Algebraic tools for dealing with the atomic shell model. I. Wavefunctions and integrals for hydrogen--like ions

Today, the 'hydrogen atom model' is known to play its role not only in teaching the basic elements of quantum mechanics but also for building up effective theories in atomic and molecular physics, quantum optics, plasma physics, or even in the design of semiconductor devices. Therefore, the analytical as well as numerical solutions of the hydrogen--like ions are frequently required both, for analyzing experimental data and for carrying out quite advanced theoretical studies. In order to support a fast and consistent access to these (Coulomb--field) solutions, here we present the Dirac program which has been developed originally for studying the properties and dynamical behaviour of the (hydrogen--like) ions. In the present version, a set of Maple procedures is provided for the Coulomb wave and Green's functions by applying the (wave) equations from both, the nonrelativistic and relativistic theory. Apart from the interactive access to these functions, moreover, a number of radial integrals are also implemented in the Dirac program which may help the user to construct transition amplitudes and cross sections as they occur frequently in the theory of ion--atom and ion--photon collisions.Comment: 23 pages, 1 figur

Two centre problems in relativistic atomic physics

The work contained within this thesis is concerned with the explanation and usage of a set of theoretical procedures for the study of static and dynamic two–centre problems in the relativistic framework of Dirac’s equation. Two distinctly different theories for handling time–dependent atomic interactions are reviewed, namely semi–classical perturbation theory and a non–perturbative numerical technique based on the coupled channel equation to directly solve the time–dependent, two–centre Dirac equation. The non–perturbative numerical technique has been developed independently and the calculations performed with it are entirely new. Calculations for ionisation cross sections and state occupancies are conducted for both these methods. The non–perturbative technique for relativistic two–centre problems is extensively explained and, given its novelty, a probity test is conducted between this technique and that of the well established perturbation theory in calculating K-and L-shell ionisation cross sections for the alpha decay of initially Hydrogen–like Polonium. To that end, an in depth outline of the perturbative technique is also made for both collision and decay processes. As well as the comparison test mentioned, this technique is also applied to the analysis of cross sections of the promotion of a single electron into the positive continuum from either a K- or L-shell due to the alpha decay of heavy, neutral nuclei (Gadolinium, Polonium and Thorium). Dirac-Coulomb eigenfunctions centred on the parent nucleus of the decay pair are taken as the basis for use in the cross section calculations utilising first order, semi-classical pertubation theory. The excellent congruence between both techniques justifies the usage of the non-perturbative algorithms in the subsequent analysis of collisions between very heavy, highly charged ions. As such, a set of calculations are performed examining the bound and continuum state occupancy of the electronic levels during a collision between U92+ -U91+, at both over–critical and non-critical projectile velocities. Overall, the non-perturbative method developed and implemented here, is shown to be reliable, compares well with available experimental data, and most importantly is flexible enough to find continued use in studies on more extreme/exotic atomic systems

Mathematical Methods in Quantum Chemistry

The field of quantum chemistry is concerned with the analysis and simulation of chemical phenomena on the basis of the fundamental equations of quantum mechanics. Since the ‘exact’ electronic Schrödinger equation for a molecule with $N$ electrons is a partial differential equation in 3$N$ dimension, direct discretization of each coordinate direction into $K$ gridpoints yields $K^{3N}$ gridpoints. Thus a single Carbon atom ($N = 6$) on a coarse ten point grid in each direction ($K = 10$) already has a prohibitive $10^{18}$ degrees of freedom. Hence quantum chemical simulations require highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together quantum chemists and the emerging and fast growing community of mathematicians working in the area, to assess recent advances and discuss long term prospects regarding the overarching challenges of (1) developing accurate reduced models at moderate computational cost, (2) developing more systematic ways to understand and exploit the multiscale nature of quantum chemistry problems. Topics of the workshop included: • wave function based electronic structure methods, • density functional theory, and • quantum molecular dynamics. Within these central and well established areas of quantum chemistry, the workshop focused on recent conceptual ideas and (where available) emerging mathematical results

