Positron-Atom Interactions Studied Using Configuration Interaction Methods

POSITRON-ATOM INTERACTIONS STUDIED USING CONFIGURATION INTERACTION METHODS

Michael William James Bromley

A thesis for the degree of Doctor of Philosophy at the Faculty of Science, Information Technology and Education Northern Territory University, Australia

Submitted on September 12, 2002.

Declaration

I hereby declare that the work herein is the result of my own investigations, and all references to ideas and research of others have been speciﬁcally acknowledged.

I certify that the work embodied in this thesis has not already been accepted in substance for any degree, and is not being currently submitted for any other degree.

Abstract

The non-relativistic configuration interaction (CI) method is applied to the study of positron interactions with either one or two valence electron atoms possessing a positron-atom bound state. Although the binding energy and other atomic properties are slowly convergent with respect to the angular momenta of the single particle orbitals used to construct the CI wavefunctions, the calculations are sufficiently large to give usefully accurate descriptions of the positronic atom structures. Calculations of the accurately known positron-atom bound states; positronic copper (e+Cu), positronic lithium (e+Li), and positronium hydride (PsH) systems were undertaken to develop and refine the numerical procedures. CI calculations confirmed the stability of three other systems; positronic beryllium (e+Be), positronic magnesium (e+Mg), and positronic zinc (e+Zn). The e+Mg calculations independently resolves the disagreement between the University of New South Wales group and a previous computational approach of the Northern Territory University group. Further CI calculations demonstrated the stability of four systems; positronic calcium (e+Ca), copper positride (CuPs), positronic strontium (e+Sr) and positronic cadmium (e+Cd). These predictions are all rigorous with respect to the underlying model Hamiltonians. CI convergence issues are examined in detail, with trends used to give estimated binding energies (in units of Hartree) of 0.037795 (PsH), 0.000886 (e+Li), 0.003083 (e+Be), 0.016151 (e+Mg), 0.016500 (e+Ca), 0.005088 (e+Cu), 0.014327 (CuPs), 0.003731 (e+Zn), 0.010050 (e+Sr) and 0.006100 (e+Cd). The CI program used for the structure calculations was adapted to perform scattering calculations via the imposition of plane-wave boundary conditions within the Kohn variational formalism. Test calculations of model potential and low-energy elastic positron-hydrogen scattering reveal that in practice, the effects of ’Schwartz’ singularities are insignificant. CI-Kohn calculations of low-energy (k = 0 0.2 a−1) elastic → 0 positron-copper scattering gave a scattering length of +13.7 a0 and a threshold Zeff of

69.3, explicitly demonstrating that Zeﬀ is not overly large for such a metal vapour. These investigations demonstrate the feasibility of using single particle orbitals centred on the nucleus to reliably describe certain classes of positronic systems with one or two valence electrons. ii List of Publications

As of the 30th March 2003, the following papers with contributions from the results presented in this thesis have been published, or are under review:

1. Conﬁguration interaction calculations of positronic atoms and ions. M W J Bromley, J Mitroy and G G Ryzhikh. Nucl. Inst. Methods. B, (2000) 171 47-59.

2. Positron binding to calcium, a conﬁguration interaction study. M W J Bromley and J Mitroy. J. Phys. B, (2000) 33 L325-L331.

3. Positronic Atoms J Mitroy, M W J Bromley, and G G Ryzhikh. Pages 199-221 in New Directions in Antimatter Physics and Chemistry. C M Surko and F A Gianturco, Eds. Kluwer Academic Publishers, The Netherlands, 2001.

4. Conﬁguration interaction calculations of PsH and e+Be. M W J Bromley and J Mitroy. Phys. Rev. A, (2002) 65 012505.

5. Conﬁguration interaction calculations of positron binding to group II elements. M W J Bromley and J Mitroy. Phys. Rev. A, (2002) 65 062505.

6. Conﬁguration interaction calculations of positron binding to Zn and Cd. M W J Bromley and J Mitroy. Phys. Rev. A, (2002) 65 062506.

7. Positron and positronium binding to atoms and ions. J Mitroy, M W J Bromley, and G G Ryzhikh. J. Phys. B, (2002) 35 R81-R116 (Topical review).

8. Positron and positronium interactions with Cu. M W J Bromley and J Mitroy. Phys. Rev. A, (2002) 66 062504.

iii 9. Variational calculations of positron-atom scattering using Conﬁguration Interac- tion type wave functions M W J Bromley and J Mitroy. Phys. Rev. A, (2003) 67 (in press).

The following papers contain contributions from M W J Bromley that are not reported in this thesis. Except for the ﬁrst paper, the contributions consisted of using the programs developed as part of this thesis to perform calculations that were essential to the published research.

1. The elastic scattering of positrons from beryllium and magnesium in the low energy region. M W J Bromley, J Mitroy and G Ryzhikh. J. Phys. B, (1998) 31 4449-4458.

2. Positron binding to a model alkali atom. J Mitroy, M W J Bromley and G G Ryzhikh. J. Phys. B, (1999) 32 2203-2214.

3. Asymptotically exact expression for the energies of the 3Se Rydberg series in a two-electron system. I A Ivanov, M W J Bromley and J Mitroy. Phys. Rev. A, (2002) 66 042507.

4. Use of Orthogonalising Pseudo-Potentials in electronic scattering calculations I A Ivanov, J Mitroy M W J Bromley. Comp. Phys. Commun., (2003) 152 9-24.

5. Comment on Positronium Scattering by Ne, Ar, Kr and Xe in the Frozen Target approximation. J Mitroy and M W J Bromley. J. Phys. B, (2003) 36 793-795.

6. Positronium Scattering from Kr and Xe at low energies J Mitroy and M W J Bromley. Phys. Rev. A, (2003) 67 034502.

iv Acknowledgements

This is a thesis spanning four years, two cities, many bottles of Australian red, and there are many to thank...

Professor Chojnacki for being kind enough to provide the exact input of their CI calculations, which allowed me to verify my program right from the outset!

Huge thanks to Dr. Jim Mitroy for teaching me with patience how to do → ∞ relentless computational physics, Dr. Gregory Ryzhikh for much initial help, and Dr. Sergey Novikov for a ﬁnal critical thesis reading. Many thanks to Dr. Bill King and Dr. Jenny Blackwood for providing copies of their thesii. Thanks to the NT Solar Energy Society and NT Power and Water Authority for their awards (and cash ;). The Queens’ Trust for Young Australians for their project support to send me to Can- berra; Prof. Buckman, Prof. McEachran and the crew of the good-ship AMPL for the hospitality.

Thanks to NTU for still existing, and the various IT techos; Prasad Gunatunge, Shane Nuessler, Bronwyn Allan and J-C Nou for happily maintaining the workstations used to perform the calculations. Also the GNU/Linux/g77 free-sourced-nerds, without whom this thesis would not have happened at NTU (to this extent). The beautiful CAPA/NTUPSA peoples for taking time out of their lives and daring to question the ways things are heading (and how they are now).

Best wishes to the following exquisite friends who all helped to keep me sane... Sparkles&Julieanne, J2T, travellingKat, Emma H-W, Luscious Lari, Shane-e, will&dani, Lauraζ, (Matt, Dan, Mary), Asco Gordon, Jenny Blackwood, Neutrino Girl and the McKeddieans, Que-bit, Sandra Thibodeaux, Mont´eand Greg, Hoges, Harri B, Bruce Rock, QueenJ, Dr. Kate, Groover Gill, and Iam&Grant&Jasper. Ups to the Jim Henson, Towards 2000, Quantum, NTU Revue, NEo, and Nude2000 crews for their Bubbles of Reality.

Thanks to my wonderful family for all things principled, ﬁnancial and culinary. Dedicated to my niece Caitlin; may she always smile, and the world upon her.

v Contents

1 Introduction 1 1.1 Positron-electroninteractions ...... 2 1.1.1 Applications and implications of positron annihilation ...... 4 1.2 Positron-atominteractions...... 6 1.3 Nomenclatureandunits ...... 8 1.4 Positronbindingtoneutralatoms ...... 9 1.4.1 Positroniumbindingtoneutralatoms ...... 15 1.4.2 WhyCI? ...... 17 1.5 ChapterOutlines ...... 19

2 Modelling Atomic Structures 21 2.1 Hartree-Fock and conﬁguration interaction ansatz ...... 22 2.2 Treatmentoffrozen-coreelectrons ...... 24 2.2.1 Core-direct and core-exchange interactions ...... 25 2.2.2 Core-polarisationinteraction ...... 26 2.2.3 Orthogonalitywiththecoreelectrons ...... 29 2.3 Choiceofbasisfunctions...... 29 2.3.1 Diagonalisation of the one-electron atoms ...... 31 2.3.2 Choice of two-electron conﬁgurations ...... 32 2.3.3 Determinationofdissociationlimits ...... 32

2.4 Oscillator strengths, sum rules and αd ...... 33 2.5 Atoms...... 34 2.5.1 Li and Li− ...... 35 2.5.2 Be+ andBe...... 36 2.5.3 Mg+ andMg ...... 39

vi 2.5.4 Ca+ andCa...... 42 2.5.5 Cu...... 44 2.5.6 Cu− ...... 45 2.5.7 Zn+ andZn...... 48 2.5.8 Sr+ andSr ...... 50 2.5.9 Cd+ andCd ...... 53 2.6 Summary ...... 56

3 Positron Binding to one-Electron Systems 57 3.1 CI method for one-electron positronic systems ...... 58 3.1.1 Subsetbasis...... 59 3.2 Electron-positronannihilation ...... 60 3.3 Partial-wave convergence and extrapolation ...... 61 3.4 Previous CI calculations of e+Li and e+Cu ...... 62 3.5 e+Liresults...... 63 3.6 e+Curesults ...... 66 3.7 e+Li and e+Cupartial-waveconvergence...... 69 3.8 Summary ...... 74

4 Positron Binding to Simple two-Electron Systems 75 4.1 CI method for two-electron positronic systems ...... 76 4.2 Choice of two-electron-positron conﬁgurations ...... 77 4.3 ResultsforPsH...... 78 4.3.1 Convergence of Ps-H scattering calculations ...... 82 4.4 e+Beresults...... 84

+ 4.5 Lint = 3 calculations of PsH and e Be...... 88 4.6 Chaptersummary ...... 89

5 Positron Binding to Group II and IIB Atoms (and CuPs) 93 5.1 Positron binding to alkaline-earth atoms ...... 94 5.1.1 e+Mgresults ...... 95 5.1.2 e+Caresults ...... 99 5.1.3 e+Srresults...... 101 5.1.4 L 3 calculations of e+Mg, e+Ca and e+Sr ...... 101 int ≤

vii 5.1.5 e+Mg, e+Ca and e+Srextrapolationissues ...... 103 5.2 PositronbindingtogroupIIBatoms ...... 109 5.2.1 e+ZnResults ...... 109 5.2.2 e+CdResults...... 114 5.2.3 e+Zn and e+Cdextrapolationissues ...... 116 5.3 Experimental veriﬁcation of positron-atom binding ...... 119 5.3.1 e+-Zn and e+-Cd model potential construction ...... 120 5.3.2 e+-Zn and e+-Cd model potential scattering...... 121 5.4 CuPs...... 127 5.5 Chaptersummary ...... 132

6 Positron Scattering Using the CI-Kohn Method 133 6.1 Heuristicsofpositron-atomscattering ...... 134 6.1.1 Methods...... 135 6.2 Kohnvariationalmethod ...... 137 6.2.1 Theory ...... 137 6.2.2 Kohnlinearequations ...... 141

6.2.3 Annihilation coeﬃcient Zeﬀ ...... 143 6.3 Schwartzsingularities ...... 144 6.3.1 Modelpotentialinvestigations...... 144 6.4 s-wave elastic e+ scatteringfromhydrogen...... 148 −1 6.4.1 Validations and calculations at k = 0.5 a0 (3.4eV) ...... 149 6.4.2 Validations and calculations at zero-energy ...... 152 6.5 s-wave elastic e+ scatteringfromcopper ...... 154

6.5.1 Convergence with Lmax ...... 154 6.5.2 Positron-copper scattering for 0

7 Conclusion 173 7.1 Speciﬁcresults ...... 176 7.2 Perspectivesforthefuture...... 178 7.2.1 Improvements...... 178

viii 7.2.2 Naturalspringboard ...... 179

A Radial basis wavefunctions 181 A.1 Slatertypeorbitals...... 182 A.2 Laguerretypeorbitals ...... 183 A.3 Gaussiantypeorbitals ...... 184 A.4 s-wavescatteringbasisfunctions ...... 184

B Matrix elements 185 B.1 One-bodymatrixelements...... 185 B.1.1 Scatteringmatrixelement ...... 186 B.2 Two-bodymatrixelements ...... 187 B.2.1 Two-bodyintegrals...... 187 B.2.2 Electron-electron anti-symmetrisation ...... 188 B.2.3 Corematrixelements ...... 189

B.2.4 Vp2 matrixelements ...... 190 B.3 Two-electron-positronsystem ...... 191 B.3.1 Reducedtwo-bodyangularintegrals ...... 191 B.4 Numericalintegrations ...... 192 B.4.1 Gridtransform ...... 192 B.4.2 Two-body radial matrix elements (running integrations) . . . . .193

C Eigenvalue problem and solving 197 C.1 Hamiltonianconstruction ...... 197 C.1.1 Two-body Hamiltonian matrix elements ...... 197 C.1.2 Two-electron-positron Hamiltonian matrix elements ...... 198 C.2 Hamiltoniandiagonalisation ...... 199 C.2.1 Davidsonalgorithm ...... 199 C.3 Kohnscatteringlinearequations ...... 201

D Expectation values 204 D.1 Electronandpositrondensitymatrices ...... 204 D.2 r and r ...... 205 h ei h pi D.3 r2 and cos(θ ) ...... 206 h epi h ep i D.4 2-γ annihilation rate and Zeﬀ ...... 207

ix E Validations 209 E.1 Boundstatevalidations ...... 209

E.1.1 Two-electron αd and f validations ...... 214 E.1.2 Gaussian type orbital calculations of PsH ...... 215 E.2 Kohn variational scattering validations ...... 217 E.2.1 Potentialscattering ...... 217 E.2.2 Elasticscatteringfromstaticatoms ...... 219 E.2.3 s-wave e+ elasticscatteringfromh ...... 222

F Laguerre exponents of positronic systems 224

G Extrapolation details 228

x List of Figures

1.1 Positronium...... 3 1.2 Static and polarisation potentials of positron-neutralatom...... 6

1.3 Resonant peaks in Zeﬀ forbutane...... 8 1.4 Energy level diagram of e+Atombinding...... 11 1.5 Structural diagrams of positron binding to atoms ...... 14

3.1 Convergence of Γ and r2 with L for e+Li...... 65 v h epi max + + 3.2 Binding energy of e Li and e Cu vs Lmax ...... 70 + + 3.3 Spin-averaged annihilation rate of e Li and e Cu vs Lmax ...... 70 + + 3.4 Extrapolation exponent pE of e Cu and e Li vs Lmax ...... 71 + + 3.5 Extrapolation exponent pΓ of e Cu and e Li vs Lmax ...... 72

4.1 PsH binding energies vs Lmax ...... 82

4.2 PsH annihilation rates vs Lmax ...... 83

+ + + 5.1 Binding energy of e Mg, e Ca and e Sr vs Lmax ...... 103 + + + 5.2 Spin-averaged annihilation rate of e Mg, e Ca and e Sr vs Lmax . . . . 105 + + + 5.3 pE(Lmax) of e Mg, e Ca and e Sr...... 106 + + + 5.4 Binding energy of e Be, e Zn and e Cd vs Lmax ...... 117 + + + 5.5 Spin-averaged annihilation rate of e Be, e Zn and e Cd vs Lmax . . . . 117 + + 5.6 δ0(k) for e -Zn and e -Cdelasticscattering ...... 122 5.7 DCS for e+-Zn and e+-Cdelasticscattering ...... 125 5.8 DCS Ratio for e+-Zn and e+-Cdelasticscattering ...... 125

5.9 Binding energy of PsH and CuPs vs Lmax ...... 129

5.10 Spin-averaged annihilation rate of PsH and CuPs vs Lmax ...... 130

6.1 Model potential scattering with 4 LTOs vs k ...... 147

xi 6.2 Model potential scattering with 28 LTOs vs k ...... 147 + 6.3 Scattering length of e -Cu vs Lmax ...... 158 −1 c + 6.4 k = 0 and 0.1 a0 Zeff e -Cu vs Lmax ...... 159 −1 v + 6.5 k = 0 and 0.1 a0 Zeff e -Cu vs Lmax ...... 160 + v 6.6 Threshold e -Cu Zeff increments vs Lmax ...... 161 −1 + v 6.7 k = 0.1 a0 e -Cu Zeff increments vs Lmax ...... 161 v 2 + 6.8 Zeff/X e -Cu increments vs Lmax ...... 162 6.9 Phase shift of low-energy s-wave e+-Cu scattering vs k ...... 166 v + 6.10 Zeff of low-energy s-wave e -Cu scattering vs k ...... 167 6.11 Experimental Z vs I E −1 ...... 168 eff | − Ps| + 6.12 Zeff of low-energy s-wave e -Cu scattering vs k ...... 170

7.1 Binding energy of known e+Atom systems vs ionisation potential . . . . 175 7.2 The ratio Γ/√ε vs √ε ...... 175

B.1 Schematic diagram of running Gaussian grids ...... 194

F.1 e+ Laguerre exponents with ℓ ...... 226

xii List of Tables

1.1 Decaypropertiesofpara-Psandortho-Ps ...... 4 − 1.2 Atomic properties of Ps, Ps , Ps2 andPsH ...... 15

2.1 Parameters deﬁning the core-polarisation potentials ...... 27 2.2 EnergylevelsofLi ...... 35 2.3 Li− ioncharacteristics ...... 36 2.4 Energy levels of Be+ ...... 37 2.5 Beatomcharacteristics ...... 38 2.6 Energy levels of Mg+ ...... 40 2.7 Mgatomcharacteristics ...... 41 2.8 Energy levels of Ca+ ...... 42 2.9 Caatomcharacteristics ...... 43 2.10EnergylevelsofCu...... 44 2.11 Cu− ioncharacteristics...... 46 2.12 Energy levels of Zn+ ...... 48 2.13 Znatomcharacteristics ...... 49 2.14 Energy levels of Sr+ ...... 51 2.15 Sratomcharacteristics...... 52 2.16 Energy levels of Cd+ ...... 53 2.17 Cdatomcharacteristics ...... 54

3.1 CI calculations of e+Li...... 64 3.2 CI calculations of e+Cu ...... 67 3.3 Partial-wave decompositions of e+Cu and e+Li ...... 73 3.4 Summary of binding energies of e+Li and e+Cusystems ...... 74

4.1 PsH Lmax calculations ...... 80

xiii 4.2 PsH Lint calculations...... 81 + 4.3 e Be Lmax calculations ...... 85 + 4.4 e Be Lint calculations ...... 87

4.5 PsH Lint =3calculations ...... 89 + 4.6 e Be Lint =3calculations...... 90 4.7 Summary of binding energies of PsH and e+Besystems ...... 91

+ 5.1 e Mg Lmax calculations ...... 97 5.2 Summary of binding energies of the e+Mgsystem...... 98

+ 5.3 e Ca Lmax calculations ...... 100 + 5.4 e Sr Lmax calculations...... 102 5.5 e+Mg, e+Ca and e+Sr L 3calculations ...... 104 int ≤ 5.6 e+Mg, e+Ca and e+Srimprovedextrapolations ...... 107

+ 5.7 e Zn Lmax calculations ...... 111 5.8 e+Zn L 3calculations...... 113 int ≤ + 5.9 e Cd Lmax calculations ...... 115 5.10 Summary of e+Zn and e+Cdbindingenergies ...... 118

+ + 5.11 Elastic e -Zn and e -Cd scattering cross-sections and Zeﬀ ...... 123

5.12 CuPs Lmax calculations ...... 128

5.13 CuPs Lint calculations ...... 131 5.14 Summary of binding energies of the 2e−-e+ systems ...... 132

6.1 Positron-atomscatteringprocesses ...... 135

−1 6.2 k = 0.5 a0 s-wave positron-hydrogen scattering ...... 151 −1 6.3 k = 0 a0 s-wave positron-hydrogen scattering ...... 153 −1 6.4 elastic k = 0 a0 s-wave positron scattering from Cu ...... 155 −1 6.5 k = 0.1 a0 elastic s-wave positron scattering from Cu ...... 156 6.6 k = 0.02 0.08 a−1 s-wave positron scattering from Cu ...... 164 → 0 6.7 k = 0.12 0.20 a−1 s-wave positron scattering from Cu ...... 165 → 0 6.8 Eﬀective range theory estimates of A and Zeﬀ ...... 169

B.1 Example inner and outer radial Gaussian grids ...... 195

C.1 Davidson iterations required for e+Cd ...... 201

xiv E.1 Diagonalisation of H atom with pure LTO basis λ = 1.0 ...... 210 E.2 Diagonalisation of H atom with pure LTO basis λ = 0.5 ...... 211 E.3 Radial expectation values of H atom with pure LTO basis λ = 1.0 . . . 212 E.4 Radial expectation values of H atom with pure LTO basis λ = 0.5 . . . 213 E.5 STOheliumatomcharacteristics ...... 214 E.6 STOcalciumatomcharacteristics...... 215 E.7 Gaussian exponents for positronium hydride ...... 216 E.8 PsHverifyingcalculations ...... 216 E.9 Electron scattering lengths of static hydrogen potential...... 218 E.10 s-wave electron scattering phase shifts of static hydrogen potential . . . 219 E.11 e− s-wavescatteringfromhelium ...... 220 E.12 Zero-energy e+ s-wavescatteringfromNe ...... 221 E.13 e+ s-wavescatteringfromNe ...... 221 E.14 e+ s-wave scattering from model hydrogen atoms ...... 223

F.1 Laguerre exponents for positronic systems ...... 225

G.1 pE forpositronicsystems ...... 229

G.2 pΓ forpositronicsystems...... 230

xv xvi Chapter 1

Introduction

“All of the true things I am about to tell you are shameless lies”

Kurt Vonnegut Jr. - The First Book of Bokonon (Cat’s Cradle) 1962

1 The positron (e+) is the anti-particle of the electron (e−). Both particles are

1 leptons, have spin of 2 , and have exactly the same mass and magnetic moment as well as carrying an opposite yet equal charge. It just so happens that due to charge- parity violation we are living in an electron-dominated universe, making the positron the exotic particle of the pair. Positrons are the easiest anti-particle to produce and are routinely used as a means to study the structure of materials and in medical applications such as positron emission tomography (PET). This thesis explores some of the fundamental interactions of positrons with matter. In particular, this is a computationally intensive search for atoms that will stably bind a positron to them. The motivation behind this can be traced back to the discovery in 1997, that lithium was able to form an electronically stable bound state with a positron [1, 2]. Since then, the existence of both positron and positronium atomic bound states has become increasingly well established, and is one of the more interesting problems within the ﬁeld of positron physics [3].

