac–Hartree–Fock–Breit (DHFB) self-consistent field equations along with algorithms for molecular properties, electron correlation, and higher order QED effects. The DHFB equations are solved by a direct method based on a relativistic generalization of the McMurchie–Davidson algorithm for molecular integrals that economizes memory requirements and is not significantly more expensive computationally than comparable nonrelativistic calculations. Some noteworthy features of this approach include the ease with which relativistic point-group symmetry can be analyzed and the ease of calculation of electromagnetic properties, for example, g factors, nuclear hyperfine interactions, nuclear magnetic resonance (NMR) shielding parameters and molecular effects of parity-violating weak interactions. The “negative energy” states, which are often regarded as a dangerous nuisance in other treatments of relativistic effects, make a vital contribution. As well as outlining the main ideas underlying our development, this study presents results for small molecules, some of which involve heavy elements.

The Dirac equation in the algebraic approximation. V. Self-consistent field studies including the Breit interaction

It is argued that the Breit interaction can be included in the self-consistent-field procedure of Dirac-Hartree-Fock calculations, and need not necessarily be treated as a first-order perturbation, as is widely believed. The matrix Dirac-Hartree-Fock equations including the frequency-independent Breit interaction are presented and discussed. Prototype calculations for helium-like and beryllium-like ions are presented.