Theory of electron capture in ion-atom collisions

Cross sections for electron capture by (^4)He(^2+) ions from ground state atomic hydrogen are presented for a (^4)He(^2+) laboratory energy range from 1 to 800 keV (0.25 to 200 keV amu (^-1)). The cross sections were calculated using a coupled channel approximation in which the electronic wavefunction was expanded in terms of a finite number of atomic orbital basis states centred upon the target and the projectile. Electron translation factors which incorporated a switching function were included in the basis states. The semi-classical impact parameter approximation was employed. The cross sections presented are for electron capture into the 2s state of (^4)Me(^+), and into the n = 2 level of (^4)He(^+) using two states and four states respectively in the basis expansion. Four functional forms of switching function were used in the translation factors. The cross sections are compared with ones calculated using two-state and four-state atomic basis expansions which used plane-wave translation factors, and also with other theoretical and experimental cross sections. For energies ≤ 2.5 keV amu (^-1) fairly reasonable agreement is obtained with other data. For energies ≥ 2.5 keV amu (^-1) the present cross sections are in poor to extremely poor agreement with other data, steady divergence of the present results from existing data being observed with increasing energy. The present results are discussed, and conclusions and suggestions for future work are made

Computational Techniques

This chapter introduces fundamental computational approaches and ideas to energy materials. These can be divided into two main streams: one dealing with the motion of atoms or ions described at a simplified level of theory and another focusing on electrons. The modeling framework, which covers both streams, is outlined. The atomistic simulation techniques discussed in the chapter are concerned with describing the energy landscape of individual atoms or ions, where classical mechanics can be usefully employed as the first successful approximation. Multiscale approaches could be the method of choice if one is interested in large molecules, inhomogeneous solids, complex environments or geometrical arrangements, systems that are far away from equilibrium or have particularly long evolution times. One of the principal objectives of atomistic simulations is to derive an accurate and coherent approach to the prediction of defect structure, energetics and properties. Two of the most widely employed methods are outlined. This edition first published 2013 © 2013 John Wiley & Sons, Ltd

High resolution for x-ray spectroscopy studies with highly charged heavy ions at the CRYRING@ESR electron cooler

In this work, we report on the first x-ray spectroscopy study associated with the RR processes for bare lead ions at the electron cooler of the CRYRING@ESR, as storage rings are currently the only facilities routinely delivering hydrogen-like ions at high-Z in large quantities. With ultra-cold electron beam temperatures and near zero electron-ion collision energies, the effective production of characteristic projectile x-rays was well demonstrated at 0 deg and 180 deg observation geometries in our experiment by decelerated 10 MeV/u hydrogen-like lead ions. To reveal the role of radiative feeding transitions in the formation of observed intense Lyman and Balmer lines, an elaborate theoretical model describing the radiative decay dynamics and each (n, l, j)-state population varying over time is put to a test. As a result, the presented rigorous treatment reproduces observed x-ray spectroscopy really well in terms of the RR transitions and characteristic x-ray lines. In addition, we found a strong enhancement for l = n − 1 states in inner shells due to radiative Yrast-cascades from high Rydberg states, that finally contribute strikingly to the observed intensities of characteristic x-ray lines. Further on the current thesis lays the basis for a successful effort to push the experimental resolution of x-ray spectroscopy for L → K ground-state transitions at high-Z of below 80 eV at about 100 keV. This was done in an RR experiment of free electrons into the bound states of initially hydrogen-like uranium ions by adopting low temperature x-ray detectors, namely metallic magnetic calorimeters. Such an experiment allowed us for the first time to resolve the substructure of the Kα2 line and partially the Kα1 line in helium-like uranium ions. The preliminary data again prove the unique potential of the experimental method based on x-ray spectroscopy at the electron cooler of the CRYRING@ESR