1.1 Positron-electron interactions

The positron ﬁrst made its theoretical appearance in 1930 with Dirac’s relativistic treatment of an electron as a necessary negative energy solution [4]. The experimental observation of positrons followed quickly in 1932 with Anderson’s series of bubble chamber experiments showing cosmic ray tracks consisting of positively charged particles with almost the same charge to mass ratio as that of an electron [5, 6, 7]. When a positron meets an electron it can lead to two rather extraordinary (but related) events which do not occur during electron-electron collisions. Firstly, under certain conditions, they can bind together and form an electronically stable complex called positronium. Secondly they can annihilate with each other, producing γ-rays with a net energy equal to the mass energy of the electron and positron. Positronium (Ps) was initially postulated in 1934 [8] using standard atomic theory, given its name in 1945 [9] and detected in an experiment in 1951 [10]. In many ways the interplay between the positive and negative charges in Ps is similar to that of the hydrogen atom, with the positron directly substituting for the proton. The diﬀerence lies in the centre-of-mass and reduced-mass considerations: for the H atom, the light electron orbits around the heavy proton, whereas in Ps, the equally massive e+ and

2 e− orbit their mutual centre of mass (see Figure 1.1).

Figure 1.1: The positronium atom consists of an electron and a positron orbiting their mutual centre of mass.

The reduced-mass diﬀerence of positronium compared to hydrogen means that the binding energy of Ps is e2/(4πε 4a ) = 0.250 Hartree 6.80285 eV [11] i.e. half 0 0 ≈ that of hydrogen. Correspondingly, the average electron-positron distance is 3.0 a0, meaning that the spatial extent of Ps is twice that of H. Also of interest is the dipole polarisability (αd), which provides a measure of how an atomic system behaves in the presence of an electric ﬁeld (and thus how the system interacts with a charged particle). 3 3 The dipole polarisability of Ps is 36 a0, i.e. 8 times larger than that of H (4.5 a0).

Whilst Ps is electrically stable with respect to the interplay of electron and positron charges, it is unstable with respect to particle-antiparticle annihilation. Ground state positronium comes in a singlet and triplet form with almost the same energy. The 1 3 hyperﬁne level splitting of the lower ( S0) and upper ( S1) states is (to lowest order in α, the ﬁne structure constant) 7α2/12 0.0000301 Hartree [12, 13]. Whilst this is ≈ larger than that seen in hydrogen, it is negligible compared to the Ps binding energy [14]. Para-Ps is the singlet state (1Ps), which has the spin of the electron and positron in opposing directions for a total spin = 0. The triplet state (3Ps) is called ortho-Ps, and has a total spin = 1.

The invariance properties of Quantum Electrodynamics mean that these forms of Ps have diﬀering decay properties which are listed in Table 1.1. Para-Ps predominantly decays into 2γ-rays (each of energy 511 keV), whereas ortho-Ps predominantly ≈ decays into 3γ-rays at a rate about 1000 times slower than para-Ps. The spin-averaged annihilation rate for Ps is 2.008 109 sec−1. Whilst this lifetime of 4.98 10−10 seconds × × may appear short, it represents a reasonably long time at the atomic scale.

3 Table 1.1: The ground state properties of para-Ps and ortho-Ps (to lowest order in

α) where me is the electron mass,h ¯ is Planck’s constant, c is the speed of light, and α is the ﬁne-structure constant.

para-Ps ortho-Ps DominantDecayMode 2 γ-rays 3 γ-rays 2 5 2 2 6 mec α 2(π −9)mec α Annihilation Rate [3] 2¯h 9π¯h 8.0325 109 sec−1 7.2112 106 sec−1 ≈ × ≈ × Lifetime 0.12449 10−9 sec 138.67 10−9 sec × ×

1.1.1 Applications and implications of positron annihilation

There are two unique practical uses of positrons: ﬁrstly for defect studies of materials and secondly in positron emission tomography. Both of these applications require positrons to be produced in a controlled fashion.

There are two ways that positrons are created in the laboratory [15]. Firstly, fast positrons are emitted during β+ radioactive decay of many radio-isotopes (β− decay emits fast electrons). Sodium-22 is most commonly used [15], with C11,O15, F18, V48, Mn52 and Co56 also available as β+ sources.

Secondly, the use of linear accelerator produced high-energy positron beams became possible in laboratory studies during the late 1960s and early 1970s. During this time investigations into high-energy positron beam interactions with foil moderators uncovered a useful property. Whilst most of the positrons annihilated, a small number of slow positrons were emitted with a narrow energy distribution around 1 eV, which could then be used in low-energy positron experiments [15].

Positrons are now a spectroscopic tool with which to study cavities in solid-state materials with [16]. They act as a non-destructive probe since the information about the positron-electron interaction is carried out of the material by the resultant γ-rays. The mean free time of the positron between collisions in materials is large compared to the lifetime of positronium. This allows a positron that has been injected into the surface to penetrate deep into the sample before it slows down to thermal equilibrium and annihilates. When materials have cavities, there is a tendency for positrons to drift into these cavities. The lifetime of the positron inside a cavity is longer since the electron density is lower in the cavity. As the number of cavities in the material

4 increases, so does the lifetime of the positrons in the material, and correspondingly the intensity of the γ-ray signal will be lengthened in time. Positrons have been recently used for a scanning positron microscope [17] to study defect concentrations of the order of one part in a million. Further to these applications, correlation measurements between the two γ-rays emitted in opposite directions (with small angular corrections) give information about the motion of the positron and electron pair just prior to annihilation. This has made possible momentum density and Fermi-surface studies of materials [18], although these are hedged with more uncertainty than the cavity studies.

This non-invasive, non-destructive property also means that positrons are now routinely used for oncological, cardiological and neurological medical imaging, i.e. positron emission tomography (PET). An example of PET in oncology [19] utilises fluorodeoxyglucose (FDG) which has a high concentration in cancer cells. Firstly β+ decaying fluorine-18 (which has a half-life of 110 min) is attached to 2-deoxyglucose to form FDG and injected into the patient. When the F18 eventually decays, the emitted positron will annihilate with a nearby electron, predominantly resulting in the creation of two γ-rays, which travel in opposite directions out of the body without affecting nearby cells etc. Using coincidence detection techniques, high-resolution three-dimensional images of the organs or even the whole body can be formed [19]. This allows for cancer detection before other, more conventional, techniques.

On a local sidenote, the Northern Territory of Australia has a lengthy history of annihilation studies. In 1978, the ﬁrst measurements of a strong signal of 511 keV γ-rays coming from the centre of our galaxy was made using balloon-borne detectors above Alice Springs [20] (the second largest metropolis of the NT, only 1500 kilometres down the road from the NT University). The interactions of high-energy charged particles with gases produce electron-positron pairs, with the resulting annihilation radiation providing unique insight into regions of space where visible light emission may otherwise be obscured by dust particles [21].

Despite the increasing applied use of positrons as both a spectroscopic and medici- nal tool, it has only been in the last few years that theoretical studies of positron-atom dynamics have been possible at a highly-detailed level.

5 1.2 Positron-atom interactions

The typical positron-atom interaction is very diﬀerent from the typical electron-atom interaction. A positron interacting with the short-range electrostatic ﬁeld of a neutral atom experiences a strong repulsive interaction. From the outset this provides a hostile environment for the positron to penetrate through to reach the interior of the atom. However, the presence of a positron causes a polarisation of the atomic electron cloud towards the positron, which creates a long-range attractive potential of the asymptotic form, αd lim Vpol(r) , (1.1) r→∞ ≈−2r4 where αd is the dipole polarisability of the atom. The combination of such short-range and long-range potentials can be seen in Figure 1.2.

Figure 1.2: Schematic diagram of positron - neutral atom potential. At low r, V (r) is dominated by the (repulsive) static potential, whereas at large r it is dominated by the (attractive) polarisation potential of Eqn. 1.1. The depth of the potential well determines whether a positron - atom bound state can be formed. The potential shown here is the static-hydrogen potential with a semi-empirical polarisation potential of the form seen later (Equation (2.18) and ρ chosen for this example to be 0.9 a0).

8 7 6 5 Vstatic 4 3 2 ≈ α 4 1 Vpol - d / 2 r V(r) (atomic units) 0 -1 -2 0 0.5 1 1.5 2 r (a0)

Whenever positrons collide with atoms or molecules there is the possibility of annihilation. As soon as the ﬁrst positron sources were created, so too commenced the ﬁrst annihilation experiments [22, 23].

6 The early positron experiments used MeV positrons introduced into a gas cell. As the positrons thermalise (i.e. rapidly slow down to thermal energies via inelastic collisions with the gas atoms), annihilation is observed. The annihilation rate during a collision with an atom or molecule is characterised by the Zeff parameter, which can be interpreted as the number of electrons available for annihilation during a collision. In the simplest model of annihilation, namely the plane-wave Born approximation, Zeff is equal to the number of electrons in the atom or molecule [24]. This is sometimes called the Dirac rate after P.A.M. Dirac who first estimated the annihilation rate of a positron immersed in an electron gas [4].

One of the salient features of the early annihilation experiments was that the measured Zeff was much larger than the Dirac rate [25]. At that time the suggestion was made that positrons forming bound states with the gas molecules were somehow responsible for the large rates [25, 26, 27]. As time evolved, further research resulted in experiments yielding ever larger values of Zeff . For example, heptane with 58 electrons has a Zeff of 242,000, and decane with 82 electrons has a Zeff of 507,000 [28]! The details of how the bound states actually increased Zeff were somewhat vague, until recently when the University of New South Wales (UNSW) group of Dzuba and collaborators advanced a model in which Zeff was proportional to the density of vibrational levels of the molecule [29, 30].

In recent years fundamental positron physics has seen tremendous experimental advances with tunable beams of cold positrons (of sub-eV energy, 30 meV energy ≈ width) now possible in the laboratory. Using this technology, the group of Surko et.al. recently performed the first experiments directly observing positrons forming bound states with molecules and greatly increasing the Zeff [31, 32]. The experiment used pulses of cold positrons (of energy < 0.5 eV, 25 meV FWHM) injected into a gas cell consisting of either butane C4H10 or deuterated-butane C4D10. By measuring the intensity of the γ-ray signal, resonant peaks in Zeff located just below the stretch- vibration energy of the carbon-hydrogen and carbon-deuterium bonds were observed as shown in Figure 1.3.

7 Figure 1.3: Resonant peaks in Zeﬀ are observed when cold positrons are injected into a gas cell containing butane or deuterated-butane [31]. The peaks are located just below the stretch-vibration energy of the carbon-hydrogen and carbon-deuterium bonds, providing direct evidence of the positrons forming bound states with the molecules (Picture from [32] provided courtesy of Dr. Cliﬀ Surko).

1.3 Nomenclature and units

Since the number of positron binding atoms and ions has expanded enormously over recent years, a systematic approach needs to be taken concerning names and chemical symbols. This work uses the recommendations originally discussed by Schrader [33] and later adopted by Mitroy and Ryzhikh [34] that any atom containing a bound positron is called positronic ”X” where ”X” is the name of the element. The symbol representing the atom is e+Y where Y is the chemical symbol of the element. For example, the system consisting of a positron bound to magnesium is positronic magnesium and should be written as e+Mg.

Another issue to be addressed concerns the use of names or symbols to give a

8 heuristic description of the structure. For example, positronic lithium can be written as e+Li or alternatively as PsLi+ [2]. Analysis of the structure of positronic lithium suggested a model consisting of a polarised Ps atom orbiting the Li+ core at a mean distance of 10 a0 [35]. Current practice in chemistry is that chemical symbols are not uniformly written as a heuristic guide to the structure (otherwise the lithium hydride cation would be written as Li+H, and not as LiH+). Hence, there seems to be no overwhelming reason to insist on chemical names that best describe the structure. Molecules containing positrons are called positronic ”X” where ”X” is the name of the molecule. The symbol representing this system is e+YZ where YZ is the chemical symbol for the molecule. For example, the system containing a positron bound to lithium hydride is named as positronic lithium hydride, which is written as e+LiH. Jiang and Schrader [36] have already appropriated the name positronic water to describe a water molecule with positrons substituted for the protons (i.e. Ps2O). Al- though one can understand the motivation behind the use of the name positronic water, the name positronic water would more properly describe a positron bound to + water, i.e. e H2O. A more appropriate name for Ps2O would be di-positronium oxide or possibly positron substituted water. All formulae and results in this thesis are presented in atomic units (a.u.), in which e = ¯h = me = 1, where e and me are the charge and mass of an electron andh ¯ is Planck’s constant divided by 2π. This results in the atomic unit for energy (called a Hartree), defined such that 0.50 Hartree = 13.6057 eV (the ionisation energy of the infinite mass hydrogen atom). Similarly, the atomic unit of length, a0 =h/ ¯ (meαc) is the radius of the first Bohr orbit of the hydrogen atom, with the dipole polarisability 3 αd given in units of a0. There are a few exceptions to this which are clearly indicated. For example, time is given in seconds, while annihilation rates (Γ) are given in s−1. All phase shifts are given in radians. For the purposes of this thesis, a binding energy, ε, that is positive implies binding, whereas a negative ε implies that the system remains unbound.

1.4 Positron binding to neutral atoms

It has been known for a long time that an extra electron can attach itself to many atoms and form a negative ion, with most of the periodic table accurately surveyed

9 [37]. In fact, the first demonstration of an extra electron binding to a neutral atom came from pioneering calculations on the H− ion by Bethe [38] and Hylleraas [39, 40]. Thus it is not so unusual that the question as to whether a positron could attach itself to a neutral atom and form a bound state was definitively settled in the affir- mative by a calculation. In 1997, two independent yet remarkably similar variational calculations using explicitly correlated Gaussian (ECG) functions both gave a rigorous demonstration that positronic lithium was stable against decay into the lowest dissociation channel, namely Li+ + Ps by 0.00217 [1] and 0.001224 [2] Hartree. The condition for positron binding to a neutral atom is dependant on the ionisation potential (I) of the parent atom. If I is greater than the binding energy of positronium (i.e. E(Ps) = 0.250 Hartree = 6.80 eV) then the energy of the positron-atom complex must be less than the ground state energy of the atom for the system to be stable against dissociation into e+ + Atom (left side of Figure 1.4). The positron affinity, which is defined as the energy gained by the positron when attached to the atom, is in this case equal to the binding energy (ε) of the positronic atom complex. However, if I is less than 0.250 Hartree, then the condition for positron binding requires that the Ps+Atom+ system be stable against dissociation. This occurs when the positron affinity of the atom exceeds 0.250 I (see right side of Figure 1.4). For − atoms with I < 0.250 Hartree the positron affinity is not equal to the binding energy (i.e. a positive positron affinity does not imply binding, and hence the term positron affinity is rarely used in this thesis). Whether positron binding to any neutral atom was even possible, was for many years one of the more intriguing unanswered problems in positron physics [41, 24, 42, 33]. Previous theoretical attempts to study e+ binding to atoms gave no hard evidence of binding, a brief history of which is:

1965, a calculation by Hoang achieved positron binding to lithium, solely as a • result of not taking into account the Pauli exclusion principle [43].

1968, Gertler et.al. disproved the existence of e+He, but remained inconclusive • with respect to e+H [44].

1970, Wang and Ache [45] were unable to achieve a bound e+He state. • 1970, Schrader [46] produced a self-consistent ﬁeld model for positron bind- • ing to closed shell electron systems which stressed the importance of treating

10 Figure 1.4: Energy level diagram for the two conditions for e+Atom binding depending on whether the ionisation potential of the parent atom is greater than or less than the binding energy of positronium (6.80 eV) e+ Binding (IP > 6.80eV) Ps Binding (IP < 6.80eV)

E(Atom+ ) E(Atom+ )

IP 6.80eV

IP E(atom)

E(Ps)=6.80eV E(Ps)=6.80eV ε E(Atom) E( e + Atom) ε E( e + Atom)

electron-positron correlations explicitly. The formalism has not been applied to any positron binding system.

1971, Aronson et.al. proved that e+H was unbound [47], (in agreement with • high-precision variational calculations of the e+-H scattering length [48]).

1973, Cavaliere et.al. [49] used very simple pseudo-potentials in their analysis • of positron binding to all of the alkali atoms. Both the pseudo-potentials and trial wavefunctions were so crude that these calculations were not regarded as a serious prediction of binding (e.g. 0.5 eV positron binding energy for caesium!)

1974, Golden and Epstein [50] using a model potential analysis suggested (with- • out giving deﬁnitive proof) that He, N and Ne were all incapable of binding a positron whilst (in order of increasing likelihood) H, O, Ar and Kr were unlikely to bind.

1976, Clary did large-scale conﬁguration interaction-Hylleraas calculations on • e+He, e+He− and e+Li, and found all to be unstable [51].

1976, Drachman et.al. [42] performed a variational calculation attempting to • bind a positron to excited He(3Se) without success (however they had a good guess as to its eventual structure).

1981, Kurtz and Jordan [52] performed static with polarisation positron scatter- •

11 ing calculations on the alkaline earths Be, Mg, Ca without making any claims of positron binding.

1984, Karl et.al. [53] used spectroscopic analysis about the AH+ and AH systems • to deﬁne potential curves for these systems. These were then applied to the e+ (and Ps binding) problems with appropriate reduced mass corrections. Results suggested that He, Li, N, O, F, Ne, Na, Al and Ar could not bind a positron.

1989, Ward et.al. [54] found the alkali atoms Li, Na, K, unstable against • Ps+Atom+ dissociation during close-coupling calculations of low-energy positron scattering.

1993 Szmytkowski using the polarised orbital scattering method within a rela- • tivistic framework found the alkaline earths (Ca, Sr, Ba, Ra) unstable against Ps-Atom+ dissociation [55]. However, Be and Mg [55] as well as Zn, Cd and Hg [56, 57] were all found to bind a positron. These calculations were restricted to only include dipole polarisation potentials, resulting in positron binding energies of questionable accuracy.

In 1995, many-body perturbation theory (MBPT) calculations by the UNSW • group [58] indicated that e+Mg, e+Zn, e+Cd and e+Hg were all stable. How- ever, the approximations made in these calculations meant that they were not universally accepted as providing rigorous evidence of binding.

1996, Yoshida and Miyako using the quantum Monte-Carlo method were unable • to establish positron binding to lithium [59].

1998, McEachran and Stauﬀer also found that Mg and Zn would bind a positron • [60]. Their polarised orbital calculations included higher order multipoles of the polarisation potential, however the underlying Hartree-Fock model overestimated the dipole polarisability of the parent atoms.

Following the initial demonstrations of positron binding to lithium using ECG basis functions [1, 2], there was a surge in the number of neutral atoms identiﬁed as being able to bind a positron. First the ab-initio Stochastic Variational Method (SVM) [61, 62, 63, 64, 35], Ryzhikh et.al. rigorously proved that Be [35] and metastable He(3Se) [65, 66] could also bind a positron. Since the optimisation of an ECG basis

12 for a system with more than five particles becomes increasingly slow, Ryzhikh et.al. developed the fixed-core SVM (FCSVM) [35], and proceeded to work through the I, II, IB and IIB atoms of the periodic table. In the FCSVM, the electrons are separated into valence and core electrons with the core electrons acting to define the field in which the valence electrons and the positron establish the bound state. The FCSVM improved the convergence of the existing positronic lithium calculations [35, 67, 68] as well as positronic beryllium [35, 69]. Further FCSVM calculations were performed demonstrating positron binding to sodium [70, 35, 67], magnesium [71, 35, 69], copper [72, 73, 68], zinc [74, 73, 68] and silver [75, 73, 68]. Whilst the FCSVM calculations are not ab-initio, they are variational with respect to their underlying model Hamiltonians. The details of the model polarisation potentials are the biggest source of uncertainty, however the predictions of binding for all of these systems are maintained in calculations with no core-polarisation potentials [35, 70, 71, 72, 74, 75]. It should also be noted that the initial calculations of positronic lithium were quickly verified by utilisation of a frozen-core model of lithium by Yuan et.al. [76], in which they also showed that positronic sodium was stable. They further suggested that potassium and the heavier alkalis would not bind a positron. In 1999 Mitroy and Ryzhikh [77] used the configuration interaction (CI) method to independently verify positron binding to copper, although the calculation was limited by the use of an ad-hoc Slater type basis. This was followed by a relativistic CI treatment of the positronic copper and positronic silver systems by Dzuba et.al. [78, 79] (also demonstrating that positron binding to gold was less likely). In these calculations by the UNSW group, the interaction between the e+ and valence e− was treated explicitly with the relativistic CI method while the core-valence interactions were handled with MBPT.

Positron-neutral atom heuristics

Emerging from the SVM and FCSVM structure calculations was the heuristic picture of a positronic atom wavefunction as a mixture of e+A and PsA+ type conﬁgurations [80]. These positronic atoms can be schematically described by a wavefunction with the general form [75]:

+ Ψ= α Φ(Atom)φ(rp)+ β Ω(Atom )ωPs(R) . (1.2)

13 The Φ(Atom)φ(rp) configuration represents a positron bound to a polarised atomic + core. The Ω(Atom )ωPs(R) configuration represents a polarised positronium (Ps) cluster moving in the field of the residual ion core. In Equation (1.2), rp is the position of the positron, while R is the position of the Ps centre of mass, both relative to the nucleus. The coefficients α and β give the relative weighting of each part of the wavefunction (α2 + β2 = 1), such that the annihilation rate for any positronic atom is approximately equal to β2 2.0 109 sec−1 [75]. All systems demonstrate a significant × × amount of virtual positronium clustering, even those with large ionisation potentials in which the valence electrons are tightly held to the nucleus and the positron orbits at a large radius [75]. In the case of a system like positronic beryllium, the dominant configuration sees the positron orbiting the polarised atom as in Figure 1.5.(a). For a system like positronic lithium, the dominant binding mechanism sees one of the outer electrons attaching itself to the positron, with the resulting positronium atom polarised by and remaining attached to the residual positively charged ion, as seen in Figure 1.5.(b).

Figure 1.5: Structural diagram of the electron and positron charge clouds in the vicinity of an atom. In (a), under the presence of the positron, the atomic electron cloud has shifted towards the positron. In (b), one of the outer electrons has attached itself to the positron, with the resulting positronium atom being polarised by the residual positively charged ion. The structure of e+Be is dominated by conﬁguration (a), whilst the structure of e+Li is dominated by conﬁguration (b).

Calculations of positron binding to a model sodium atom (with an adjustible short- range potential added to provide a continuum of ionisation potentials of the model alkali atom) [81] further conﬁrmed the ideas of Eqn. 1.2. These calculations also provided loose upper and lower limits of the ionisation potentials (0.1767 I 0.479 ≤ ≤ Hartree) for which positron binding to a one-electron atom is possible. This also

14 − Table 1.2: Properties of the polyelectron states: Ps, Ps , Ps2 and PsH. The binding energy, ε, is given in units of Hartree. The mean electron-positron distance, r and h epi the mean electron-electron distance, r , are given in units of a . Also given is the h eei 0 9 −1 spin-averaged 2γ annihilation rate, Γt in units of 10 s .

System Model ε r r Γ h epi h eei t Ps Exact 0.250 3.000 2.0081 Ps− Hylleraas [88] 0.01200507 5.490 8.549 2.080

Ps2 SVM [89] 0.0160038 4.487 6.033 4.465 PsH SVM [35] 0.0391944 3.480 3.574 2.4520 explained the failure of the FCSVM in establishing positron binding to potassium [81], (the ionisation potential of potassium is 0.159516 Hartree [82]).

1.4.1 Positronium binding to neutral atoms

As early as 1946 Wheeler raised the possibility of the formation of ’polyelectron’ bound states involving multiple electrons and/or positrons after using a variational expansion to demonstrate the stability of the two-electron-positron system Ps− [83]. Similarly, Hylleraas conﬁrmed the stability of Ps− [84] and along with Ore demonstrated the stability of the Ps2 molecule (two electrons and two positrons) [85]. Ore further improved the calculations of Ps2 [86], and then proved that positronium hydride (PsH) was stable [87]. The properties of these systems are summarised in Table 1.2, but in hindsight their stability is not so surprising given their structural similarities with H− and H2. Somewhat surprisingly, more has been known about the ability of positronium rather than positrons to bind to neutral atoms. Positronium binding to a neutral atom should be more easily achieved, since the positron is interacting with a negative ion, which is an environment that is more conducive to the formation of a bound state:

At the Hartree-Fock level of accuracy, calculations between 1949 and 1978 demon- • strated the stability of Ps binding to the halides. Simons [90, 91] established the stability of PsCl. Cade and Farazel [92, 93] investigated positronium binding to F, Cl and Br. Kurtz and Jordan investigated PsF [94].

15 1981, Ps− was first detected in an experiment by Mills [95], with later measure- • ments of its annihilation rate as (2.09 0.09) 109 s−1 [96]. ± × 1984, Karl et.al. [53] used spectroscopic analysis about the AH+ and AH systems • to define potential curves and applied to the Ps (and e+ binding) problems with appropriate reduced mass corrections. Over half of the 42 neutral atoms investigated were identified as candidates for Ps binding.

1992, Schrader et.al. [97] in a beam experiment colliding positrons and methane • measured the PsH binding energy as (1.1 0.2) eV. ± 1992, the predictions of Ps binding to the halogens was strengthened with the • application by Schrader et.al. of a model potential quantum Monte Carlo (QMC) method to PsCl [98], and later to PsF and PsBr [99].

1995, Saito et.al. used second order perturbation theory based calculations in • ﬁnding LiPs unstable, but PsF and PsCl bound [100, 101]. Further calculations in 1998 predicted the stability of PsBr and PsI [102].

1996, Harju et.al. using an ab-initio diﬀusion quantum Monte Carlo (DMC) • method found that LiPs was unstable [103].

1997, Yoshida and Miyako [104] found LiPs was stable using DMC. However, • their binding energy was double that of other calculations [105, 69].

1998, ab-initio DMC calculations of Bressanini et.al. [106] demonstrated Ps bind- • ing to the ﬁrst row atoms, Li, C, O and F. Later calculations on LiPs improved their estimate of the binding energy [105].

1998, Jiang and Schrader [36] showed that PsO was stable in their study which • demonstrated the stability of the di-positronic system Ps2O (named more for form than substance as positronic water).

In 1999, exotic systems consisting of two-positrons such as e+PsH [107], Li+Ps • 2 + and Na Ps2 [67, 108] were shown to be stable.

1998, Ps binding to a number of alkali atoms was established by Ryzhikh and • Mitroy ﬁrstly using the ab-initio SVM to rigorously demonstrate the stability of LiPs [109, 69], and then using the FCSVM to obtain the binding energies and

16 annihilation characteristics of LiPs [35, 69], NaPs [71, 35, 69] and KPs [110]. The binding energy of LiPs, 0.012341 Hartree is probably known to an accuracy of 1-2% [69], that for NaPs, 0.008419 Hartree, is most likely accurate at the 15% level [69], while the binding energy of KPs was so far from convergence that it would not be surprising if the actual binding energy was twice as large as the current estimate of 0.003275 Hartree [110].

A variety of diﬀerent techniques have been used to study the structure of PsH, • with its binding energy and annihilation rate now known precisely [66, 89, 111].

1.4.2 Why CI?

That quantum mechanical calculations involving positrons are much more demanding than their electron-only counterparts, is exemplified by the nearly 70 year lag between the first definitive negative ion and positronic atom calculations. The attractive nature of the electron-positron 1/rij interaction enhances the probability of both being located at the same point, whilst the repulsive nature of the electron-electron interaction plus the Pauli exclusion principle decreases the chance of electron co-location. The optimal wavefunction is one that explicitly includes inter-particle coordinates, with Hylleraas-type wavefunctions having been used to high accuracy for up to three particle systems. Beyond that, due to the presence of rij terms in the trial wavefunction, evaluation of the Hamiltonian matrix elements becomes increasingly tedious. Other traditional atomic physics techniques such as Hartree-Fock (HF), configuration interaction (CI) and many-body perturbation theory (MBPT) that are so successful for electron only systems, struggle to describe the strong electron-positron correlations. The limitations of perturbative methods such as the polarised orbital and MBPT in describing positron-atomic bound states, means that the binding energies resulting from these calculations are not expected to be very reliable. It is necessary treat the e+-e− correlations with a minimum of approximation in order accurately determine the binding energy of positron binding systems. The strong electron-positron correlations makes the application of orthodox perturbation theory somewhat problematic and therefore the MBPT calculations of Dzuba et.al. used two different orthogonal manifolds of states in the expansion over intermediate states [58]. This approach is believed to overestimate the strength of the polarisation correlation potential [69].

17 It is known that single-centre positron-atom scattering calculations show a slow convergence of the phase shift with respect to the orbital angular momentum of the target states included in the close-coupling expansion [112, 113]. Polarised orbital calculations of positron-atom scattering have also had to deal with a similar problem [114]. The main issue in applying the single-centred CI method to positron binding systems arises from the attractive electron-positron interaction which leads to the formation of a Ps cluster (i.e. something akin to a positronium atom weakly bound to the rest of the atom). The accurate representation of a Ps cluster using only single particle orbitals centred on the nucleus requires the inclusion of orbitals with high angular momenta [115, 77]. In Schrader’s overview of known positron-atom bound states (submitted in September 1997) he stated [33]: ”the wavefunction for a one-positron, many-electron system must somehow include the effects of the highly correlated motion of each electron-positron pair in such a way as to reproduce positronium embedded in the electron sea. The configuration interaction method, applied so successfully to purely electronic systems, has been demonstrated to be much more slowly convergent for mixed electron-positron systems owing to the effect of virtual positronium in the wavefunction, and it has never been successfully applied to a calculation of a positronium binding energy or an annihilation rate for a positronic compound”.

So far, the two most successful approaches have been the stochastic variational method [61, 64, 35] and the quantum Monte Carlo methods [98, 106, 116, 105, 117]. Both the SVM and QMC are able to describe such correlations since they have no predisposed bias in treating the interactions between any pairs of particles. This explains the remarkable success of both methods in calculating the binding energies and structures of positron bound states.

While proponents of the QMC argue that is able to treat larger systems with comparatively smaller increases in computational expense it does suﬀer from one major drawback. One of the fundamental characteristics of any positron binding systems is its lifetime against electron-positron annihilation. However, the annihilation expectation value is notoriously diﬃcult to extract from a QMC simulation (see [118, 105, 119]).

For a number of reasons, it is increasingly tedious to apply the SVM and its ﬁxed- core variant to heavier systems [69]. For example, calculations upon e+Zn [74] and KPs [110] each took almost one year of computer time, and even then the binding energies and annihilation rates were far from converged. The FCSVM uses an orthogonalising

18 pseudo-potential to enforce orthogonality between the core and valence electrons. As the number of core orbitals increases, not only does the time taken for FCSVM matrix element evaluation increase by a couple of orders of magnitude, but the trial-and-error search takes longer to construct the more complicated nodal structures. In addition to this, FCSVM calculations increasingly suﬀer from round-oﬀ errors since the matrix elements of the orthogonalising operator are always positive and often large, whereas the binding energy is small (and negative) [68].

These FCSVM convergence problems partly motivated us to study the application of the conﬁguration interaction method [120] towards describing the interactions of low- energy positrons with atoms. In order to successfully apply the CI to positron binding systems it was necessary to develop procedures to include an orbital basis with at least 100 electron and 100 positron orbitals [77, 78]. It was quickly determined that the heavier two-electron atoms would be more accessible to a CI-based investigation than an FCSVM-based approach. Importantly, it was found that the CI convergence problems associated with using a large number of single particle orbitals does not become signiﬁcantly more severe as the number of orbitals in the core gets larger, i.e. CI calculations are as slowly (or rapidly) convergent for e+Be as they are for e+Zn.

1.5 Chapter Outlines

Firstly in Chapter 2, conﬁguration interaction calculations of the oscillator strengths and energies of the one- and two-electron atomic systems validate the accuracy of the various core model potentials used throughout the remainder of the thesis. The dipole polarisabilities of the ground states of these systems are also computed since the polarisation potential is the primary dynamical mechanism leading to positron binding for the majority of the positronic systems under investigation.

Chapter 3 outlines the application of the CI method to systems with one valence electron and a positron. The radial dependence of the single particle orbitals is predominantly described by a systematic basis of Laguerre type orbitals. The frozen-core CI calculations of positronic lithium were undertaken purely to determine what it would take to achieve binding for this atom, and remained far from converged. The calculations of positronic copper are closer to convergence, with 94% of the estimated binding energy obtained by explicit calculation. Despite the positron orbiting the cop-

19 per atom at a good distance away from the valence electrons, the convergence of the annihilation rate for positronic copper remains somewhat slow. This is a feature which is seen in the CI description of every positronic system. Chapter 4 outlines the application of the CI method to the bound state systems with two electrons and a positron. Convergence issues were explored by performing CI calculations of positronium hydride. Frozen-core positronic beryllium calculations are also presented, independently verifying the bound state predictions of both the SVM [35] and the FCSVM [69]. The use of a configuration selection rule based on preferential treatment of the electron-positron as opposed to electron-electron correlations is shown to reduce the dimensionality of the CI calculations by about 50%, whilst reducing the overall accuracy of the calculations by only about 1%. Chapter 5 presents the results of calculations of positron binding to the group II and IIB atoms, (Mg, Ca, Sr, Zn and Cd), as well as the binding of positronium to copper. The e+Mg results verify the accuracy of the FCSVM calculations of Mitroy et.al. [69], helping to resolve the standing issue as to whether the MBPT calculations [58] overestimated the binding energy. The stability of e+Ca, e+Sr, e+Zn, e+Cd and CuPs is demonstrated by explicit calculation. These calculations not only give very convincing evidence that these exotic atoms are all electronically stable, but also provide a reasonable description of the structures of these systems. Chapter 6 describes the synthesis of the CI method with the Kohn variational method to describe low-energy s-wave positron scattering off one-electron atoms. Cal- culations of positron scattering from hydrogen are firstly used to validate the CI-Kohn method. This is then applied to positron scattering off copper giving estimates of the scattering length, low-energy s-wave phase shifts and low-energy annihilation parameter Zeff . The present calculation of Zeff is particularly timely as there has been renewed interest in understanding the dynamics of positron annihilation during collision processes [29, 121, 122, 30, 123, 124]. In spite of problems with slow convergence, it will be seen that the CI method is a good tool with which to perform usefully accurate investigations of positron- atom interactions. Since the CI method is a variational method, it will give rigorous predictions of positron binding with respect to the underlying model Hamiltonians. Just as important, the calculations needed to satisfy the requirements for positron binding are manageable, and within the capability of modern computer workstations.

20 Chapter 2

Modelling Atomic Structures

What did the Zen Buddhist say to the hotdog seller?

Can you make me one with everything?

- anonymous via spam email

21 The examination of positron-atom interactions from the outset requires an accurate treatment of the atomic system itself. This chapter outlines the procedures chosen to describe the underlying atomic structures. The frozen-core approach is used to simplify the structure problem by partitioning the electrons into two groups; the core electrons which are largely inert and provide the ﬁeld in which the active valence electrons move. The reliability of each frozen-core model is validated with calculations of the structure and properties of one- and two- valence electron atoms. The results of each model atom are collated in this chapter.

2.1 Hartree-Fock and conﬁguration interaction ansatz

The Hartree-Fock (HF) and CI methods are two of the standard approaches for computing atomic structures [120, 125] and are so often used in conjunction that it makes sense to refer to the combined approach as the CI-HF method. As such, only a brief description of each will be given here. The time-independent Schr¨odinger equation is

H Ψ; LS = E Ψ; LS , (2.1) | i | i where Ψ; LS is the wavefunction and E is the energy of a system with a total angular | i momentum L and total spin S. The non-relativistic Hamiltonian for an atom with a nucleus of charge Z and with Ne electrons is:

Ne Ne 1 2 Z 1 H = i + , (2.2) − 2∇ − ri rij Xi=1 Xi

22 parameters is to search for the optimised parameters such that the resultant variation in the energy (δE) δE = δ Ψ H Ψ = 0 . (2.4) h t| | ti is minimised with respect to the variations in the trial wavefunction. The trial energy,

Et, of the system is correct to second-order with respect to the variation in the trial wavefunction itself. However, the other expectation values of the system, e.g. the average radius of the electrons r , are accurate to ﬁrst order. h ei Atomic structure calculations typically begin with some form of central-ﬁeld approximation which uses single-particle orbitals centred on the nucleus to describe the system. The HF wavefunction of a N-electron closed-shell system includes antisymmetry of the electrons via construction of the wavefunction using a Slater deter- minant of the form:

φ1(r1)s(µ1) ... φ1(rN )s(µ1) 1 . .. . Ψ= . . . . (2.5) √N!

φN (r1)s(µN ) ... φN (rN )s(µN )

The spin function of each single particle orbital is given by s(µ i). The spatial part of the single particle orbitals, φi(rj), that compose the wavefunction are written as a product of a radial function and a spherical harmonic, viz.

P (r) φ (r)= iℓi Y (ˆr) . (2.6) i r ℓimi

The set of single particle orbitals φi(r) are mutually orthogonal since the radial functions Piℓi (r) for a given ℓ are chosen to be orthogonal. It is also assumed that Piℓi (r) are independent of spin angular momentum projections, which is known as the restricted HF approximation. The radial functions are further expanded in a discrete set of basis functions, the rationale behind the choice of which is discussed later in Section 2.3. The CI method [120] follows naturally from the Hartree-Fock solution. A LS- coupled wavefunction is built as a combination of conﬁgurations that can be instruc- tively organised as

23 φα(r), from the HF configuration removed and replaced with a virtual orbital φi(r) (which may or may not be of the same spin and angular momentum, but the overall coupling must still equal LS). The Φij ; LS configurations have two such orbital | αβ i replacements performed simultaneously. Thus, the trial wavefunction consists of a superposition of configurations [126, 120]. For the purposes of this chapter, the CI discussion is restricted to the two-electron case. Firstly, a two-electron configuration state function with good L and S quantum numbers is defined by coupling single-particle electron states centred on the nucleus using the usual Clebsch-Gordan coupling coefficients ;

φ φ ; LS = ℓ m ℓ m LM 1 µ 1 µ SM φ (r )s(µ )φ (r )s(µ ) . (2.8) | i j i h i i j j| Lih 2 i 2 j| si i 1 i j 2 j m ,m µ ,µ Xi j Xi j An antisymmetrised two-electron atomic wavefunction Φ ; LS is taken to be a linear | I iA combination of such conﬁguration state functions:

1 Φ ; LS = φ φ ; LS +( 1)ℓi+ℓj +L+S φ φ ; LS . (2.9) | I iA | i j i − | j i i 2(1 + δij)

The CI wavefunction isp constructed as a linear combination of NCI two-electron con- ﬁgurations: NCI Ψ; LS = c Φ ; LS . (2.10) | i I | I iA XI=1 The expansion coeﬃcients, cI , are determined from a standard eigenvalue problem

NCI (H ES )c =0 : J = 1 ...N , (2.11) IJ − IJ I CI XI=1 where the Hamiltonian matrix elements are deﬁned as

H = Φ ; LS H Φ ; LS . (2.12) IJ h I | | J i A standard eigenvalue problem is obtained if the overlap, S = Φ ; LS Φ ; LS = δ , IJ h I | J i IJ which is guaranteed when using a set of single particle orbitals which are orthogonal.

There are then NCI solutions to Equation (2.11) each with an eigenvalue E and a corresponding eigenvector deﬁned by a coeﬃcient set c . { I }

2.2 Treatment of frozen-core electrons

The frozen-core procedure used in this thesis closely follows that previously used for both the FCSVM [35] and other frozen-core CI calculations [127, 77]. The advantage is

24 that instead of working with the full N-electron problem, large atoms (i.e. those having more than two electrons) can be dealt with by freezing the core electrons so that the valence electrons move in an eﬀective model core potential that closely approximates the full N-electron behaviour.

A model Hamiltonian, Hm is introduced and the Schr¨odinger equation solved for the model problem;

H Ψ; LS = E Ψ; LS . (2.13) m| im m| im

The model Hamiltonian for an atom consisting of Ne valence electrons outside a frozen- core was:

Ne Ne 1 2 Z 1 Hm = i +Vdir(ri)+Vexc(ri)+Vp1 (ri) + Vp2 (ri, rj) . (2.14) − 2∇ − ri rij − Xi=1 Xi

In this expression, both ri and rj refer to the valence electron coordinates. The core potential consists of a direct core potential (Vdir), a core-exchange operator (Vexc), and a core-polarisation interaction (Vp1 and Vp2 ).

2.2.1 Core-direct and core-exchange interactions

The Coulombic core potential Vcore, generated by the spherical core of electrons is in eﬀect a one-body potential since it only acts on one valence electron at a time, i.e.

Ncore δℓ ,ℓ 1 φ V φ = i j (2L + 1)(2S + 1) φ φ ; L S φ φ ; L S , h i| core| ji ˆ ˆ c c h c i c c|r | c j c ciA 2(ℓi ℓj) c=1 12 X LXc,Sc (2.15) where n runs over the closed shell orbitals, and the shorthand notation ℓˆ= √2ℓ +1 has been employed. The sum is over all LcSc coupled pairs formed from one core and one valence electron. Upon expanding the antisymmetric wavefunction, the Vcore potential is eﬀectively two potentials which are generated separately. Firstly there is the direct potential Vdir which is simply the static Coulomb potential between the Hartree-Fock core electrons and the valence electron;

Ncore δℓ ,ℓ 1 φ V φ = i j (2L + 1)(2S + 1) φ φ ; L S φ φ ; L S . (2.16) h i| dir| ji ˆ ˆ c c h c i c c|r | c j c ci 2(ℓi ℓj) c=1 12 X LXc,Sc Due to the antisymmetric properties of the wavefunctions of both the Hartree-Fock core and the valence electrons, Vcore further consists of a short-range (non-local) exchange

25 component Vexc, where

Ncore δℓ ,ℓ φ V φ = i j (2L + 1)(2S + 1)( 1)ℓc+ℓj +Lc+Sc h i| exc| ji ˆ ˆ c c − 2(ℓi ℓj) c=1 X LXc,Sc (2.17) 1 φcφi; LcSc φjφc; LcSc . ×h |r12 | i The sign of this interaction depends on the symmetry relation between the wavefunctions of both the core and valence electrons. Both the direct and exchange matrix elements are computed without approximation from the HF wavefunctions of the core electrons, using simpliﬁed forms to those of Equations (2.16) and (2.17). The details of which is left for Appendix B.2.3.

2.2.2 Core-polarisation interaction

The operators Vp1 and Vp2 of Equation (2.14) are the one- and two-body core-polarisation potentials. These describe the additional attractive potential created by the presence of the valence particle(s) distorting the core electron cloud. Inclusion of core-polarisation potentials improves the agreement of the frozen-core energies with experiment by about an order of magnitude [128, 129, 130, 131, 127]. This is seen throughout the remainder of this chapter when the energies of the one- and two-electron atoms with polarisation potentials are compared with experiment.

The one-body polarisation potential (Vp1 ) is a semi-empirical polarisation potential derived from an analysis of the spectrum of the parent atom or ion. It has the form α g2(r) V (r)= d ℓ ℓm ℓm . (2.18) p1 − 2r4 | ih | Xℓm 2 The factor αd is the static dipole polarisability of the core electrons and gℓ (r) is a cutoﬀ function designed to make the polarisation potential both smooth and ﬁnite at 2 the origin. In this work, gℓ (r) uses the form adopted by Norcross and Seaton [132]:

g2(r) = 1 exp r6/ρ6 , (2.19) ℓ − − ℓ where ρℓ is an adjustable, ℓ-dependent cutoff parameter. These adjustable parameters unintentionally incorporate into the one-body polarisation potentials some of the relativistic effects not treated by the present non-relativistic approach (of more significance for large Z atoms). In those cases where there are two valence electrons, the response of the core depends on the positions of both electrons. When both electrons are on the same side

26 of the atom, distortion of the core is enhanced. When they are on opposite sides, the distortion of the core is inhibited. This modiﬁcation to the polarisation potential can be included by the addition of a two-body polarisation potential [133]

αd Vp2 (ri, rj)= 3 3 gp2 (ri)gp2 (rj)(ri rj) . (2.20) − −ri rj ·

The relevence of including Vp2 to accurately model correlated electrons has been discussed by Norcross and Seaton [132]. The two-body polarisation potential cutoffs are denoted by ρp2 to differentiate against the one-body ρℓ. Following the work of Mitroy [127], the cutoff parameter in the two-body polarisation potential was chosen to be independent of ℓ and is set to the average of the ρℓ’s.

Core-polarisation parameters

The core dipole polarisabilities are taken from various sources and are shown in Table 2.1. The cutoff parameters, ρ, chosen for each atomic core are also collated in Table 2.1. The cutoff parameters for both Li+ and Be2+ were taken from [35], while those 2+ for Ca are taken from [127]. The values of ρℓ for the remaining systems were set by minimising the differences between the model potential and experimental energies for the one-electron systems described later in this chapter. The values of ρℓ for ℓ > 3, and the two-body ρp2 were taken as the arithmetic mean of ρ0, ρ1, ρ2 and ρ3.

3 Table 2.1: Core dipole polarisabilities αd (in a0) and one-body cutoﬀ parameters ρℓ

(in a0) deﬁning the various semi-empirical core-polarisation potentials. The two-body polarisation potential cutoﬀ parameter used in Equation (2.20) is given by ρp2 . The citation in the αd column gives the source of the core dipole polarisability.

Core αd ρ0 ρ1 ρ2 ρ3 ρ≥4 ρp2 Li+ 0.1925 [134] 1.40 1.40 1.40 1.40 1.40 1.40 Be2+ 0.0523 [135] 0.95 0.95 0.95 0.95 0.95 0.95 Mg2+ 0.4814 [136] 1.1795 1.302 1.442 1.52 1.361 1.361 Ca2+ 3.16 [136] 1.6516 1.6594 1.9324 1.77 1.77 1.75 Cu+ 5.36 [137] 1.9883 2.03 1.83 1.80 1.91 1.91 Zn2+ 2.294 [74] 1.63 1.80 2.30 1.60 1.83 1.83 Sr2+ 5.813 [137] 1.755 2.0174 2.714 2.402 2.2221 2.2221 Cd2+ 4.971 [137] 1.68 1.97 2.52 1.79 2.00 2.00

27 Naming convention of computations

One issue that should be clariﬁed is that, so far, CI has referred to the CI method itself. In the context of describing a speciﬁc calculation that has employed the CI method, the convention adopted is that CI denotes that the calculation involved no polarisation potentials, while CIpol denotes that this calculation included a semi-empirical polarisation potential.