Development and applications of quantum chemistry to open shell systems

This thesis investigates the applicability of a range of computational techniques across a range of open shell chemical systems from the geometrically simple but electronically complex to the geometrically complex but electronically simple. Initially an investigation into a range of geometrically simple but electronically complicated systems is presented. The Monte Carlo Configuration Interaction method (MCCI) is applied to challenging transition metals dimers such as ScNi in order to establish the ground state potential energy surface, from equilibrium bond lengths through to dissociation using highly compact wavefunctions compared to Full Configuration Interaction (FCI). It shall be demonstrated that the ScNi dimer represents the current limit of this technique. Software development of MCCI is then undertaken in order to perform calculations of spin-orbit coupling interactions. Results on B, C, O, F, Si, S, F, Cl, OH, NO, CN and C2 species are shown to be comparable with other techniques using the one-electron Breit-Pauli Hamiltonian. The application of quantum chemistry to geometrically complex but electronically simple systems is then considered. Density Functional Theory (DFT) is used to investigate the mechanism and energetic barriers leading to ring inversion of the biscalix[4]arene supra-molecule. A minimum barrier height of 19.31 kcalmol−1 to inversion is elucidated along with details of the complete mechanistic pathway to inversion. The focus then moves to polymetallic clusters of calix[4]arene. A DFT study is made of the preferential binding of calix[4]arene towards first row transition metals of various oxidation and spin states. Results indicate that Cu3+ (singlet) species will preferentially bind to the lower rim over other metals in the study. The final DFT-related work presented is a study of the preferential binding at the upper rim of polymetallic calix[4]arene clusters towards a range of important small gas molecules. It was found that gases such as NH3 and SO2 bind most strongly to the upper rim with the inclusion of a transition metal at the lower rim providing strengthening of the host-guest binding

Direct Methods for Solving the Schrödinger Equation

The notions of electron correlation and correlation problem arising in the framework of approximate solutions to the Schrödinger equation are presented. Then, we briefly review the original ideas of explicit inclusion of the interelectronic distance, r12, into the wavefunction as a solution to this problem. Exemplifying the efficiency of the explicit correlation for achieving high accuracy, we analyze the Nakatsuji's free-complement (FC) method. We demonstrate that at each FC order, fewer number of complement functions is required to get lower energies compared with those resulting from the conventional FC method. Applying the FC method to the triplet excited state of the He atom, we have discovered the appearance of permanents in addition to the determinants in the FC expansion of the wavefunction. These permanents are shown to be important for the energy convergence. To achieve a better understanding about the explicitly correlated methods, especially, the R12 and F12 methods, we analyzed three possible candidates with various correlation functions F(r_{12}) for a compact and efficient ansatz. Our main focus on the linear correlation factor r12 has led this analysis to the investigation of the correlated molecular orbital (CMO) theory of the Frost and Braunstein (FB). We revisit CMO theory within both restricted (R) and unrestricted formalisms (U). We also introduce the unrestricted FB (UFB) ansatz for the first time and derive the necessary expressions for both RFB and UFB overlap, kinetic, nuclear-attraction and interelectronic Coulomb repulsion matrix elements. All integrals have been obtained in closed form except one for which, we have used an accurate one-dimensional quadrature. Finally, we investigate the potential energy curve (PEC) of UFB for H2 at small, intermediate and large internuclear distances. Then, we compare its performance with that of RFB, restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and configuration interaction (CI) wavefunctions. Reproducing the RFB results for a much wider range of bond lengths in H2 reveals that the calculations of FB contain significant errors. We have also found a pole in the RFB linear correlation coefficient. Our UFB ansatz provides significant improvement over the RFB where passing the symmetry breaking point it completely removes the hump in the RFB PEC. The UFB ansatz also shows surprising features such as the presence of multiple solutions, non-smooth PEC, symmetry-broken solutions that are higher in energy than the restricted solution and RFB->UFB stability in the presence of lower UFB solutions. These phenomena can have significant impacts on the explicitly correlated calculations such as R12 and F12 within the unrestricted framework. Also, a detailed discussion on the large-$R$ asymptotic analysis of these five wavefunctions shows that none of these PECs has the correct R^{-6} decay within the minimal basis model. The UFB energy, however, demonstrates dispersion-like O(R^{-8}) decay which is an improvement over the CI and UHF with exponential decays. Considering the generalized FB (GFB) wavefunction where r12^n is the correlation factor and $n$ is a positive integer, we have shown that no analytic function of r12 can capture the dispersion within the minimal basis