For the purposes of comparing the present CIpol calculations against other calculations, where the other calculations employed a similar polarisation potential to that adopted for the CIpol calculations, they will also be similarly named. For example,

FCSVMpol refers to a FCSVM calculation that includes a semi-empirical polarisation potential. However, depending on the context, FCSVM refers to either the method itself, or a FCSVM calculation that does not include a core-polarisation potential. If a diﬀerent method for including polarisation has been used, this will be explicitly rel denoted, e.g. CIpol-MBPT denotes a relativistic CI treatment with MBPT used to generate core-polarisation potentials.

FCSVM core-polarisation treatment

Although the present core-polarisation treatment is very similar to that adopted for the FCSVM calculations [35], the present CI treatment has two advantages. Firstly, irrespective of the angular momentum state of the valence particles, the

FCSVMpol is restricted to a common ρ to describe the polarisation potential of Equa- tion (2.18). This is the main reason why the FCSVMpol binding energies of the purely electronic systems do not agree as well with experiment as the present CI binding energies. This is most evident for the atoms with larger core polarisabilities.

A second diﬀerence results from the FCSVMpol using a linear combination of Gaus- sians to approximate the radial functions g(r). However, the impact of using the Gaussian expansion is small [35, 69].

MBPT core-polarisation treatment

An alternative method to deﬁne the core-polarisation potentials is to use many-body perturbation theory. While the CI-MBPT method of the UNSW group provides an ab- initio method of doing this, it comes at the cost of increased computational complexity

28 [78, 79]. It should be noted that semi-empirical factors were still required to improve the agreement of the energies resulting from the ab-initio MBPT generated potentials with the experimental energies [78, 79].

2.2.3 Orthogonality with the core electrons

To prevent the wavefunctions of the valence electrons collapsing into a conﬁguration which includes core electron orbitals, orthogonalisation is required. A Gram-Schmidt orthogonalisation of the orbital set is performed to ensure that all of the single particle orbitals used to represent the valence electrons were orthonormal to the core. This is trivially done for the present CI calculations which solely consist of single particle orbitals centred on the nucleus. Constructing an FCSVM wavefunction that has zero-overlap with the core is non- trivial as explicit orthogonalisation would be very complicated. The FCSVM used an orthogonalising pseudo-potential projection (OPP) operator Pˆ [138, 139, 35], that adds an energy penalty to the model Hamiltonian if there is any overlap of the core and valence wavefunctions, i.e.

Ncore H = H + λPˆ = H + λ φ φ . (2.21) OPP m m | cih c| c=1 X When the constant λ is made suﬃciently large, a variational method will implicitly seek to minimise any overlap with the occupied core orbitals [140], The ability to use an OPP was implemented in the present CI program, but was not used for any of the results reported in this thesis.

2.3 Choice of basis functions

The radial functions of the single particle orbitals, Piℓi (r), of Equation 2.6 are further expanded in a discrete set of radial basis functions χα,ℓi (r). These can be diﬀerent for both electron and positron orbitals, and are written as

i Piℓi (r)= dα rχα,ℓi (r) , (2.22) α X where the coeﬃcients, dα, provide the relative weighting of each radial basis function in each orbital. The aim is to use a discrete set of L2 square-integrable basis functions that can be eﬀectively expanded towards completeness [141]. There are a number of

29 different sets of radial basis functions that are popular in atomic structure calculations. Examples include Slater type orbitals [142, 143], Gaussian type orbitals [144], Laguerre type orbitals [145] and B-splines [146]. Although some of these basis functions have the property that their matrix elements can be evaluated analytically, it was decided to compute all matrix elements numerically using tabulations of the orbitals on a radial r-space grid. A consequence of the choice to perform all integrations using numerical quadrature is the ability to use a Gaussian, Slater or a Laguerre basis with equal facility (and even mix the types of basis functions). The procedures used to construct the grid and evaluate the matrix elements (in particular the 1/rij Slater integrals) can be found in Appendix B. The starting point for all these CI calculations was a HF calculation for the ground states of the neutral atom (or singly-charged ion) using the program of Mitroy [147]. These calculations express the HF orbitals as a linear combination of Slater type orbitals (STOs), χ (r)= N rnα−1 exp( λ r) , (2.23) α,ℓ α − α where n ℓ + 1. The definition of the normalisation factor N is given in Appendix α ≥ α section A.1. The set of optimised STO exponents, λα, for each atomic core were taken from existing data compilations [142, 148, 149]. It is known that Slater (and Gaussian) type basis sets can suffer from severe linear dependence problems when the basis set is made arbitrarily large. However, a Laguerre type basis set has superior linear dependence properties, and can be increased to a very large size without any linear dependence issues arising. The form of the Laguerre type orbital (LTO) adopted in this work was

ℓ (2ℓ+2) χ (r)= Nαr exp( λαr)L (2λαr) , (2.24) α,ℓ − nα−ℓ−1 again where n ℓ + 1. The deﬁnitions of the normalisation factor N , as well as the α ≥ α associated Laguerre polynomials L(2ℓ+2) (2λ r) are given in Appendix section A.2. nα−ℓ−1 α As is usual with a Laguerre basis, the LTO functions used a common exponential parameter, λα for a given ℓ, which ensures orthogonality. Since the Laguerre basis is characterised by a single λα for each ℓ, it is simply easier to optimise the energies with respect to variations in the Laguerre basis. The main problem with a pure Laguerre basis is that it can take a large number of terms to represent the behaviour of the valence orbital wavefunctions both near to

30 and far from the nucleus. For example, calculations of electron scattering from sodium by Bray [150] showed that an inordinately large number of LTOs were required to represent the 3p and 4s orbitals with any accuracy. The optimal exponent needed to represent the wavefunction close to the nucleus is different from the exponent needed to represent the wavefunction far from the nucleus. The approach taken here to the representation of the valence orbitals was based on pragmatism. A linear combination of STOs and LTOs are used to describe the radial dependence of those orbitals with the same angular momentum as those in the atomic ground state. The STOs used to represent the Hartree-Fock core orbitals were retained in the calculation and orbitals were added to span the space defined by the STO set. These STOs give a good representation of the valence wavefunction in the interior region. The LTOs are then added to describe the wavefunction further from the nucleus [81]. The set of orbitals, φ , completely spans the space defined by the raw STO and { i} LTO basis functions since the total number of orbitals was equal in dimension to that of the combined STO + LTO basis. It should be emphasised that the mixed basis was only used for the partial-waves that have core electron orbitals of the same symmetry, all other symmetries (and later, all of the positron orbitals) used a pure LTO basis.

2.3.1 Diagonalisation of the one-electron atoms

The CI diagonalisation of a set of single-electron orbitals (i.e. conﬁgurations) outside a spherical HF frozen-core is equivalent to a frozen-core HF calculation. To reﬂect this, the single-electron frozen-core CI and CIpol calculations are designated as FCHF and

FCHFpol. The one-electron systems; Li, Be+, Mg+, Ca+, Cu, Zn+, Sr+ and Cd+, required the calculation of the s, p, d and f Rydberg excited state energies for comparison against the experimental binding energies. Since the cutoff parameter ρ for the ℓ 3 ℓ ≤ polarisation potentials were tuned using these single-electron calculations, the ℓ 3 ≤ mixed Slater-Laguerre basis included a very large number of LTOs. About 30 LTOs per ℓ ensured at least a six or seven significant figure level of accuracy in the energy of the lowest s, p, d and f states. All of the one-electron energies presented through this thesis are given relative to the energy of the electron core (i.e. the system with one valence electron removed).

31 2.3.2 Choice of two-electron conﬁgurations

The electron basis used for the two-electron calculations were generally the same as the electron basis optimised for use in the positronic atom systems. This permits the comparison of the different CI energies for the positronic atom and the parent atom without having to worry about basis set effects. The two-electron CI basis included all the possible configurations with total angular momentum LT that could be formed by allowing the two electrons to populate all the single particle orbitals with ℓ L with one restriction; an additional parameter ≤ max Lint was defined and used to eliminate configurations involving the simultaneous excitation of both electrons into high ℓ states. Suppose, ℓ1 and ℓ2 are the orbital angular momentum of the two electrons in a given CI basis function, then the rule

min(ℓ ,ℓ ) L , (2.25) 1 2 ≤ int

1 e was applied. For example, a two-electron S system (LT = 0) with Lint = 0 will only allow ℓ = 0 single particles states to be included, which produces a model CI atom somewhat similar to a two-electron HF atom (however, a Lint = 0 CI type calculation can still include electron-electron correlations that a HF calculation is unable to!). This cumbersome notation is introduced here to maintain compatible notation with that adopted for the positronic atom CI calculations. All of the two-electron energies presented through this thesis are given relative to the energy of the electron core (i.e. the system with the two electrons removed).

2.3.3 Determination of dissociation limits

The calculated CI energy of any atomic state depends on both the details of the model Hamiltonian, and the details of the orbital basis used for the CI calculation. For computational reasons the orbital basis sets used for the calculations of a particular atomic species with different numbers of valence particles (e.g. neutral Cu, e+Cu and CuPs) are sometimes not the same. This raises the question of whether the calculation that gives the best energy is used for each individual species or whether the dissociation energy is computed using exactly the same orbital basis for each species. The viewpoint taken in this thesis is that a consistent orbital basis is used for all of the different species. The extreme example of this is that there are three different electron basis sets used for copper calculations in this thesis; the extended Cu basis used to tune the ρℓs,

32 secondly the e+Cu set and thirdly the smallest CuPs set. The energy shifts associated with using these diﬀerent basis sets are typically very small, e.g. of order 10−6 Hartree. Instances where this occurs for speciﬁc systems will be explicitly mentioned in the text.

2.4 Oscillator strengths, sum rules and αd

An accurate description of the polarisation potential of the atom (core + valence) is critical in describing the long-range interaction of a charged particle with the atom or ion. As a positron (electron) approaches a charge cloud, the atomic electrons will be attracted to (repelled from) the incoming particle (given that the positron is moving so slowly that the atomic electrons have time to react). This is particularly important here, since positrons bind to neutral atoms largely as a result of the polarisation interactions between the positron and the neutral atom [151] (although in the case of positronic atoms such as e+Li, e+Ca and e+Sr it is better to think of the polarisation interaction between Ps and a residual positive ion). It is known that the leading term in the attractive polarisation potential at large r 4 is αd/2r , where αd is the static dipole polarisability of the atom [152]. The next term ′ 6 ′ in the long-range interaction is αq/(2r ), where αq has contributions from the static quadrupole polarisability and the non-adiabatic dipole polarisability [152, 153]. Terms of order 1/r6 (and higher) were not included in the model core polarisation potentials used in this thesis. The dipole polarisability can be most conveniently calculated from the transition probability for atomic excitation by the absorption of dipole radiation [120]. The absorption line strength Sif between an initial state i and a ﬁnal state f is given by

S = Ψ O Ψ 2 , (2.26) if |h ik k f i| where the length and velocity forms of the transition operator O are given by

∇j Ol = rj and Oν = , (2.27) Eif Xj Xj where the sum is over the active electron/s coordinates j. In this work only the length form was employed, since an in-depth analysis of atomic transition rates was not the focus of the thesis (Note that the diﬀerences between the two are often small in a large basis CI calculation [120, 127]).

33 When there is a core-polarisation potential employed, a modiﬁed dipole length operator (r′) of the form: α r′ = r 1 d 1 e(−r6/σ6) , (2.28) − r3 − p is required [154, 155, 127]. The value of σ was taken to be the same as the cutoﬀ

ρp2 used for the di-electronic polarisation potential for each atom (since the present approach is to set ρp2 to be the mean of the individual ρℓ’s).

Of particular interest is the absorption oscillator strength, fif given by

2 Eif Sif fif = , (2.29) 3 (2Si + 1)(2Li + 1) where E = E E , and L and S are the quantum numbers of initial state. The if f − i i i convention used throughout the thesis is that the experimental Eif is not used as is sometimes the case to improve agreement with experiment [120, 127]. At all times here the model Hamiltonian energy diﬀerence is used. The static dipole polarisability of an initial state i is calculated using a sum rule over all ﬁnal states [156] arising from the diagonalisation of the model Hamiltonian in a large L2 basis that are accessible via a dipole transition:

fif αd = 2 . (2.30) Eif Xf This sum rule was evaluated by computing the oscillator strengths between the 1Se ground state and the 1P o bound and continuum states obtained from the diagonalisation of the Hamiltonian in exactly the same basis. The present program has been veriﬁed using test calculations of hydrogen and helium (see Appendix E). Further to this, the core-polarisation treatment and oscillator strength corrections were validated by repeating the neutral Ca atom CIpol calculations of Mitroy [127], using exactly the same HF core orbitals, core-polarisation potentials, and STO valence orbitals. These results are given in Appendix Table E.6.

Note that the FCHFpol and CIpol αd’s reported in this thesis always include the contribution from the core dipole polarisability.

2.5 Atoms

This section tests the accuracy of the core potentials with calculations upon the single valence electron systems, and their ground state dipole polarisability. A further test of

34 the underlying model potential is provided by the calculation of the energies of the two- electron 1Se ground and lowest lying nsnp 1P o excited states, the resonant oscillator strength connecting these two states, and the ground state dipole polarisability.

2.5.1 Li and Li−

The model Hamiltonian for lithium uses almost exactly the same model potential as that used for the earlier FCSVMpol calculations [35, 67]. The accuracy with which this type of model describes the structure of both neutral Li (Table 2.2) and Li− (Table 2.3), has been discussed in these previous works [35].

Table 2.2: Theoretical and experimental energy levels (in Hartree) of the low-lying Li atomic states. Energies are given relative to the energy of the Li+ core. The column FCHF gives the energies when only the static terms are included in the core potential, while FCHFpol adds the core-polarisation potential. The experimental energies for the spin-orbit doublets are statistical averages. The last row is the dipole polarisability (in 3 a0) of the 2s ground state, with the FCHFpol result including the contribution from the core. The uncertainty in the last digit of the experimental αd is given in parentheses.

Level FCHF FCHFpol Exp. [157] 2s -0.1963045 -0.1981150 -0.198142 2p -0.1286368 -0.1300152 -0.130235 3s -0.0737970 -0.0741622 -0.074182 3p -0.0567714 -0.0571560 -0.057236 3d -0.0555617 -0.0556118 -0.055606 4d -0.0312406 -0.0312642 -0.031274 4f -0.0312500 -0.0312536 -0.031243

αd 168.99 163.92 164(4) [158]

Table 2.2 also demonstrates that the inclusion of the (even relatively small) Li+ core-polarisation leads to the valence electron being more tightly bound (i.e. an increased one-electron binding energy), and correspondingly the overall dipole polarisability of the model atom is reduced.

35 The electron aﬃnities (EA) of lithium in Table 2.3 are given relative to the one- + electron energy of the e Li basis (i.e. FCHFpol = -0.1981150 Hartree and FCHF = -0.1963045 Hartree), which are also identical to the 2s energies of the extended LTO basis used in Table 2.2. One general trend that is apparent from Table 2.3 is that there is hardly any diﬀerence between the Lint = 3 and Lint = 10 calculations when it comes to representing the two-electron ground state. This is observed in all of the two-electron calculations and this is exploited for the positronic atom calculations.

Table 2.3: Energy and electron aﬃnity (EA) (in Hartree) and dipole polarisability 3 − (in units of a0) of the Li ion as a function of Lint. Energies are given relative to the energy of the Li+ core (uncertainty in the last digit of the experimental energies are given in parentheses). Model calculations when only the static terms are included in the core potential are denoted by CI, while CIpol adds the core-polarisation potential.

The number of conﬁgurations for each symmetry are denoted by NCI . The CIpol αd includes both core and valence contributions.

1 e 2 1 e 1 o Model Lint NCI ( S ) E(2s S ) EA NCI ( P ) αd CI 10 1353 -0.2190045 0.0227000 2280 763.49 FCSVM [35] -0.219017 0.022713

CIpol 0 153 -0.2013918 0.0032768 255 2875.2

CIpol 1 273 -0.2206811 0.0225661 480 752.16

CIpol 2 393 -0.2207815 0.0226665 705 755.63

CIpol 3 513 -0.2208112 0.0226962 930 755.29

CIpol 10 1353 -0.2208322 0.0227172 2280 754.95

FCSVMpol [35] -0.220852 0.022731 Exp. [159] -0.220837(7) 0.022695(7)

2.5.2 Be+ and Be

The model potential calculations of beryllium by Norcross and Seaton [132] were the ﬁrst calculations to use one- and two-body core-polarisation potentials, demonstrating the ability of this type of calculation to accurately describe the spectrum of a two- electron atom. The model Hamiltonian for beryllium is almost exactly the same as the model potential used for calculations of electron-Be+ scattering by Mitroy and

36 + Norcross [160], as well as the FCSVMpol e Be calculations by Ryzhikh et.al. [35, 69]. That this model accurately describes the structure of both Be+ (Table 2.4) and neutral Be (Table 2.5) has been discussed in these previous works, and is not discussed in any detail in this thesis.

Table 2.4: Theoretical and experimental energy levels (in Hartree) of the low-lying Be+ ion states. Energies are given relative to the energy of the Be2+ core. The organisation of the table is otherwise the same as Table 2.2.

Level FCHF FCHFpol Exp.[157] 2s -0.6660704 -0.6691460 -0.669248 2p -0.5194244 -0.5231700 -0.523749 3s -0.2664913 -0.2671650 -0.267233 3p -0.2284143 -0.2293554 -0.229577 3d -0.2222936 -0.2224982 -0.222477 4s -0.1428683 -0.1431221 -0.143153 4p -0.1276598 -0.1280347 -0.128132 4d -0.1250413 -0.1251353 -0.125124 4f -0.1250002 -0.1250157 -0.125008

αd 24.817 24.445

As mentioned earlier, the two-electron calculations shown in Table 2.5 use exactly the same electron orbital basis as that used for the e+Be calculations. Also included in the table is the recent high-precision ECG result by Komasa [161] of αd = 37.755 3 3 a0, which agrees with the present dipole polarisability of 37.710 a0 to around 0.1%.

The use of an ℓ-dependent ρℓ could improve the agreement with experiment, but this was not investigated since any improvement would be marginal. Of importance is that the L = 0 calculation has an anomalously large α 60 int d ≈ 3 2 2 a0. This demonstrates the need to include (2s +2p ) conﬁguration mixing to describe the neutral beryllium ground state. Using the Lint = 0 type structure to describe the parent atom in a positronic beryllium calculation will be seen to lead to an anomalously large positron binding energy.

37 3 Table 2.5: Energy (in Hartree) and dipole polarisability (in units of a0) of the Be atom as a function of Lint. Energies are given relative to 2+ the energy of the Be core. Model calculations when only the static terms are included in the core potential are denoted by CI, while CIpol

adds the core-polarisation potential. The number of conﬁgurations for each symmetry are denoted by NCI . The energy diﬀerence (Ediff in Hartree) and oscillator strength (f in dimensionless units) are of the 2s2 1Se 2s2p 1P o transition. res → 1 e 2 1 e 1 o 1 o Model Lint NCI ( S ) E(2s S ) NCI ( P ) E(2s2p P ) Ediff fres αd CI 3 162 -1.0077397 CI 10 414 -1.0078247 FCSVM [35] -1.007906

CIpol 0 45 -0.9699289 81 -0.7909728 0.1789561 1.8132 59.598 38

CIpol 1 90 -1.0111469 153 -0.8159961 0.1951508 1.3635 36.959

CIpol 2 126 -1.0116326 217 -0.8170582 0.1945744 1.3787 37.583

CIpol 3 162 -1.0117576 281 -0.8173203 0.1944373 1.3799 37.663

CIpol 10 414 -1.0118442 665 -0.8175045 0.1943397 1.3804 37.710

FCSVMpol [69] -1.011953

CIV3CV [162] 820 2524 0.19459 1.3744 CIV3/MCHF [162] 1.375(7) ECG [161] 0.193918 37.755 Beam Foil Exp. [163] 1.34(5) Exp. [157] -1.011848 -0.817908 0.193940 2.5.3 Mg+ and Mg

The present magnesium model Hamiltonian uses core-polarisation cutoff parameters similar to those used in calculations of electron-Mg+ scattering by Smith et.al. [164]. The overall agreement between the theoretical and experimental Mg+ single particle energies can be seen in Table 2.6. The results of Mg calculations are shown in Table 2.7, which used exactly the same electron orbital basis as that used for the e+Mg calculations. The advantage of using different ρℓ for different ℓ are apparent for the two-electron magnesium system.

The CIpol energies are closer to experiment than the energies from the FCSVMpol calculations which used a core-polarisation potential with a single ρ set to 1.25 a0 [35]. There is also reasonable agreement with the experimental and theoretical determinations of the two-electron oscillator strengths. At the present time the most sophis- ticated theoretical treatments of the oscillator strengths for the alkaline-earth atoms use a relativistic CI approach (CIrel) to treat the two valence electrons, while core- rel polarisation is treated using MBPT (CIpol-MBPT) [165, 166]. The present oscillator strength of 1.7294 agrees with that of Porsev et.al. [165], 1.725, at a 1% level of accuracy. 3 The dipole polarisability of the largest CIpol calculation was 71.7 a0. However, the ﬁrst and largest term (about 95% of the total) in the sum rule is from the resonant oscillator strength of 1.7294. Assuming that fres = 1.725 of [165] is more accurate, then using this in the sum rule gives an improved estimate of the Mg dipole polarisability of 3 71.4 a0. Irrespective of which αd is to be preferred, the small diﬀerence suggests that the present core-polarisation potentials give an accurate description of the Mg atom.

39 Table 2.6: Theoretical and experimental energy levels (in Hartree) of the low-lying Mg+ ion states. Energies are given relative to the energy of the Mg2+ core. The organisation of the table is otherwise the same as Table 2.2.

Level FCHF FCHFpol Exp.[157] FCSVMpol[35] 3s -0.5418735 -0.5525357 -0.552536 -0.552169 3p -0.3843914 -0.3897370 -0.389736 -0.390024 4s -0.2317993 -0.2343235 -0.234481 -0.234226 3d -0.2249516 -0.2268043 -0.226801 -0.227070 4p -0.1834773 -0.1850145 -0.185114 -0.185114 5s -0.1286736 -0.1296672 -0.129751 4d -0.1265485 -0.1273728 -0.127381 -0.127510 4f -0.1250113 -0.1251535 -0.125153

αd 38.486 35.006 34.62(26) [167]

40 3 Table 2.7: Energy (in Hartree), oscillator strengths (dimensionless) and dipole polarisability (in units of a0) of the Mg atom as a function 2+ of Lint. Energies are given relative to the energy of the Mg core. The organisation of the table is otherwise the same as Table 2.5 (except the resonant transition is the 3s2 1Se 3s3p 1P o transition). → 1 e 2 1 e 1 o 1 o Model Lint NCI ( S ) E(3s S ) NCI ( P ) E(3s3p P ) Ediff fres αd CI 3 306 -0.8187735 CI 10 558 -0.8188771 FCSVM [35] -0.818863

CIpol 0 120 -0.8037026 210 -0.6541418 0.1495609 1.8995 98.417

CIpol 1 225 -0.8316042 336 -0.6727845 0.1588197 1.6854 70.232

CIpol 2 270 -0.8326645 408 -0.6735284 0.1591361 1.7244 71.542

41 CIpol 3 306 -0.8328500 472 -0.6736919 0.1591582 1.7277 71.639

CIpol 10 558 -0.8329654 856 -0.6738040 0.1591614 1.7294 71.687

FCSVMpol [69] -0.832072

CIpol [136] -0.83293 70.74

CIpol [168] 0.1584 1.771 CIrel with core excitations [169] 0.160766 1.709 rel CIpol-MBPT [165] -0.833556 -0.674226 0.159330 1.725 Hanle eﬀect Beam Exp. [170] 1.80(5) e− Beam Exp. [171] 1.86(30) Beam Foil Exp. [172] 1.75(9) Beam Foil Exp. [173] 1.83(9) Exp. [174] 75.0 (35) Exp. [157] -0.833530 -0.673824 0.159705 2.5.4 Ca+ and Ca

2+ 3 The dipole polarisability of the Ca core is 3.16 a0 [136], and has a signiﬁcant eﬀect on the valence electron behaviour. Calcium has been the subject of previous calculations employing similar model polarisation potentials [175, 127, 176]. The Ca2+ core orbitals were obtained from a HF calculation of the Ca+ 4s 2Se state. The present model

Hamiltonian used the same ρℓ cutoﬀ parameters of FCHFpol-based calculations of the spectrum of Ca+ [175]. The agreement between the theoretical and experimental Ca+ single electron energies can be seen in Table 2.8. Also given are the energies of Mitroy

[127] as well as previous experimental [167] and theoretical [176] results for αd. Table 2.9 shows the results of the neutral calcium atom calculations using the same

Lmax = 10 electron basis as that used for the positronic calcium calculations. These oscillator strengths compare well with the extensive CI/B-spline (with model core rel polarisation potential) calculations of Hansen et.al. [177], as well as the CIpol-MBPT calculations of Porsev et.al. [165]. Also shown are some older experimental results, as well as the recent photoassociation experiment of Zinner et.al. [178] which produced a 4s2 1Se 4s4p 1P o oscillator strength about 1-2% larger than the calculations. → 3 The present value of the dipole polarisability was 161.8 a0. As was done for Mg, using the calcium resonant oscillator strength of Porsev et.al. [165] is expected to give 3 an improved estimate of the dipole polarisability for Ca as 158.6 a0.

Table 2.8: Theoretical and experimental energy levels (in Hartree) of the low-lying Ca+ ion states. Energies are given relative to the energy of the Ca2+ core. The organisation of the table is otherwise the same as Table 2.2.

Level FCHF FCHFpol Exp. [82] FCHFpol [127] 4s -0.4163292 -0.4362865 -0.436278 -0.436278 3d -0.3375829 -0.3738577 -0.373917 -0.373918 4p -0.3098305 -0.3208439 -0.320820 -0.320820 5s -0.1931239 -0.1982929 -0.198588 -0.198032 4d -0.1699716 -0.1751333 -0.177246 -0.175151 5p -0.1566758 -0.1600597 -0.160230 -0.160066 4f -0.1251902 -0.1261893 -0.126188 -0.126188

αd 96.331 75.488 70.89(15) [167]

αd 73.8 [176]

42 3 Table 2.9: Energy (in Hartree), oscillator strengths (dimensionless) and dipole polarisability (in units of a0) of the Ca atom as a function of 2+ Lint. Energies are given relative to the energy of the Ca core. The organisation of the table is otherwise the same as Table 2.5 (except the resonant transition is the 4s2 1Se 4s4p 1P o transition). The calculations of Mitroy [127] validated the program (see Appendix Table E.6). → 1 e 2 1 e 1 o 1 o Model Lint NCI ( S ) E(4s S ) NCI ( P ) E(4s4p P ) Ediff fres αd CI 3 277 -0.6332878 CI 10 529 -0.6333537

CIpol 0 120 -0.6371922 165 -0.5270567 0.1101355 2.2450 203.31

CIpol 1 186 -0.6576142 275 -0.5528235 0.1047908 1.6254 157.30

CIpol 2 241 -0.6594776 355 -0.5531316 0.1063460 1.7343 161.67 43 CIpol 3 277 -0.6596207 419 -0.5531891 0.1064316 1.7370 161.73

CIpol 10 529 -0.6597053 803 -0.5532334 0.1064719 1.7386 161.76

CIpol[127] 4 135 -0.659701 186 -0.553569 0.106132 1.820

CIpol[177] 7 4239 -0.661057 7111 -0.553847 0.107210 1.7453 rel CIpol-MBPT [165] -0.661274 -0.553498 0.107776 1.732

CIpol [136] 156.0 Hanle effect Exp. [171] 1.75(23) Hanle effect Exp. [170] 1.74(06) Photoassociation Exp. [178] 1.754(10) Electric field deflection Exp. [179] 169(17) Exp. [82] -0.660932 -0.553164 0.107768 2.5.5 Cu

There are large uncertainties in the frozen-core model for Cu due to the low excitation energy of electrons in the 3d shell. This weak binding of the 3d-shell electrons means 3 that copper has a relatively large core dipole polarisability of 5.36 a0 [137]. The core orbitals used here were taken from a HF calculation of the 3d104s (2Se) ground state. The overall agreement between the theoretical and experimental single particle energies using the Cu core-polarisation potential of Table 2.1 can be seen in Table

2.10. Also shown in Table 2.10 are the single particle energies of the FCSVMpol model

Hamiltonian [72] which used a single ρℓ = 2.0 a0 for all ℓ.

Table 2.10: Theoretical and experimental energy levels (in Hartree) of the low-lying neutral Cu atomic states, oscillator strength of the 4s 4p transition and dipole → 3 polarisability of the ground state (in a0). Energies are given relative to the energy of the Cu+ core. The organisation of the table is otherwise the same as Table 2.2

Level FCHF FCHFpol Exp. [180] FCSVMpol [72] 4s -0.2384806 -0.2839423 -0.283939 -0.283267 4p -0.1249049 -0.1440384 -0.144056 -0.144054 5s -0.0807047 -0.0862654 -0.087392 -0.086180 5p -0.0548389 -0.0588431 -0.058933 -0.058957 4d -0.0551747 -0.0564024 -0.056399 -0.056364 6s -0.0408761 -0.0426884 -0.043143 5d -0.0309209 -0.0314841 -0.031564 -0.031561 4f -0.0312539 -0.0313564 -0.031391 -0.031356

αd 75.676 41.647 41.1 [181] 42.469 49 [158] 49.3 [182]

f4s→4p 0.9619 0.7064 0.659(6) [183]

In addition, Table 2.10 presents theoretical and experimental oscillator strengths of the primary 4s 4p transition and ground state dipole polarisability. The dipole → polarisability of neutral copper with the FCSVMpol core-potential was calculated using 3 the present CI program as 42.469 a0, which is slightly larger than the value obtained 3 for the dipole polarisability, 41.647 a0. It will be seen that this partly explains the diﬀerences in binding energy for the FCSVMpol and CIpol calculations of positronic

44 3 copper. The value obtained for the dipole polarisability, 41.647 a0, can be used to help resolve the existing uncertainty over the dipole polarisability of Cu. The tabulation of 3 3 [184] gives values of 41.1 a0 by Doolen [181] and 49.3 a0 by Gollisch [182]. It is known that the description of the 3d104p state is complicated by conﬁguration interaction with core excited 3d94s4p states [185, 186], which is not included in the present frozen-core model. An assessment of the accuracy of the present polarisability was done by examining the oscillator strength for the resonant 4s 4p collision. The → present oscillator strength of 0.7064, is reasonably close to the experimental value of 0.659 0.006 [183]. The experimental oscillator strength was derived from the life- ± time with a small correction due to an alternate decay path. The Cu 3d104p 2P o level can decay to either the 3d104s 2Se ground state or the 3d94s2 2De meta-stable state. The experimental lifetimes for the 3d104p 2P o level [183] were converted to oscillator strengths using the oscillator strengths for the 3d104p 2P o 3d94s2 2De transition → state quoted in [187]. The corrections were of the order of about 1-2%. Therefore, comparison with the experimental oscillator strength suggests that the present polar- 3 isability of 41.647 a0 may still be slightly too large, with the preferred value being that 3 of Doolen [181] (41.1 a0). Although no explicit calculations upon Ag have been done, the present Cu calculations also have implications for the dipole polarisability of Ag. Once again, there 3 3 are two recommended values [184], they are 48.4 a0 by Doolan [181] and 57.8 a0 by Gollisch [182]. The good agreement obtained with the result by Doolen for Cu suggests 3 that 48.4 a0 should be adopted as the preferred polarisability for Ag.

2.5.6 Cu−

The results of a series of calculations for the Cu− ground state are listed in Table 2.11. The electron orbitals used here were not the same as the electron orbitals used for the + e Cu calculations. Instead, the Lmax = 10 electron orbitals used for the CuPs system were adopted for the calculation of the electron aﬃnity (EA), since that is used later to discuss the electron-electron structural component of the CuPs calculations.

The present EA of 0.033449 Hartree is marginally smaller than the FCSVMpol rel electron aﬃnity of 0.034267 Hartree [188] and about 25% smaller than the CIpol-MBPT EA of Dzuba et.al. [78], 0.0447 Hartree. The experimental EA is 0.04541 Hartree [189].

45 Table 2.11: Energy and electron aﬃnity (in Hartree) and dipole polarisability (in units 3 − of a0) of the Cu ion as a function of Lint. Energies are given relative to the energy of the Cu+ core. Model calculations when only the static terms are included in the core potential are denoted by CI, while CIpol adds the core-polarisation potential. NCI columns denote the number of conﬁgurations for each symmetry. The CIpol αd includes both core and valence contributions. One CIpol calculation had the Vp2 potential rescaled by a factor of 0.7.

1 e 2 1 e 1 o Model Lint NCI ( S ) E(4s S ) EA NCI ( P ) αd CI 3 325 -0.2651266 0.0266461 CI 10 577 -0.2651804 0.0266998 CI [77] 4 419 -0.265169 0.026689 CIrel [78] -0.26424 0.02594

CIpol 0 120 -0.3047555 0.0208139 195 400.07

CIpol 1 211 -0.3160159 0.0320744 351 281.04

CIpol 2 289 -0.3170142 0.0330726 447 277.15

CIpol 3 325 -0.3172601 0.0333186 511 275.64

CIpol 10 577 -0.3173903 0.0334488 895 274.82 CI (0.7 V ) 10 577 -0.3219972 0.0380556 pol × p2 CIpol [77] 4 419 -0.316946 0.033679

FCSVMpol [188] -0.317198 0.034267 rel CIpol-MBPT [78] -0.31802 0.04130 rel CIpol-MBPT Rescaled [78] -0.32869 0.04475 Exp. [189, 180] -0.329354 0.04541

The different way these calculations treat core-polarisation can explain a major part of their differences in the EA. The FCSVMpol [188] and CIpol calculations define the short-range cut-off factor empirically, and use it in the one- and two-body polarisation potentials. However, the usage of an ℓ-dependent cut-off parameter in the CI calculations only results in a minor change to the EA.

46 rel Discrepancy between CIpol and CIpol-MBPT EAs of copper

The ab-initio MBPT core-polarisation potential of the UNSW group underestimates the strength of the core-polarisation potential [78]. Therefore the core-polarisation potentials for individual ℓ values were rescaled to bring the single particle binding energies into agreement with experiment. Dzuba et.al. [78] adopted the following scaling factors, 1.18 for ℓ = 0, 1.42 for ℓ = 1, and 1.8 for ℓ = 2. However, Dzuba et.al. treat one- and two-body polarisation potentials differently since they do not appear to multiply the two-body polarisation potential by any sort of equivalent factor (no explicit statement about this is made in [78], but later calculations upon Ag− and Au− by the same group [79] state no scaling is done to the two-body potential). Therefore their one- and two-body polarisation potentials could be inconsistent in the asymptotic region. The a-priori justification for usage of a polarisation potential that is rescaled for just one part of the core-polarisation potential is uncertain. The effect of the two-body polarisation potential is to decrease the EA. So rel although the CIpol-MBPT calculations do give an electron affinity in agreement with experiment, this has been obtained by the somewhat expedient decision to strengthen only that part of the core-polarisation potential that increases the electron affinity. The electron affinity obtained by Dzuba et.al. when they used their purely ab-initio core-polarisation potential was 0.4130 Hartree, about 10% smaller than experiment. The procedure of Dzuba et.al. was mimicked by weakening the strength of the di-electronic part of the core-polarisation by multiplying it by a scaling factor of 0.70. When this was done, the EA increases to 0.0380556 Hartree (see the CI (0.7 V ) pol × p2 row of Table 2.11). Thus about half of the difference between the CIpol calculated EA and experiment can be recovered by weakening the strength of the two-body potential. The remainder of the difference can probably be attributed to effects not taken into account by the present CIpol calculation. They are relativistic effects, inclusion of other polarities of the core-polarisation potential, and other dynamical effects due to the weak binding of electrons in the 3d10 core.

47 2.5.7 Zn+ and Zn

The Zn2+ core orbitals were obtained from a HF calculation of the neutral Zn 4s2 1Se state. Table 2.12 gives a comparison of the theoretical and experimental binding energies for Zn+. The inclusion of the core-polarisation potentials again dramatically improves the agreement of the frozen core model Hamiltonian with experiment. Also shown in Table 2.12 are the dipole polarisabilities of the 4s 2Se state, which compare well against the extensive model potential calculations of Laughlin [190] (in which the experimental results of Kompitsas et.al. [191] were called into question with regard to the accuracy of the measurements of the higher 4snf energies).

Table 2.12: Theoretical and experimental energy levels (in Hartree) of the low-lying Zn+ ion states. Energies are given relative to the energy of the Zn2+ core. The organisation of the table is otherwise the same as Table 2.2.

Level FCHF FCHFpol Exp. [180] FCHFpol[74] 4s 0.6162460 0.6603015 0.660180 0.656336 4p 0.4142650 0.4368332 0.436629 0.439296 5s 0.2477912 0.2553738 0.257230 0.254613 4d 0.2149902 0.2187392 0.218488 0.219699 5p 0.1913679 0.1963276 0.197577 0.196972 6s 0.1348695 0.1376028 0.138488 4f 0.1251556 0.1258735 0.125887 0.125867 5d 0.1211429 0.1225399 0.122605 0.12304

αd 24.389 18.089

αd(other) 16.73 [190] 15.54(81) [191]

The neutral zinc energies are listed in Table 2.13, with a CIpol two particle energy about 1% smaller than experiment [180]. Of more signiﬁcance is that the energy diﬀerence for the resonant ns2 1Se nsnp 1P o transition (0.2039895 Hartree) is about → 5% smaller than experiment (0.212988 Hartree [180]). Since zinc is a moderately heavy atom, relativistic corrections could be making contributions to the binding energies. While usage of the semi-empirical polarisation potentials will partly compensate for such energy shifts, some residual errors for the excited states can be expected.

48 3 Table 2.13: Energy (in Hartree), oscillator strengths (dimensionless) and dipole polarisability (in units of a0) of the Zn atom as a function 2+ of Lint. Energies are given relative to the energy of the Zn core. The organisation of the table is otherwise the same as Table 2.5 (except the resonant transition is the 4s2 1Se 4s4p 1P o transition). → 1 e 2 1 e 1 o 1 o Model Lint NCI ( S ) E(4s S ) NCI ( P ) E(4s4p P ) Ediff fres αd CI 3 328 -0.9323831 CI 10 580 -0.9325358 FCSVM [35] -0.932496

CIpol 0 136 -0.9736892 192 -0.7762939 0.1973953 1.6865 50.457

CIpol 1 214 -0.9935236 336 -0.7897786 0.2037450 1.5116 40.658

CIpol 2 292 -0.9949422 432 -0.7910883 0.2038539 1.5373 41.182 49 CIpol 3 328 -0.9953609 496 -0.7914038 0.2039572 1.5417 41.232

CIpol 10 580 -0.9955939 880 -0.7916044 0.2039895 1.5437 41.254

FCSVMpol [35] -0.988586 rel CIpol [130] 0.2131 1.590 CIpol [192] 0.21299 1.4004 CI with core-excitations [192] 0.21508 1.563 rel CIpol [193] 0.213244 1.570 39.12 Level-crossing Exp. [194] 1.49(5) Level-crossing Exp. [195] 1.46(4) Exp. [196] 1.55(8) Refractive Index Exp. [197] 38.8(8) Exp. [180] -1.005410 -0.792422 0.212988 The tendency to underestimate the energy difference for the resonant transitions can be expected to lead to polarisabilities that are somewhat too large. This does occur, but the effect is not as large as would be expected by consideration of the 3 energy differences alone. The CIpol polarisability for Zn is about 2.5 a0 larger than experiment. There are probably some fortuitous cancellations with errors in the dipole matrix elements partly compensating for the errors in the energy difference. Again, it is possible to try and correlate different determinations of the oscillator strength and dipole polarisability with each other. The relativistic pseudo-potential calculation of Ellingsen et.al. [193] gave an oscillator strength of 1.57 while reproducing the experimental polarisability of (38.8 0.8) a3 [197]. The oscillator strength from a CI ± 0 calculation explicitly allowing for core excitations and applying relativistic shifts to the energies was 1.563 [192]. These values are slightly larger than two early experimental measurements [194, 195], while agreeing with the most recent measurement of 1.55 ± 0.08 by Martinson et.al. [196]. Since the dipole polarisability is strongly influenced by the oscillator strength of the resonant transition (the present calculations indicate that 90% of the polarisability comes from the resonant transition, 5.5% from the core, and only 4.5% from the rest of the valence excitations), compatibility of the oscillator strength with the dipole polarisability requires an oscillator strength of about 1.56.

2.5.8 Sr+ and Sr

The Sr2+ core orbitals were obtained from a HF calculation of the Sr 5s2 1Se state. 3 Strontium has a core-polarisability of 5.813 a0 [137], which is the most polarisable core of all the atoms investigated in this thesis. The improvement of the theoretical Sr+ single particle energies with the inclusion of the core-polarisation potentials can be seen in Table 2.14.

Table 2.15 shows the results of the neutral Sr atom calculations using the Lmax = 10 electron basis as used for the positronic strontium calculations. These two-electron rel calculations again compare well with the extensive CIpol-MBPT calculations of Porsev et.al. [165]. The CIpol oscillator strength of 1.847 is within experimental error of the experimental determination of 1.81 0.08 by Kelly et.al. [198]. The present dipole ± 3 polarisability of 204.3 a0, is slightly larger than the electric deﬂection experiment result of 186 15 a3 by Schwartz et.al. [199]. Again using the resonant oscillator strength ± 0 3 (1.831) of Porsev et.al. [165], an improved estimate of αd would be 198.5 a0.

50 Table 2.14: Theoretical and experimental energy levels (in Hartree) of the low-lying Sr+ ion states. Energies are given relative to the energy of the Sr2+ core. The organisation of the table is otherwise the same as Table 2.2.

Level FCHF FCHFpol Exp. [157] 5s -0.3797134 -0.4053501 -0.405350 4d -0.3181485 -0.3382608 -0.338262 5p -0.2830460 -0.2948613 -0.294861 6s -0.1806224 -0.1874207 -0.187846 5d -0.1581028 -0.1612391 -0.162323 6p -0.1464697 -0.1501526 -0.150369 4f -0.1255792 -0.1274513 -0.127451

αd 127.470 90.143

51 3 Table 2.15: Energy (in Hartree), oscillator strengths (dimensionless) and dipole polarisability (in units of a0) of the Sr atom as a function 2+ of Lint. Energies are given relative to the energy of the Sr core. The organisation of the table is otherwise the same as Table 2.5 (except the resonant transition is the 5s2 1Se 5s5p 1P o transition). → 1 e 2 1 e 1 o 1 o Model Lint NCI ( S ) E(5s S ) NCI ( P ) E(5s5p P ) Ediff fres αd CI 3 370 -0.5782734 CI 10 622 -0.5783316

CIpol 0 136 -0.5934657 192 -0.4912292 0.1022365 2.3479 245.94

CIpol 1 214 -0.6103234 336 -0.5148031 0.0955203 1.7062 199.31 52

CIpol 2 292 -0.6127920 480 -0.5151487 0.0976433 1.8424 204.29

CIpol 3 370 -0.6129434 576 -0.5151999 0.0977435 1.8451 204.31

CIpol 10 622 -0.6130266 960 -0.5152385 0.0977881 1.8465 204.30

CIpol [168] 0.09614 1.81 rel CIpol-MBPT [165] -0.614409 -0.515901 0.098508 1.831 Level-crossing Exp. [195] 1.92(6) Exp. [198] 1.81(8) Electric ﬁeld deﬂection Exp. [199] 186(15) Exp. [157] -0.614602 -0.515736 0.098866 2.5.9 Cd+ and Cd

With Z = 48, the neutral cadmium atom exhibits significant relativistic effects, and as such has attracted considerable theoretical interest. Previous theoretical approaches include both relativistic [200] and non-relativistic CI [128], relativistic Hartree-Fock (HFR) [201], its multi-configuration variant (MCHFR) [200], and also using the multi- configuration Dirac-Fock method (MCDF) [201]. In the present non-relativistic approach, the inclusion of the semi-empirical core-polarisation potentials will, to an extent, also substitute for some of the relativistic effects arising from the core electrons. The Cd2+ core orbitals were obtained from a HF calculation of the Cd 5s2 1Se 3 state. Cadmium also has a significant core dipole polarisability of 4.971 a0 [137]. The inclusion of the core-polarisation potentials improves the agreement between the theoretical and experimental binding energies of Cd+, which can be seen in Table 2.16.

Table 2.16: Theoretical and experimental energy levels (in Hartree) of the low-lying Cd+ ion states. Energies are given relative to the energy of the Cd2+ core. The organisation of the table is otherwise the same as Table 2.2.

Level FCHF FCHFpol Exp. [202] 5s -0.5521406 -0.6211981 0.621369 5p -0.3808608 -0.4128233 0.412730 6s -0.2286586 -0.2409554 0.243236 5d -0.2065028 -0.2122923 0.212292 6p -0.1796120 -0.1867128 0.187792 7s -0.1259070 -0.1307400 0.132470 4f -0.1255450 -0.1273414 0.127340

αd 36.237 21.824

53 3 Table 2.17: Energy (in Hartree), oscillator strengths (dimensionless) and dipole polarisability (in units of a0) of the Cd atom as a function 2+ of Lint. Energies are given relative to the energy of the Cd core. The organisation of the table is otherwise the same as Table 2.5 (except the resonant transition is the 5s2 1Se 5s5p 1P o transition). → 1 e 2 1 e 1 o 1 o Model Lint NCI ( S ) E(5s S ) NCI ( P ) E(5s5p P ) Ediff fres αd CI 3 361 -0.8388343 CI 10 613 -0.8389819

CIpol 0 120 -0.9240624 210 -0.7394516 0.1846108 1.6618 57.458

CIpol 1 225 -0.9370806 392 -0.7509092 0.1861714 1.4836 49.416

CIpol 2 316 -0.9385767 509 -0.7525354 0.1860413 1.5059 50.037

54 CIpol 3 361 -0.9391903 581 -0.7529619 0.1862284 1.5109 50.073

CIpol 10 613 -0.9394978 965 -0.7532148 0.1862830 1.5129 50.087

CIV3pol [128] 1.42(4) rel CIpol [200] 0.204140 1.57

MCHFRpol [200] 0.203802 1.39

HFRpol [201] 1.388 MCDF with core excitations [201] 0.198255 1.455 Level-crossing Exp. [203] 1.42(4) Phase-shift Exp. [204] 1.12(8) Beam foil spectroscopy Exp. [205] 1.30(10) Refractive Index Exp. [206] 49.65 1.65 ± Exp. [202] -0.951880 -0.752803 0.199078 The present cadmium calculations shown in Table 2.17 gave a dipole polarisability 3 of 50.087 a0 that is about 1% larger than experiment [206]. However, this calculation also underestimates the energy diﬀerence for the resonant transition by about 6.5%. Therefore, it would seem that an oscillator strength of 1.65 or larger would have to used in the oscillator strength sum-rule to get agreement with the experimental dipole polarisability of (49.65 1.65) a3 by Goebel and Hohm [206]. The experimental os- ± 0 cillator strength measurements are smaller than 1.45 [203, 205]. Similarly, relativistic calculations gave resonance oscillator strengths of 1.39 and 1.455 [201], although the basis sets used for these calculations were moderate in size when compared with the present calculations. There are obvious incompatibilities between the experimental oscillator strength and dipole polarisability. An earlier measurement of the polarisability by Goebel et.al. [207] gave a smaller value, namely (45.3 0.2) a3, but Goebel et.al. regard this ± 0 measurement as being less reliable [206]. The oscillator strength measurements are about 30 years old and the situation obviously demands a modern determination of the oscillator strength to resolve this conundrum. My own opinion on the matter is that the dipole polarisability of Cd is probably 3 about 45 a0 (or even lower). There is no reason to believe that the present CIpol calculation should give the correct polarisability for cadmium when a similar model overestimates the polarisability of zinc by about 6%. Based on this analysis, it is expected that the present CI calculations will slightly overestimate the strength of the attractive interaction between a positron and cadmium (or zinc).

55 2.6 Summary

Comparison with the experimental single electron energies indicates the model potential single electron energies are accurate at the 0.1-0.2% level. The calculation of the dipole polarisability of the ground state of neutral copper is particularly timely since the standard polarisability tabulation [184] recommended two different values. Tests upon the structure of Li−, and the group II atoms (Be, Mg, Ca and Sr) reveals that the underlying model Hamiltonian for the valence electrons gives a description of these atoms that gives two-electron energies to within 1% of experiment. The oscillator strengths are generally within 1 2% of the state-of-the-art calculations. The level of − accuracy achieved for the group I and II elements could hardly be any better. The Cu− ion and the group IIB atoms, Zn and Cd, all present a more difficult case for the present frozen-core electron treatment, with fres values of Zn and Cd accurate to around 5 10%. Whilst inclusion of the semi-empirical core polarisation potentials − improves agreement of the two-electron energies with experiment, discrepancies do remain that are unable to be identified by the present frozen-core treatment. This could be a result of not explicitly including relativistic effects or could be due to inaccuracies arising from a frozen-core treatment of the 3d (or 4d) core orbitals.

56 Chapter 3

Positron Binding to one-Electron Systems

“When people run around and around in circles we say they are crazy. When planets do it we say they are orbiting”.

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57 This chapter outlines the application of the CI method to positronic systems with one valence electron. The method is tested with calculations of two positronic systems, namely positronic lithium and positronic copper. Both of these systems have been the subject of previous SVM-based calculations [1, 35, 72, 67, 73, 68], allowing for a detailed assessment of the accuracy of the present CI approach. Although the CI method is one of the most commonly used methods in the calculation of atomic structures, until recently it had not been applied to positronic systems on a large scale. The ﬁrst CI calculation upon PsH did demonstrate the stability of the system, but only yielded 0.3% [208] of the 3-body binding energy. This was improved to 35% by Strasburger and Chojnacki [115] which concluded ”the energy of the complex can be reproduced relatively well at the CI level ... However, it is practically impossible to obtain reliable values of the collision probability [annihilation rate]”. The ﬁrst CI calculation to give a positive result for positron binding to a neutral atom was a calculation on the ground state of e+Cu [77].

3.1 CI method for one-electron positronic systems

The CI wavefunction Ψ; LS is taken to be a linear combination of states created | i by coupling single electron orbitals to single particle positron orbitals using the usual Clebsch-Gordan coupling coeﬃcients ;

Ψ; LS = c l m l m LM 1 µ 1 µ SM φ (r )s(µ )φ (r )s(µ ) . | i i,j h i i j j| Lih 2 i 2 j| si i 1 i j 0 j m ,m µ ,µ Xi,j Xi j Xi j (3.1) The main problem in applying the CI method to positron binding systems arises in attempting to describe positronium clustering using only single particle orbitals centred on the nucleus [115]. To do so requires the inclusion of single particle orbitals with high angular momenta [77]. The model Hamiltonian for an atom consisting of one valence electron and a positron was

1 2 1 2 Z Z Hm = 1 0 + + Vdir(r1)+ Vexc(r1)+ Vp1 (r1) −2∇ − 2∇ − r1 r0 1 Vdir(r0)+ Vp1 (r0) + Vp2 (r1, r0) . (3.2) − − r10

In this expression, r1 refers to the electron coordinate and r0 refers to the positron coordinate. The treatment of the frozen core electrons is the same as that outlined

58 in Chapter 2.2. Whilst the direct potential (Vdir) representing the direct Coulomb interaction with the Hartree-Fock core is the same magnitude for both the valence electron and the positron, it is repulsive for an electron and attractive for a positron.

It should also be emphasised that the exchange potential (Vexc) only operates between the valence electron and the Hartree-Fock core.

2 With regard to the core-polarisation potentials, the same cutoﬀ function gℓ (r) of Equation (2.19) as well as the same polarisation cutoﬀ parameters have been adopted for both the electron and the positron. There is much evidence in the literature that suggests that the positron-atom polarisation potential is stronger than the electron- atom polarisation potential [209]. However, there is almost no information that can be used to independently tune the strength of the positron-atom polarisation.

Once the eigenvalue/vector problem has been constructed and solved (methods for this are outlined in Appendix C), the set of c coeﬃcients composing the CI { i,j} wavefunction of Equation (3.1) are obtained. From this, expectation values can be computed to provide information about the structure of the positronic atom ground state. The mean distance of the electron and positron from the nucleus are denoted by r and r respectively, while the square of the electron-positron distance, r2 , was h ei h pi h epi also computed for some of the systems. The details of calculating these expectation values from the wavefunctions are presented in Appendix D.

3.1.1 Subset basis

As described in Figure 1.4, whether a positron binds to a neutral atom is dependent on the ionisation potential of the atom. Since the binding energies of these systems are typically small compared to the ionisation potential of the atom [35], it is crucial that the model atom description be realistic. As a positronic atom calculation changes the details and/or size of the electron basis, the ionisation potential of the underlying model atom may change and thus the positron binding energy needs to be re-evaluated relative to the modiﬁed model atom. The energy of the underlying model atom is calculated from the electron subset of the same electron/positron orbitals used for the positronic atom.

59 3.2 Electron-positron annihilation

During the 1950’s Shirokov [210] and Lee [211] demonstrated that electron-positron annihilation can be described using the wavefunction resulting from the application of standard non-relativistic Schr¨odinger mechanics (also see the review of Ferrell [23]). The positron-atom system will decay by positron-electron annihilation and the 2γ annihilation rate is computed to give an estimate of the lifetime. The rate is computed from the overlap between the positron and the electron wavefunctions using the Dirac function δ(r r ). The 2γ rate also gives information about the tendency for the e − p electron and positron to form a Ps cluster. The annihilation rate for the 2γ decay summed over all possible ﬁnal states is written in SI units as [212, 21, 66]

Ne Γ = 4πcr2 Ψ(r ,..., r ; r ) Oˆs δ(r r ) Ψ(r ,..., r ; r ) 2γ 0 1 Ne 0 i i − 0 1 Ne 0 i=1 (3.3) X 2 2 3 ˆs = 4πcr0Ne d τ ONe Ψ(r1,..., rNe ; rNe ) . Z

2 2 ˆs where the classical electron radius r0 = e /(4πε0mc ) in SI units. The operator Oi is a spin projection operator that selects spin-0 combinations of the ith electron and the positron, which has the following form

Oˆs = 1 1 S2 . (3.4) i − 2 i,0 The details of calculating Equation (3.3) from δ(r r ) are left for Appendix D.4. All i − 0 of the annihilation rates given in this thesis are spin-averaged, e.g. the 2γ annihilation − rates for Ps, Ps , Ps2 and PsH were given back in Table 1.2. Since the electrons are partioned into two sets, the annihilation rate for the core electrons (Γc) and the valence electrons (Γv) are computed separately. The core annihilation rate is calculated simply as the overlap between the positron and core electron densities. It has been noted previously [35, 81], that this frozen-core approach does not include correlations between the core electrons and the positron. As a consequence, the calculated core annihilation rates Γc are expected to be too small.

60 3.3 Partial-wave convergence and extrapolation

Investigations of the convergence of both bound state and scattering calculations with respect to the angular momenta of the orbital set began with the two-electron investigations of Schwartz [213]. Due to the particularly slow convergence for systems involving positrons, the CI calculations included orbitals with quite high angular momenta. The parameter Lmax is used to denote the maximum angular momentum of any single particle orbital included in the CI calculation. An extrapolation procedure is employed to estimate the contributions from those partial-waves with ℓ>L not explicitly included in the calculation. The L max max → limits were estimated with a simple technique. Making the assumption that the ∞ successive increments, X to any expectation value X scale as 1/Lp for suﬃciently L h i large L, one can write

L ∞ max 1 X = lim XL +∆ p . (3.5) h i Lmax→∞ L LX=0 L=LXmax+1 Given the expectation values of three successive values of Lmax, the power series is simple to evaluate. The coeﬃcient ∆ is deﬁned as

p ∆= XLmax (Lmax) , (3.6) and the exponent p can be derived from L p X max = Lmax−1 . (3.7) L 1 X max − Lmax There is a degree of uncertainty attached to the extrapolation since the asymptotic form in Lmax (i.e. p) is not known for many operators. However, Gribakin and Ludlow [214] have used second order perturbation theory to establish the asymptotic dependence of the increments to the energy pE = 4, and annihilation pΓ = 2 with L.

As will be seen, the asymptotic region for pE is not reached for Lmax as large as 18 in the case of e+Cu or even 30 in the case of e+Li. Irrespective of the uncertainties in p, the errors in making any extrapolations can kept to a reasonable size by explicitly making Lmax as large as possible. Suppose 80% of the energy is given by explicit calculation, and also suppose that the error in the extrapolation correction (of the remaining contribution to the energy) is 20%, then the net error in the energy will be

4%. Given that the annihilation rate is much more slowly convergent with Lmax (i.e. pΓ

61 3.4 Previous CI calculations of e+Li and e+Cu

The bound state of a positron and a neutral copper atom was shown to be stable with a binding energy of 0.005518 Hartree in an application of the FCSVMpol [72]. Subsequently the CI method was used to conﬁrm this prediction of positron binding to copper [77]. The notable feature of the CIpol calculation was that despite using a very large orbital basis with terms up to ℓ = 14 to represent the correlated electron-positron pair, the explicit binding energy of 0.003694 Hartree was 35% under the FCSVMpol binding energy.

A better CI calculation was performed by the UNSW group of Dzuba et.al. [78] who solved the Schrödinger equation in a finite range box of radius 30 a0 while using a B-spline basis to represent the radial dependence of the wavefunction. The advantage of the B-spline basis was that the convergence of the energy with the number of basis functions could be studied systematically. Confining the system inside a box meant that the convergence of the energy with respect to the number of ℓ-terms in the single particle basis was accelerated. The final energy quoted by Dzuba et.al., 0.00625 Hartree incorporated a correction to the energy which took the finite size of the box into consideration.

There have also been two previous applications of CI methods to positronic lithium that failed to achieve binding. Clary [51] applied a mixed CI-Hylleraas technique and more recently a precursor to the present CIpol calculation used a Slater type orbital basis [77, 215].

The e+Li and e+Cu calculations of Mitroy and Ryzhikh [77, 215] used basis sets that can be described as a (roughly) even tempered basis set of Slater type orbitals. The e+Li basis set consisted of a minimum of 13 STOs for the partial-waves from ℓ = 0 14. The e+Cu basis set consisted of a minimum 14 STOs for each of the → partial-waves from ℓ = 0 10. For both calculations, the degree of STO exponent → optimisation was relatively superﬁcial and almost no optimisation was performed for the STOs with ℓ 6. In addition, the linear dependence issues associated with using ≥ an ad-hoc STO basis set were becoming a problem and it was clear that accurate energies could only be obtained by choosing a basis set that could be systematically improved [77, 215].

The present CI calculations of e+Li and e+Cu used a mixed Slater/Laguerre basis

62 which addresses both of these issues. Firstly, problems arising from linear dependence of a large basis which are minimised in the present treatment. Secondly, it was a much simpler proposition to optimise the single exponent (per partial-wave) of the Laguerre basis instead of the multiple exponents that characterise the Slater basis.

3.5 e+Li results

Calculations of positronic lithium (e+Li) by the CI method were very taxing since e+Li consists of a strongly clustered e−-e+ pair located far from the nucleus [67]. Positronic lithium is one of the best examples of a positronic atom that is not particularly suitable to treatment by the CI method. The e+Li system was treated in the frozen core model with one active electron and a positron. Since the one-electron energy of the e+Li electron basis was -0.19811500 Hartree (relative to the energy of the Li+ core), the lowest energy dissociation channel for the positronic system is the Li+ + Ps channel. The results of a series of successively larger calculations are reported in Table 3.1. The largest calculation included angular terms up to ℓ = 30, had a minimum of 15 LTOs per spherical harmonic, with up to a total of 7022 conﬁgurations. The calculations with L 29 satisfy the formal requirement for binding, since the e+Li max ≥ energy is more negative than the energy that of the Li+ + Ps dissociation channel (E = 0.250 Hartree relative to the energy of the Li+ core). The largest calculation − gave an energy of -0.25010782 Hartree (relative to the energy of the Li+ core).

Despite the largest Lmax = 30 calculation including such a large number of single particle orbitals (467 e− and 466 e+ orbitals, requiring the calculation of over 5 108 × electron-positron matrix elements), the condition for binding is only just satisﬁed by

0.00010782 Hartree, only 4% of the FCSVMpol binding energy (0.002477 Hartree) [67]. Due to the size of the basis the exponents of the LTOs for each ℓ are not particularly well optimised. A rough optimisation of the exponents was done when Lmax = 20 (the LTO exponents for the positronic atoms are collated in Appendix F). However, during the course all of the CI e+Li calculations it had been noticed that the optimal values of the LTO exponents for a given ℓ generally changed (exponents decrease) as Lmax was increased and the Ps cluster drifts out from the nucleus. Thus, the binding energy of 0.00010782 Hartree does not represent the variational limit for Lmax = 30.

63 + Table 3.1: Results of CI calculations for e Li up to a given Lmax. The total number of electron and positron orbitals are denoted by Ne and Np. The two-body energy, E(e+Li), is given relative to the energy of the Li+ core (in Hartree). The mean electron-nucleus distance r and the mean positron-nucleus distance r are given h ei h pi in a0. The spin-averaged 2γ annihilation rates for the core and valence electrons are given separately as Γ and Γ (in units of 109 sec−1). The results in the row are c v ∞ from the L extrapolation, with the row p being the extrapolation exponent max → ∞ for each expectation value of the 28, 29, 30 series.

L N N N E(e+Li) r r Γ Γ max e p CI h ei h pi c v 0 17 16 272 -0.19162466 3.83143 15.5651 0.0001971 0.00110 1 32 31 497 -0.19983084 3.95147 10.6469 0.0019038 0.03097 2 47 46 722 -0.21015081 4.15697 7.90811 0.0040191 0.12475 3 62 61 947 -0.21912372 4.37159 7.07809 0.0047141 0.23492 4 77 76 1172 -0.22590348 4.57564 6.80673 0.0047280 0.33816 5 92 91 1397 -0.23093608 4.76634 6.72874 0.0045208 0.42997 6 107 106 1622 -0.23471290 4.94383 6.73272 0.0042612 0.51051 7 122 121 1847 -0.23759628 5.10893 6.77565 0.0040073 0.58104 8 137 136 2072 -0.23983662 5.26260 6.83793 0.0037774 0.64301 9 152 151 2297 -0.24160576 5.40570 6.90968 0.0035751 0.69772 10 167 166 2522 -0.24302309 5.53893 6.98541 0.0033992 0.74626 12 197 196 2972 -0.24511608 5.77797 7.13712 0.0031143 0.82834 14 227 226 3422 -0.24654913 5.98306 7.27907 0.0028997 0.89479 16 257 256 3872 -0.24756164 6.15664 7.40536 0.0027369 0.94954 18 287 286 4322 -0.24829423 6.30132 7.51396 0.0026126 0.99527 20 317 316 4772 -0.24883426 6.42048 7.60525 0.0025171 1.03385 22 347 346 5222 -0.24923859 6.51773 7.68083 0.0024433 1.06665 24 377 376 5672 -0.24954542 6.59667 7.74280 0.0023859 1.09472 25 392 391 5897 -0.24967085 6.63032 7.76938 0.0023620 1.10725 26 407 406 6122 -0.24978110 6.66060 7.79336 0.0023408 1.11889 27 422 421 6347 -0.24987828 6.68785 7.81501 0.0023220 1.12971 28 437 436 6572 -0.24996415 6.71238 7.83454 0.0023053 1.13979 29 452 451 6797 -0.25004024 6.73448 7.85218 0.0022904 1.14919 30 467 466 7022 -0.25010782 6.75441 7.86811 0.0022770 1.15796 p 3.50 3.05 3.00 3.31 2.04 -0.2508864 7.0358 8.0995 0.002111 1.4038 ∞ CI-Hy [51] 84 -0.229499 0.880103 14 [215] 2522 -0.24208711 5.5108 6.9914 0.00351 0.73328

FCSVMpol [67] -0.252477 9.108 9.966 0.00158 1.741

64 The behaviour of the mean positron radius, r is not monotonic. Initially, the h pi positron drifts towards the nucleus as Lmax is increased. Then, having achieved a minimum value, the positron starts to drift away from the atom. This outward drift is accompanied by an outward drift in the mean radius of the electron r . The FCSVM h ei calculations demonstrated that the e+Li system consisted of a deformed Ps atom orbiting the Li+ core [67], with r2 = 16.244. The tendency for the CI wavefunction h epi + to increasingly resemble Ps orbiting a Li core as Lmax increased can be seen in the variation of r2 and Γ with L shown in Figure 3.1. At L = 30, r2 = 18.004 h epi v max max h epi 2 2 a0, which extrapolated to 17.61 a0 (with a p = 3.38). The other notable feature about

Table 3.1 and Figure 3.1 is the very slow convergence with Lmax. Building up the wavefunction for a Ps cluster located at approximately 10 a0 from the nucleus requires a very large partial-wave expansion.

Figure 3.1: The convergence of the valence annihilation rate Γv (left axis, in units of 109 sec−1) and r2 (right axis, in units of a2) with L for e+Li. h epi 0 max 1.4 26

1.2 24 1 ) ) -1 22 -2 0 s 0.8 9 > (a

0.6 2 ep (10 20 v

2 0.4 v 18 0.2

0 16 0 5 10 15 20 25 30 Lmax

The extrapolations to the L using Equations (3.5), (3.6) and (3.7) are max → ∞ only included in Table 3.1 for completeness. Given that the variational optimisation of the radial basis is uncertain, the L projections should not be regarded as max → ∞ a rigorous estimate of the binding energy and annihilation rate. The e+Li calculations were undertaken purely and simply to determine what it would take to get an explicit prediction of positron binding to Li with the CI method, and to highlight the diﬃculties of performing CI calculations upon such systems. The

65 message from this is that CI calculations of systems containing positrons are just going to have to be big. The program development necessary to handle the exacting e+Li calculations had one useful by-product. The numerics of the program had to be made very robust to handle orbitals with ℓ = 30 and consequently the CI calculations upon e+Cu were straightforward by comparison.

3.6 e+Cu results

Since the ionisation potential of neutral Cu (experimentally -0.283939 Hartree [157]) is greater than 0.250 Hartree, the positron binding energy ε is calculated by the identity

ε = E(Cu) E(e+Cu) , (3.8) − with positron binding occuring when ε is positive. The e+Cu ground state was treated as a system with one active electron and a positron. The 28 electrons occupying the core were ﬁxed with the core orbitals being taken from a HF calculation of the Cu 4s 2Se ground state. The properties of e+Cu as given by the current CI calculations with core-polarisation potentials are summarised in Table 3.2. The impact of not including core-polarisation has been seen in previous CI calculations [72, 78]. The same electron basis as that used for the e+Cu calculation gave the energy of the ground state of the model copper relative to the Cu+ core as E(Cu) = 0.2839422 Hartree. Positron binding was thus achieved when L = 3, − max where there was also a large decrease in r . h pi The e+Cu system has an extrapolated binding energy of 0.0050879 Hartree, which rel is slightly smaller than the binding energies given by both the FCSVMpol and CIpol- MBPT calculations. The explicit calculation gives about 95% of the binding energy with the remainder coming from the extrapolation of L . The present energy max → ∞ is relatively close to converged. The Lmax = 18 binding energy changed by about 0.00007 Hartree when the number of radial functions for ℓ 4 was increased from 12 ≥ to 14. It should be emphasised that the positron binding energy given in Table 3.2 (and indeed throughout this thesis) is never itself extrapolated. Rather, the energy of the positronic atom is extrapolated, and the binding energy is then calculated relative to the dissociation threshold (which may also have been extrapolated, as will be seen for e+Be).

66 + + Table 3.2: Results of CI calculations for e Cu up to a given Lmax. The two-body energy, E(e Cu), is given relative to the energy of the Cu+ core (in Hartree). The binding energy according to Equation (3.8) is in column ε (also in Hartree). The organisation of the rest of the table is the same as Table 3.1. The L extrapolation results in the rows and p are from the 16, 17, 18 series. max → ∞ ∞ L N N N E(e+Cu) ε r r Γ Γ max e p CI h ei h pi c v 0 22 18 396 -0.28247030 -0.00147180 3.03527 29.2747 0.000485 0.000196 1 42 35 736 -0.28291330 -0.00102879 3.04439 25.5876 0.001954 0.002248 2 60 51 1024 -0.28344767 -0.00049442 3.06801 21.2510 0.005680 0.011400 3 75 66 1249 -0.28420650 0.00026441 3.11537 16.7758 0.011904 0.035541 4 89 80 1445 -0.28504186 0.00109976 3.17428 13.7698 0.017930 0.071126 5 103 94 1641 -0.28580710 0.00186501 3.23062 12.0665 0.022345 0.109618 6 117 108 1837 -0.28644318 0.00250109 3.27937 11.0933 0.025298 0.146081 7 131 122 2033 -0.28695012 0.00300803 3.32015 10.5028 0.027249 0.178819 8 145 136 2229 -0.28734778 0.00340569 3.35388 10.1226 0.028558 0.207537

67 9 159 150 2425 -0.28765867 0.00371658 3.38167 9.86580 0.029453 0.232482 10 173 164 2621 -0.28790234 0.00396025 3.40458 9.68544 0.030080 0.254076 11 187 178 2817 -0.28809435 0.00415226 3.42348 9.55483 0.030528 0.272762 12 201 192 3013 -0.28824667 0.00430458 3.43912 9.45784 0.030855 0.288954 13 215 206 3209 -0.28836839 0.00442629 3.45210 9.38430 0.031097 0.303021 14 229 220 3405 -0.28846636 0.00452427 3.46290 9.32756 0.031281 0.315277 15 243 234 3601 -0.28854581 0.00460372 3.47193 9.28311 0.031421 0.325992 16 257 248 3797 -0.28861068 0.00466859 3.47950 9.24783 0.031529 0.335388 17 271 262 3993 -0.28866402 0.00472193 3.48588 9.21950 0.031615 0.343658 18 285 276 4189 -0.28870815 0.00476606 3.49127 9.19653 0.031683 0.350960 p 3.31 - 2.94 3.66 4.04 2.18 -0.2890301 0.0050879 3.5385 9.0522 0.03205 0.45863 ∞ 10 [77] 152 156 2156 -0.286961 0.003694 3.3696 9.9319 0.02999 0.21650 [77] -0.287768 0.004496 3.456 9.475 0.0317 0.339 ∞ FCSVMpol [72, 188] 0.005597 3.578 8.663 0.0339 0.544 rel CIpol-MBPT [78] 0.00625 The L = 18 annihilation rate, Γ was only 0.351 109 sec−1. Upon extrapolation max v × with p = 2.18 a value of 0.459 109 sec−1 is obtained. This is about 20% smaller than Γ × the FCSVMpol annihilation rate. Only a small part of this difference can be attributed to the different binding energies of these two models. It has been shown [32, 123] that the ratio Γ2/ε 6.4 1019 s−2/Hartree for positronic atoms with a parent ionisation ≈ × potential greater than 6.80 eV. Positronic atoms with a smaller binding energy have a longer exponential tail which means the positron is less likely to annihilate with the valence electrons. Therefore a calculation that has a binding energy that is 10% smaller will generally give an annihilation rate which is 5% smaller. We suspect that the bulk of the difference with the FCSVMpol annihilation rate is related to the radial basis. It has been noted during the course of these calculations that the annihilation rate is more sensitive to the inclusion of additional orbitals than is the binding energy. The value of pΓ = 2.18 is larger than the asymptotic value of 2 suggested by the analysis of Gribakin and Ludlow [214]. This could be an indication that further optimisation of the radial basis is desirable.

The present binding energy is 10% smaller than the FCSVMpol binding energy rel of 0.005597 Hartree and about 20% smaller than the CIpol-MBPT binding energy of

0.00625 Hartree. The energy diﬀerence between the FCSVMpol and CIpol calculations is mainly due to the diﬀerent core-polarisation potential. As seen earlier in Table

2.10, the polarisability of the Cu ground state computed with the FCSVMpol core- 3 polarisation potential was 42.5 a0, which is slightly larger than the polarisability when 3 computed with the CIpol model potential of 41.7 a0 (see Table 2.10). Therefore, the slightly smaller e+Cu binding energy could be a consequence of a model atom with a slightly smaller dipole polarisability. This has been checked by repeating the CIpol calculations with ρℓ = 2.0a0 for all ℓ which is equivalent to the core- polarisation potential used for the FCSVMpol calculation. When this was done, even without reoptimisation of the LTO exponents, the extrapolated L positron max → ∞ binding energy increased to about 0.00544 Hartree. The remaining discrepancy to the

FCSVMpol results can probably be attributed to the limited dimension of the LTO radial basis.

rel The diﬀerences with the CIpol-MBPT binding energy of 0.00625 Hartree [78] are also likely to be the consequence of two diﬀerent core interactions. As mentioned earlier, the MBPT core-polarisation potential was scaled by multiplying the one-body

68 potentials by scaling factors between 1.18 and 1.80. Since a corresponding scaling factor was not applied to the two-body potential it is likely that strength of the two- body potential is probably too small. Since the two-body e+-e− core-polarisation potential generally decreases the positron binding energy it is not surprising that the rel CIpol-MBPT binding energy is larger than the other predictions of the binding energy. rel It should also be noted that the CIpol-MBPT calculation was a relativistic calculation. These three different calculations with different model Hamiltonians give an indication of the uncertainty in the positron binding energy. The small differences in the binding energies further strengthen the evidence for the stability of e+Cu. One of the largest areas of uncertainty is the specification of the core-polarisation potential acting on the positron. It is quite likely that the present calculation, with a polarisation potential tuned to the electron-core interaction, underestimates the strength of this potential. Comparisons of the scattering length for He, Ne and Ar for electron and positron scattering [124] suggest that the positron core-polarisation potential is stronger than the electron core-polarisation potential. Similarly, the correlation- polarisation potential computed using MBPT (this assumes that a Ps-type configuration is not included in the MBPT expansion of intermediate states) underestimates the strength of the actual polarisation potential [29]. To briefly check this, a CIpol calculation was also performed which strengthened the polarisation potentials for the positron. This was done by setting the ρ (e+)= ρ (e−) 0.1 a , and correspondingly ℓ ℓ − 0 + the Vp2 cutoff ρp2 was decreased by 0.05 a0. Using exactly the same e Cu basis (of Table 3.2) results in a slight increase in the extrapolated binding energy from 0.00509 Hartree to 0.00514 Hartree.

3.7 e+Li and e+Cu partial-wave convergence

The behaviour of the energy and valence annihilation rate shows a slow rate of convergence towards an asymptotic form. This can be seen in Figures 3.2 and 3.3, in which + the binding energy and valence annhilation rates as a function of Lmax for both e Li and e+Cu are displayed.

69 + + Figure 3.2: The binding energy (in Hartree) of e Li and e Cu as a function of Lmax. 0.006 0.004 e+Cu 0.002 0 -0.002 e+Li (Hartree) -0.004 ε -0.006 -0.008 -0.01 0 5 10 15 20 25 30 Lmax

Figure 3.3: The 2γ spin-averaged annihilation rate (in units of 109 sec−1) of e+Li and + e Cu as a function of Lmax for the valence electrons only. 1.2

1 e+Li

) 0.8 -1 s 9 0.6

(10 + 2γ e Cu Γ 0.4

0.2

0 0 5 10 15 20 25 30 Lmax

70 Figure 3.4: The extrapolation exponent pE, relating two separate energy increments + + using Equation (3.7) as a function of Lmax for e Li, and e Cu. The analysis of Gribakin and Ludlow [214] suggests a limiting value of 4 as L . max → ∞ 4 e+Cu 3 e+Li 2 E p 1

-1 5 10 15 20 25 30 Lmax

Figure 3.4 shows values of pE computed using Equations (3.5), (3.6) and (3.7) + + as a function of Lmax for both e Li and e Cu. The convergence of the incremental contributions to the expected asymptotic form with pE = 4 is rather slow. It is entirely probable that the successive increments to the binding energy only achieve the asymptotic form when the binding energy is already converged for all practical purposes. In the later calculations of positron binding to the alkaline-earth elements, the fact that the energy increments are not even close to their asymptotic form meant that a more involved approach to the extrapolation correction was warranted. The extra complexity is hardly warranted for e+Cu since only 5% of the binding energy comes from the extrapolation. The e+Li exponents for the LTOs were optimised at

Lmax = 20, and one would expect that the energy increments decrease faster than they should at the highest Lmax values. A more variationally optimised basis would probably lead to slightly smaller values of pE at the largest Lmax values.

The behaviour of pΓ as a function of Lmax is shown in Figure 3.5. The value of pΓ increases steadily as a function of Lmax and is slightly larger than 2 at Lmax = 30. This inconsistency with the analysis of Gribakin and Ludlow [214] could be due to a radial basis that was not fully optimised.

71 Figure 3.5: The extrapolation exponent, pΓ, relating two separate annihilation rate + + increments using Equation(3.7) as a function of Lmax for e Li, and e Cu. Gribakin and Ludlow [214] suggest a limiting value of 2. 3 e+Cu 2 e+Li 1 Γ

p 0

-1

-2

-3 5 10 15 20 25 30 Lmax

The slow convergence of the CIpol wavefunction is also apparent in the partial- wave decomposition of the valence wavefunction. For an electron-positron system, the percentage of the wavefunction comprising orbitals with ℓ = J is deﬁned as

d = d3r d3r c δ δ ℓ m ℓ m LM 1 µ 1 µ SM J 0 1 i,j Jℓi Jℓj h i i j j| Lih 2 i 2 j| Si× Z i,j mi,mj µi,µj X X X (3.9) 2

φi(r1)s(µi)φj(r0)s(µj) .

The decompositions shown in Table 3.3 reveal that only 33% and 30% of the

+ Lmax = 30 e Li CIpol wavefunction comes from the J = 0 and J = 1 partial-waves. A + projection of the e Li FCSVMpol wavefunction gave 25% and 26% of the wavefunction in these partial-waves [215]. The difference between these percentages was expected + since the CIpol wavefunction could obviously be improved. However, the present e Li + wavefunction is significantly improved over the earlier (STO based) CIpol e Li wavefunction which decomposed as 43% and 32% for the J = 0 and J = 1 terms [77, 215]. + The breakdown of the Lmax = 18 e Cu wavefunction shows that the high ℓ terms still contribute towards a significant part of the wavefunction. The J =0(nsns′) type configurations and J =1(npnp′) terms comprise 84.9% and 9.67% of the CI wavefunction. These are close those obtained from the projections of the latest FCSVMpol wavefunction, given the differences in the polarisation potentials.

72 Table 3.3: The partial-wave decomposition of the e+Cu and e+Li ground states expressed as a percentage (i.e. d 100). For comparison the FCSVM partial-wave J × pol decompositions [215, 188] are given. However, due to the complications of projecting the FCSVM wavefunctions onto spherical harmonics, only the ﬁrst ﬁve partial-waves were computed. The row J , which is the sum of the partial-wave contributions, thus provides a measure of theP contribution from the ℓ> 4 partial-waves in the FCSVMpol wavefunctions. e+Cu e+Li

J CIpol FCSVMpol CIpol FCSVMpol 0 84.85704 82.7838 32.73329 25.0850 1 9.673688 10.7793 30.16426 25.9023 2 3.083015 3.5271 16.07055 16.2376 3 1.272178 1.5042 8.859242 10.4985 4 0.554761 0.6762 4.948303 6.8839 5 0.261329 2.821204 6 0.131096 1.647775 7 0.069416 0.987213 8 0.038535 0.606339 9 0.022294 0.381435 10 0.013372 0.245440 11 0.008285 0.161458 12 0.005281 0.108363 13 0.003451 0.074172 14 0.002305 0.051631 15 0.001569 0.036501 16 0.001085 0.026202 17 0.000761 0.019057 18 0.000538 0.014045 19 0.010480 20 0.007905 21 0.006022 22 0.004631 23 0.003591 24 0.002807 25 0.002209 26 0.001750 27 0.001395 28 0.001117 29 0.000898 30 0.000723

J 100 99.2706 100 84.6073

P 73 3.8 Summary

Calculations of positronic copper and positronic lithium summarised in Table 3.4 highlight the pros and cons of using the CI method to describe positronic systems. The e+Cu calculation is close to convergence. Using an angular basis that included terms up to ℓ = 18 and with the radial dependence of the orbitals described by a large basis of Laguerre type orbitals, about 94% of the expected e+Cu binding energy was obtained by explicit calculation. The convergence of the e+Cu annihilation rate was somewhat slower with ℓ, but about 75% annihilation rate was obtained by explicit calculation. There remains an issue with respect to the precise binding energy of positronic copper. The three calculations with different treatments of the core-polarisation have all resulted in binding energies ranging from 0.0051 to 0.0063 Hartree. The extent to which the core-polarisation potential is also compensating for relativistic shifts in the energy also requires clarification. Whilst the CI method struggles to describe the structure of e+Li, binding has nonetheless been achieved with the inclusion of single particle orbitals of angular momentum up to ℓ = 30. This has demonstrated that a system with a well-defined Ps cluster orbiting at approximately 10 a0 from the nucleus can only be described with a very large partial-wave expansion. The CI method can be expected to be even more slowly convergent for e+Na which has the Ps cluster located even further away from the nucleus [67]. In order to describe such systems with any degree of precision, one should be resigned to performing CI calculations with orbital basis sets that have dimensions exceeding 1000!

Table 3.4: Binding energies (in Hartree) of positronic lithium and positronic copper. Only the latest calculations of a given type by a particular group are listed in this table.

System PresentCIpol FCSVMpol Other Explicit ∞ e+Li 0.000108 0.000886 0.002477 [67] 0.002471 a e+Cu 0.004766 0.005088 0.005597 b 0.00625 c

a Latest SVM calculation of Mitroy and Ryzhikh [73]. b FCSVMpol calculation of Mitroy and Ryzhikh [72], improved in [188]. c rel CIpol-MBPT calculations of Dzuba et.al. [78].

74 Chapter 4

Positron Binding to Simple two-Electron Systems

“I want to take my version of reality from inanimate objects”

- Fred Smith from The Elephant Song, (Soapbox album) 1998

75 In this chapter, the CI method is applied to the calculation of the positronium hydride (PsH) and positronic beryllium (e+Be) ground states. Since accurate binding energies and wavefunctions have been reported for both of these systems [111, 69], they represent an ideal computational laboratory with which to study the application of the CI method to positronic systems with two active electrons. Another reason for investigating these two systems is that they have completely different structures. The PsH system consists of a reasonably well defined Ps atom bound to a H atom, somewhat similar to a light isotope of the H2 molecule [216]. In contrast, positronic beryllium finds the positron orbiting a polarised neutral Be atom at a relatively large distance from the nucleus [69]. Although there are fundamental difficulties associated with the CI treatment of the Ps cluster, the present results indicate it is possible to compute energies and annihilation rates that are sufficiently well converged to be useful. The extrapolated binding energies are within 2% of the best previously computed values, while the annihilation rates are within 10% of the expected values.

4.1 CI method for two-electron positronic systems

The conﬁguration state wavefunction Φ ; LT ST for a multi-electron positronic atom | α i is created by coupling the anti-symmetrised atomic electron states with total quantum numbers LiSi, to single particle positron states using the usual Clebsch-Gordan coupling coeﬃcients;

T T T 1 T Φ ; L S = L M ℓ m L M T S M µ S M T | α i h i Li j j| L ih i Si 2 j| S i M m M µ XLi j XSi j (4.1) Φ (Atom)s(M )φ (r )s(µ ) . × i Si j 0 j For the two-electron-positron systems considered in the remainder of the thesis, each Φi(Atom) consists of anti-symmetrised two-electron conﬁguration state wavefunctions of Equation (2.9) with total quantum numbers LiSi, i.e.

Φ (Atom) = φ φ ; L S i | ai bi i iiA (4.2) 1 ℓa +ℓb +Li+Si = φa φb ; LiSi +( 1) i i φb φa ; LiSi . 2(1 + δ ) | i i i − | i i i ai,bi p The CI wavefunction is constructed as a linear combination of these atomic wavefunc-

76 tions coupled to a single positron orbital (φp(r0)s(µp)), i.e.

NCI NCI Ψ; LT ST = c Φ ; LT ST = c φ φ [L S ] φ ; LT ST , (4.3) | i I | I i I | aI bI I I A pI i XI=1 XI=1 where the Clebsch-Gordan coeﬃcients are implicitly included. For the two-electron- positron systems investigated here, the spin of the intermediate two-electron states is always set to SI = 0.

The model Hamiltonian for an atom consisting of Ne valence electrons and a positron orbiting a closed HF electron core was

Ne Ne 1 2 Z 1 2 Z Hm = i 0 + + Vdir(ri)+ Vexc(ri)+ Vp1 (ri) − 2∇ − ri − 2∇ r0 Xi=1 Xi=1 N N e 1 e 1 Vdir(r0)+ Vp1 (r0)+ Vp2 (ri, rj) + + Vp2 (ri, r0) . (4.4) − rij − − ri0 Xi

The additional terms represent the electron-electron interaction 1/rij and the electron- electron di-electronic core-polarisation potentials V (r , r ). The details of calculat- − p2 i j ing the two-electron-positron matrix elements are presented in Appendix B. Due to the larger dimensionalities of the two-electron-positron problem compared to the electron-electron and electron-positron problems, the resultant Hamiltonian matrix is very large with the number of conﬁgurations approaching 100,000. This matrix has the properties of being both real and symmetric, but fortunately is also sparse, which permits the use of an iterative diagonalisation procedure such as the Davidson algorithm [217], of which the implementation of Stathopoulos and Froese Fischer [218] was used here. This is discussed further in Appendix C.2.

4.2 Choice of two-electron-positron conﬁgurations

Two different approaches were taken to the construction of the CI basis for the two electron-positron systems. In the first, all the possible total LT = 0 configurations that could be formed by letting the two electrons and the positron populate all the single particle orbitals with ℓ L were included in the basis, i.e. ≤ max

max(ℓ1,ℓ2,ℓ0)= Lmax , (4.5) where the orbital angular momenta of the two electrons are denoted by ℓ1 and ℓ2, and the positron ℓ0. The convergence of the binding energy, and other system properties

77 could then be studied as a function of Lmax, thus permitting extrapolation to the L limit. max → ∞ The second approach used to construct the CI basis recognises the fact that the electron-positron correlations are much stronger than the electron-electron correlations. It is the electron-positron correlations that largely mandate the inclusion of orbitals with large values of ℓ. A selection rule, min(ℓ ,ℓ ) L , was deﬁned and used to 1 2 ≤ int restrict the size of the CI basis (see also Equation (2.25)). A basis with Lint = Lmax has no restrictions upon orbital occupancy.

This Lint rule can be motivated by writing the CI expansion of Equation (4.1) in an alternate close-coupling type expansion [219], written heuristically as

Ψ; LS = c φJ (r ) [φPs(r , r )] = c φJ (r )[φ (r )φ (r )] . (4.6) | i i,m i 1 m 2 0 J i,(j,k) i 1 j 2 k 0 J Xim i(mX=j,k) One electron and a positron are coupled to form a state with net angular momentum J, which is then coupled to the second electron (occupying a single particle state with angular momentum J). Suppose it is wished to include a state analogous to the Ps ground state with a centre-of-mass angular momentum of zero coupled to the rest of the system. Then the partial-wave expansion of φPs(r r ) would be written as m 0 − 2 jk [φj(r2)φk(r0)]J=0 with the net angular momentum of the coupled orbital product equalP to zero [219]. Thus, the Lint parameter is equivalent to the maximum orbital angular momentum of the Ps-type state (or H-type state) that would be included in a close-coupling expansion involving products of H-type and Ps-type states.

4.3 Results for PsH

Since the initial demonstration that positronium hydride (PsH) was stable in 1951 [87], a variety of computational methods have been used to study the structure of PsH with the result that both its binding energy and annihilation rate are now known very precisely [66, 89, 111]. The condition for binding is that the energy of the PsH ground state should be lower than the energy of the H(1s) + Ps(1s) dissociation channel. The binding energy (in Hartree) for a particular basis is thus deﬁned as ε = E(PsH) 0.750, and binding | |− occurs when ε is positive. The PsH system uses an exact Hamiltonian consisting of two active electrons and

78 a positron, 2 2 1 2 Z 1 2 Z 1 1 H = i 0 + + . (4.7) − 2∇ − ri − 2∇ r0 r12 − ri0 Xi=1 Xi=1 The PsH calculations give a rigorous lower bound to the energy of the system. The formalism and numerics of the two-electron-positron part of the CI program were initially validated by reproducing the results of a previous CI calculation of PsH by Strasburger and Chojnacki [115]. Their best CI calculation gave a total energy of 0.76369386 Hartree [220]. Only trivial additions to the present CI program were − required to run with a Gaussian basis since all matrix elements were calculated numerically. Using exactly the same Gaussian type orbital basis, the program reproduced their results to all significant figures (these results are presented in Appendix E.1.2). Table 4.1 gives energies and expectation values for a series of calculations on PsH with no restrictions upon orbital occupancy (i.e. Lint = Lmax) using a purely LTO basis. The number of Laguerre orbitals included in each calcualation and the LTO exponents of each ℓ are also listed in the table. The largest calculation included single particle orbitals up to Lmax = 9. The Laguerre exponents were optimised by hand, and it was found the best energy occurred when the electron and positron exponents were the same (these are collated for all systems in Appendix Table F.1). That the electron and positron orbits should be the same for large values of ℓ is understandable since the ℓ(ℓ+1)/(2r2) centrifugal barrier dominates the nuclear attraction or repulsion for large ℓ. A slight improvement over the present calculation could be achieved by letting the electron and positron exponents for the ℓ = 0 and ℓ = 1 orbitals be slightly different, but this was not done. The most notable feature of Table 4.1 is the slow convergence of both the binding energy and annihilation rate with Lmax. Even though the largest calculation had a CI dimension of 95324, only 93.8% of the expected binding energy was achieved (the latest estimates of the PsH binding energy of 0.039197 Hartree [89, 111] are expected to be accurate to 6 significant digits). The convergence of the annihilation rate was worse, with only 72.6% of the SVM annihilation rate [66] being achieved by the Lmax = 9 calculation. The slow convergence of the PsH binding energy with Lmax, and the even slower convergence of Γ is consistent with the CI calculations on positronic copper and positronic lithium.

79 Table 4.1: Results of CI calculations for PsH up to a given Lmax. The three-body energy of the PsH system is given by E(PsH), while ε gives the binding energy against dissociation into H + Ps. Both energies are given in units of Hartree. The mean electron-positron distance r2 is given in a2. The organisation of the rest of the table is the same as Table 3.1. For comparison purposes, ε and Γ are also tabulated h epi 0 v as a % of the SVM result [66]. The L extrapolation results in the rows and p are from the 7, 8, 9 series. max → ∞ ∞ L N N λ N E(PsH) ε ε(%) r r r2 Γ Γ (%) max e p CI h ei h pi h epi v v 0 13 12 1.52 1092 -0.69133422 -0.0586658 - 2.11759 3.86611 24.5069 0.37393 15.14 1 24 23 1.75 3457 -0.74704709 -0.0029529 - 2.08537 3.50096 17.1677 0.78021 31.60 2 34 33 2.02 7837 -0.76616882 0.01616882 41.25 2.12089 3.44727 15.6591 1.07518 43.55

80 3 43 42 2.12 13660 -0.77507853 0.02507853 63.98 2.15569 3.45769 15.2140 1.28204 51.92 4 51 50 2.25 20836 -0.77984446 0.02984446 76.14 2.18393 3.48150 15.0831 1.43063 57.94 5 59 58 2.40 29900 -0.78262966 0.03262966 83.25 2.20596 3.50570 15.0628 1.54138 62.43 6 67 66 2.60 41620 -0.78436101 0.03436101 87.66 2.22282 3.52659 15.0809 1.62649 65.87 7 75 74 2.85 56044 -0.78548799 0.03548799 90.54 2.23556 3.54341 15.1095 1.69369 68.60 8 83 82 3.05 73956 -0.78624814 0.03624814 92.48 2.24519 3.55665 15.1386 1.74754 70.78 9 91 90 3.25 95324 -0.78677609 0.03677609 93.83 2.25249 3.56696 15.1646 1.79129 72.55 p 3.09 - - 2.35 2.13 0.98 1.76 - -0.7887952 0.0387952 98.98 2.2975 3.6441 16.504 2.2792 92.31 ∞ SVM [66] -0.7891961 0.0391961 100 2.311 3.662 15.58 2.4691 100 Equations (3.5) to (3.7) were used to extrapolate the Lmax = 7, 8, 9 results listed in Table 4.1 to the L limit. The extrapolated binding energy is 1% smaller than max → ∞ the expected value of 0.039196 Hartree [111], while the annihilation rate is 8% smaller than the SVM annihilation rate [66]. The extrapolated r and r are also quite h ei h pi reliable. The extrapolation of r2 was not reliable. However, a visual inspection of h epi the sequence of r2 values in Table 4.1 suggests this expectation value has not yet h epi reached its asymptotic region.

Table 4.2 shows the impact that conﬁguration selection through use of the Lint parameter can have in restricting the size of the calculation without any major degra- dation in the quality of results. The data presented in Table 4.2 were computed with the same Lmax = 9 single particle orbital basis while Lint was increased from 0 to 9.

Table 4.2: Results of a sequence of CI calculations with increasing Lint for the PsH system. The conﬁgurations were constructed from the full Lmax = 9 orbital list of Table 4.1. The organisation of the table columns is otherwise the same as in Table 4.1.

(Note that the Lint = 9 calculation also appears in Table 4.1).

L N E(PsH) ε r r r2 Γ int CI h ei h pi h epi v 0 10010 -0.77503385 0.02503385 2.33985 3.66546 15.4781 1.78637 1 23276 -0.78492947 0.03492947 2.26276 3.58356 15.2911 1.78106 2 37926 -0.78623786 0.03623786 2.25520 3.57183 15.2035 1.78689 3 51660 -0.78656743 0.03656743 2.25348 3.56882 15.1797 1.78929 4 63492 -0.78668177 0.03668177 2.25292 3.56778 15.1713 1.79032 5 73788 -0.78673019 0.03673019 2.25269 3.56734 15.1678 1.79081 6 82548 -0.78675361 0.03675361 2.25258 3.56714 15.1661 1.79106 7 89196 -0.78676604 0.03676604 2.25253 3.56703 15.1651 1.79120 8 93668 -0.78677285 0.03677285 2.25249 3.56696 15.1647 1.79127 9 95324 -0.78677609 0.03677609 2.25249 3.56696 15.1646 1.79129

The most startling aspect of Table 4.2 is the stability of Γ for all values of Lint from 0 to 9. This is not entirely unexpected since the underlying idea behind the Lint selection procedure was to start with a basis that gave the best possible description of a single electron-positron pair (within the constraints of the orbital basis). Even the

Lint = 0 calculation will do a good job of representing the Ps cluster and describing its interaction with the rest of the system.

81 Other properties of the PsH system also show a degree of stability with respect to the variations in L . The mean positron distance, r decreases by 2% when L is int h pi int increased from 0 to 1, but thereafter it changes by less than 0.5%. As expected, the energy shows a monotonic decrease as Lint increases and also shows a reasonably quick pattern of convergence. The energy of the Lint = 2 calculation, is within 1.5% of the

Lint = 9 energy even though it only includes 40% of the conﬁgurations.

The slow convergence of ε and Γ with Lmax is further illustrated in Figures 4.1 and 4.2. The rapid convergence with respect to Lint is readily apparent in Figure 4.1.

Figure 4.2 also illustrates the tendency for Γ to be insensitive to the value of Lint.

Figure 4.1: The PsH binding energy (ε) for a sequence of calculations with diﬀerent values of Lint and Lmax. The SVM binding energy [66] is also shown.

0.04 SVM

Lint 0.03

Lmax

0.02

Lmax (with Lint=Lmax) (units of Hartree) ε 0.01 Lint (with Lmax=9) SVM

0 0 1 2 3 4 5 6 7 8 9 Lmax or Lint

4.3.1 Convergence of Ps-H scattering calculations

One of the areas of recent activity in positron physics is in the scattering of positronium from atoms. In particular, there has been a lot of interest on the positronium-hydrogen scattering problem [219, 221, 222, 223, 224]. Accurate estimates of the Ps-H scattering lengths have only been obtained very recently [223, 224] despite Ps-H being such an apparently simple scattering system. The initial determination of the scattering length was achieved by using a rather unorthodox adaption of the SVM to treat scattering problems [223]. The SVM technique is currently restricted in scope to the scattering length region and therefore a more general solution of the Ps-H scattering problem

82 Figure 4.2: The PsH annihilation rate (Γ in units of 109 s−1) for a sequence of calculations with diﬀerent values of Lint and Lmax. The SVM annihilation rate [66] is also shown.

2.5 SVM

2 Lint ) -1

s 1.5 9

(10 Lmax 2γ

Γ 1

Lmax (with Lint=Lmax) 0.5 Lint (with Lmax=9) SVM 0 0 1 2 3 4 5 6 7 8 9 Lmax or Lint will probably be achieved with more traditional scattering techniques based upon the close-coupling (CC) ansatz [219]. The present PsH calculations can give insight into the size of calculation required to obtain a converged solution with a close-coupling type scattering wavefunction.

For example, the (Lint = 0, Lmax = 9) calculation achieved 68.4% of the (Lint = 9,

Lmax = 9) PsH binding energy. The Lint = 0 calculation would be roughly equivalent to a CC type calculation with a basis written schematically as

Ψ = c H(ns)Ps(nℓ)F Ps(R)+ c H(nℓ)Ps(ns)F Ps(R) , (4.8) | i i ℓ i ℓ Xi Xi Ps where Fℓ (R) describes the motion of the Ps centre-of-mass with orbital angular momentum ℓ. Given that the relative accuracy in the PsH binding energy scales in the same way as the threshold cross section (σ A2 1/ε [124]), and using Table 4.2 ∝ ∝ as a rough guide, inclusion of H states with ℓ = 1 will be required to achieve Ps-H scattering cross-sections accurate at the 5% level. To achieve accuracy at the 1% to 2% level, H type states with ℓ = 2 will be needed. This is consistent with the recent R-matrix Ps-H scattering calculations of the

Belfast group [219, 225, 224]. Using a wavefunction similar to the Lint = 0 calculation, (their CC calculation restricted the hydrogen state to always be in its ground state)

83 59.4% of the PsH binding energy was achieved [219]. Further large-scale calculations using 9 H and 9 Ps states included ℓ = 2 H pseudostates, achieved 93.2% of the PsH binding energy [224]. They concluded ”the convergence of the last 10% of the exact energy seems to be slow” [224]. The present CI calculations suggest that an R-matrix calculation which explicitly includes ℓ = 2 H and Ps pseudostates should be capable of achieving 98% of the exact binding energy.

4.4 e+Be results

The chemical stability of e+Be was rigorously established with an ab-initio SVM calculation that gave a e+Be energy lower than the best variational calculation of neutral Be by 0.001687 Hartree [35]. However, the best description of the structure of e+Be was made using the FCSVMpol which gave a binding energy of 0.003147 Hartree [69]. Both the SVM and FCSVM calculations showed that the e+Be system consists of a positron weakly bound to a polarised beryllium atom. Since the ionisation potential of neutral Be is greater than 0.25 Hartree (experimental I = 0.342603 Hartree [157]), the positron binding energy ε is calculated by the identity

ε = E(Be) E(e+Be) . (4.9) − Here, E(e+Be) is the energy of the positronic beryllium system. The energy of the model neutral beryllium, E(Be), for a given Lmax was computed using a subset of the basis used for e+Be, i.e. the positron orbitals are omitted and exactly the same set of electron orbitals are included in the two-electron calculation. In eﬀect, ε should be interpreted as the energy associated with the binding of a positron to a particular model of Be. There are of course some uncertainties associated with using a neutral Be binding energy that is not the lowest possible energy, but it will be seen that this procedure leads to a CI calculation scheme which probably gives the most sensible estimates of the positron binding energy. Table 4.3 illustrates the convergence of the e+Be energy, annihilation rate, and + other properties as a function of Lmax(= Lint). Table 4.3 for e Be is the analogue of Table 4.1 for PsH. Table 4.3 does contain an additional column of results, the two- electron energy of neutral Be, i.e. E(Be) as a function of Lmax, which is needed for determination of ε for each model Be atom. Once again, the binding energy and the annihilation rate converge slowly to their asymptotic values.

84 + + Table 4.3: Results of CIpol calculations for e Be for orbital basis sets with a given Lmax. The three-body energy of the e Be system is given by E(e+Be), while the column E(Be) gives the two-body energy of the model neutral beryllium atom. Both E(e+Be) and E(Be) are given relative to the energy of the Be2+ core. The binding energy, ε, is calculated from Equation (4.9). All energies are given in units of Hartree.

The electron and positron LTO exponents for each ℓ = Lmax are in the λe and λp columns. The organisation of the rest of the table is the same as Table 3.1. The L results in the rows , and the values of p used to make the extrapolation, are from the 8, 9, 10 series. max → ∞ ∞ L N N λ λ N E(e+Be) E(Be) ε r r Γ Γ max e p e p CI h ei h pi c v 0 9 12 1.22 0.45 540 -0.96834672 -0.96992889 -0.0015822 2.65179 29.2551 0.0000360 0.000196 1 18 21 1.45 0.90 1809 -1.00990757 -1.01114689 -0.0012393 2.57290 26.5269 0.0000918 0.002100 2 26 29 1.60 1.52 4113 -1.01102414 -1.01163259 -0.0006085 2.57907 21.4781 0.0003594 0.015939 85 3 34 37 1.75 1.75 7649 -1.01186972 -1.01175755 0.00011216 2.59068 17.0856 0.0007908 0.048275 4 42 45 2.00 2.00 13073 -1.01259858 -1.01180257 0.00079601 2.60339 14.3437 0.0011878 0.089160 5 50 53 2.20 2.20 20481 -1.01316632 -1.01182229 0.00134403 2.61411 12.8270 0.0014700 0.127624 6 58 61 2.40 2.40 30577 -1.01358379 -1.01183216 0.00175163 2.62235 11.9701 0.0016564 0.160033 7 66 69 2.65 2.65 43393 -1.01388503 -1.01183762 0.00204742 2.62851 11.4524 0.0017799 0.186474 8 74 77 2.80 2.80 59697 -1.01410238 -1.01184084 0.00226154 2.63310 11.1214 0.0018633 0.207821 9 82 85 2.95 2.95 79457 -1.01426049 -1.01184285 0.00241765 2.63654 10.8981 0.0019213 0.225115 10 90 93 3.10 3.10 103505 -1.01437685 -1.01184415 0.00253271 2.63913 10.7461 0.0019618 0.239117 p 2.91 4.12 - 2.72 3.65 3.39 2.00 -1.0149307 -1.0118477 0.0030830 2.6529 10.244 0.002112 0.37120 ∞ FCSVMpol [69] -1.015100 -1.011953 0.003147 2.654 9.842 0.00222 0.41578 Another notable feature of Table 4.3 is the result that the positron does not bind to Be until Lmax is equal to 3. This indicates that the dipole part of the polarisation potential is not able to bind a positron to beryllium, and it has previously been suggested that this is a general feature of positron binding systems [151]. Since e+Be has a weaker Ps cluster than PsH, it is somewhat surprising that the + convergence of the binding energy and annihilation rate with Lmax is slower for e Be than for PsH. At Lmax = 10, the CIpol calculation has recovered only 80% of the expected binding energy and about 60% of the annihilation rate. This occurs even though the mean radius for the positron, about 10 a0, is well outside the Be charge cloud. A plausible explanation for this slow convergence in Lmax is now advanced. One of the distinguishing features of all positron binding systems is the attractive electron- positron interaction which leads to the formation of a positronium cluster. This cluster can be expected to occur wherever the electron and positron charge clouds overlap. In the case of e+Be this overlap will occur in the outer valence region, e.g. at a radius of about 3 to 4 a0. So although the overlap of the positron and electron charge clouds is less pronounced in e+Be than in PsH, the pile-up of the electron charge cloud around the positron is just as strong in the region of overlap. Therefore, this localisation of the electron charge cloud around the positron plays an important part in binding the positron to the atom, and probably contributes just as much to the strength of the annihilation rate in e+Be as it does in PsH.

Table 4.4 shows the results of calculations using orbital basis with Lmax = 10, while the Lint parameter was increased from 0 to 10. This once again shows the importance of giving preferential treatment to electron-positron as opposed to the electron-electron correlations. Some interesting trends are apparent in the tabulation of the binding energy ε versus Lint. Firstly, although there is a small downward creep in E(Be) and E(e+Be) for L 4, ε hardly changes at all. The decrease in E(e+Be) that occurs int ≥ for L 4 arises mainly as a consequence of an improved description of the neutral int ≥ Be atom, and is not the result of a stronger positron-atom attraction.

86 + Table 4.4: Results of a sequence of CIpol calculations with increasing Lint for the e Be system. The conﬁgurations were constructed from the

full Lmax = 10 orbital list of Table 4.3. The organisation of the table columns is the same as in Table 4.3, with the exception that the dipole 3 polarisability of the ground state of the model Be atom (see Table 2.5) is given in the column αd (in units of a0). Note that the Lint = 10 calculation also appears in Table 4.3.

L N E(e+Be) E(Be) α ε r r Γ Γ int CI d h ei h pi c v 0 6453 -0.98273994 -0.96992889 59.598 0.01281105 2.80337 6.85905 0.0038746 0.546956 1 17217 -1.01335113 -1.01114689 36.959 0.00220424 2.63618 11.2066 0.0018118 0.221284

87 2 29553 -1.01407337 -1.01163259 37.583 0.00244077 2.63835 10.8708 0.0019181 0.233634 3 43201 -1.01425393 -1.01175755 37.663 0.00249638 2.63883 10.7957 0.0019443 0.236845 4 57137 -1.01431834 -1.01180257 37.688 0.00251577 2.63899 10.7700 0.0019535 0.238020 5 69825 -1.01434641 -1.01182229 37.699 0.00252412 2.63906 10.7582 0.0019576 0.238555 6 80977 -1.01436042 -1.01183216 37.704 0.00252826 2.63909 10.7526 0.0019596 0.238821 7 90081 -1.01436812 -1.01183762 37.707 0.00253051 2.63911 10.7491 0.0019608 0.238975 8 97137 -1.01437264 -1.01184084 37.708 0.00253180 2.63912 10.7472 0.0019614 0.239062 9 101633 -1.01437537 -1.01184285 37.709 0.00253252 2.63913 10.7463 0.0019618 0.239106 10 103505 -1.01437685 -1.01184415 37.710 0.00253271 2.63913 10.7461 0.0019618 0.239117 The Lint = 0 calculation in Table 4.4 gives a positron binding energy of 0.012811

Hartree, which is 5 times larger than that of the Lint = 10 calculation. This arises because the Lint = 0 structure model of the neutral Be atom does not take into consideration the 2s2 + 2p2 configuration mixing that has a major influence on the structure of neutral beryllium. Restricting the basis to Lint = 0 means the positron interacts with a Be atom which is described by a wavefunction which is only marginally better than the restricted Hartree-Fock wavefunction. Such wavefunctions are known to give predictions of the dipole polarisabilities that are too large [158]. For example, 3 αd for the neutral Be atom calculated within the Lint = 0 formalism was 59.598 a0, 3 which decreases to 37.710 a0 for the Lint = 10 calculations (see Tables 2.5 and 4.4). The use of a structure model which overestimates the polarisability leads to an excessively strong attractive interaction between the positron and the atom. The other expectation values listed in Table 4.4 also show a discontinuity between L = 0 and the other values of L . Calculations with L 1 show moderate int int int ≥ variations in the expectation values that range from 5 to 15%. All of the expectation values are within 1% of their final values at Lint = 3.

+ 4.5 Lint =3 calculations of PsH and e Be

The quick convergence for all properties with respect to Lint suggests that the most eﬃcient way to do a CI calculation for a positronic system containing two valence electrons is to pick a moderately sized value of Lint, say 2 or 3, then increase Lmax systematically to the largest possible value.

To test this hypothesis, PsH calculations were performed constraining Lint = 3, and then increasing Lmax systematically up to 9. This gave extrapolated binding energies and annihilation rates very close to the full Lmax = Lint calculations found in Table 4.1. For example, the extrapolated binding energy was 0.038585 Hartree, about 0.5% smaller than the extrapolated energy from the full CI expansion (0.038795 Hartree), while the extrapolated annihilation rate of 2.2798 109 s−1 is within 0.1% of the rate × obtained from the full calculation.

A similar conclusion is also true for positronic beryllium. The series of Lint = 3 calculations for a succession of Lmax values are shown in Table 4.6. Applying this + procedure results in extrapolated e Be results close to those of the Lmax calculations

88 Table 4.5: Results of a sequence of Lint = 3 CI calculations with increasing Lmax for the PsH system. The conﬁgurations were constructed from the Lmax orbital list of

Table 4.1, with an intermediate coupling selection rule Lint = 3 applied to the allowed conﬁgurations. The organisation of the table is otherwise the same as in Table 4.1. Note that the L = 0 3 calculations also appear in Table 4.1, while the L = 9 max → max calculation also appears in Table 4.2.

L N E(PsH) ε r r r2 Γ max CI h ei h pi h epi v 0 1092 -0.69133422 -0.0586658 2.11759 3.86611 24.5069 0.37393 1 3457 -0.74704709 -0.0029529 2.08537 3.50096 17.1677 0.78021 2 7837 -0.76616882 0.01616882 2.12089 3.44727 15.6591 1.07518 3 13660 -0.77507853 0.02507853 2.15569 3.45769 15.2140 1.28204 4 19756 -0.77974822 0.02974822 2.18439 3.48219 15.0881 1.43004 5 25948 -0.78247753 0.03247753 2.20664 3.50687 15.0722 1.54023 6 32268 -0.78418049 0.03418049 2.22364 3.52805 15.0929 1.62497 7 38732 -0.78529252 0.03529252 2.23647 3.54508 15.1231 1.69194 8 45196 -0.78604433 0.03604433 2.24616 3.55845 15.1532 1.74563 9 51660 -0.78656743 0.03656743 2.25348 3.56882 15.1797 1.78929 p 3.08 - 2.37 2.15 1.08 1.76 -0.7885847 0.0385847 2.2981 3.6447 16.215 2.2798 ∞

of Table 4.3. For example, the extrapolated binding energy of the Lint = 3 series of calculations was 0.003043 Hartree, which is only 1% smaller than the Lint = 10 result

(0.003083 Hartree). The extrapolated annihilation rate from the Lint = 3 series was 0.369821 109 s−1, i.e. within 1% of the L = 10 result. × int

4.6 Chapter summary

These PsH and e+Be results give insight into the business of performing a CI calculation of positronic systems with more than one valence electron. It is clear that electron-positron correlations are diﬃcult to treat with an orbital basis centred on the nucleus. To describe the electron-positron correlations with any degree of accuracy, a very large orbital basis, with the inclusion of orbitals with large ℓ, is required.

89 + Table 4.6: Results of a sequence of Lint =3CIpol calculations with increasing Lmax for the e Be system. The conﬁgurations were constructed

from the Lmax orbital list of Table 4.3, with an intermediate coupling selection rule Lint = 3 applied to the allowed conﬁgurations. The organisation of the table is otherwise the same as in Table 4.3. Note that the L = 0 3 calculations also appear in Table 4.3, while the max → Lmax = 10 calculation also appears in Table 4.4.

L N E(e+Be) E(Be) ε r r Γ Γ max CI h ei h pi c v 0 540 -0.96834672 -0.96992889 -0.0015822 2.65179 29.2551 0.0000360 0.000196 1 1809 -1.00990757 -1.01114690 -0.0012393 2.57290 26.5269 0.0000918 0.002100 2 4113 -1.01102414 -1.01163260 -0.0006085 2.57907 21.4781 0.0003594 0.015939

90 3 7649 -1.01186972 -1.01175760 0.00011212 2.59068 17.0856 0.0007908 0.048275 4 12065 -1.01254844 -1.01175760 0.00079084 2.60339 14.3587 0.0011846 0.088913 5 16865 -1.01308617 -1.01175760 0.00132857 2.61402 12.8624 0.0014612 0.126826 6 21953 -1.01348593 -1.01175760 0.00172833 2.62219 12.0102 0.0016444 0.158814 7 27265 -1.01377640 -1.01175760 0.00201880 2.62831 11.4948 0.0017658 0.184911 8 32577 -1.01398692 -1.01175760 0.00222932 2.63285 11.1674 0.0018476 0.205959 9 37889 -1.01414057 -1.01175760 0.00238297 2.63626 10.9468 0.0019044 0.223012 10 43201 -1.01425393 -1.01175760 0.00249633 2.63883 10.7957 0.0019443 0.236845 p 2.89 - - 2.71 3.60 3.35 1.99 -1.0148010 -1.0117576 0.0030434 2.6526 10.285 0.002096 0.36982 ∞ However, it is desirable that a larger part of the computational effort be devoted to the treatment of electron-positron as opposed to electron-electron correlations. This can be achieved by using the largest possible single orbital basis, whilst restricting the CI expansion to exclude those configurations that would have both electrons oc- + cupying orbitals with large values of ℓ. The Lint = 3 calculations of PsH and e Be summarised in Table 4.7 clearly show that imposing the restriction Lint = 3 (which reduced the dimension of the CI basis by a factor of 2), has less than a 1% effect upon the extrapolated binding energies. Despite both PsH and e+Be possessing extremely different structures, this restriction, when combined with the application of extrapolation corrections to the sequence of CI calculations, leads to binding energy estimates that are accurate to within 5% (and annihilation rates at the 15% level).

Given that the restriction Lint = 3 has a minimal eﬀect on the energy and annihilation rates, the CI calculations performed in the next chapter are all based on CI basis sets with Lint = 3. However, calculations with Lint < 3 were still done to give some indication of the convergence of each system with respect to Lint.

Table 4.7: Binding energies (in Hartree) of positronium hydride and positronic beryllium. Only the latest calculations of a given type by a particular group are listed.

PsH PsH e+Be e+Be L 9 10 max → ∞ ∞ Lint = 3 0.036567 0.038585 0.002496 0.003043

Lint = Lmax 0.036776 0.038795 0.002533 0.003083 SVM [89, 69] 0.039197 0.001687 a FCSVMpol [69] 0.003147 DMC [226, 117] 0.03917(1) 0.0012(4) b Hylleraas [111] 0.039197

a The FCSVMpol calculations of Mitroy and Ryzhikh are expected to be closer to convergence than the equivalent SVM calculations [69]. b The error associated with the fixed-node diffusion quantum Monte Carlo (with two-configuration state function) calculations of Mella et.al. [117] is only statistical. A binding energy of 0.0037(2) Hartree is also reported when the trial wavefunction is limited to a single-configuration [117].

91 92 Chapter 5

Positron Binding to Group II and IIB Atoms (and CuPs)

“FORTRAN: The infantile disorder, by now nearly twenty years old, is hopelessly inadequate for whatever computer application you have in mind today: it is now too clumsy, too risky, and too expensive to use.” (1982).

- Edsger Wybe Dijkstra (1930-2002)

93 The application of the FCSVM to heavier systems suffers from two main problems, in that, the presence of the core slows down the calculations dramatically, and furthermore makes the calculations more susceptible to round-off error. This means that the CI method is an increasingly attractive method to apply to positron binding systems. In this chapter CI calculations demonstrating positron binding to the alkaline-earth atoms (magnesium, calcium and strontium) and the Group IIB atoms (zinc and cadmium) are presented. This implies that the positron scattering lengths of both Zn and Cd are positive, and at low energies the differential cross-sections are found to be larger at backward angles than at forward. This raises the possibility of experimental confirmation of the existence of positron-atom bound states. In addition, evidence is presented in favour of the ability of positronium to bind to copper.

5.1 Positron binding to alkaline-earth atoms

The initial prediction of positron binding to magnesium was made by the polarised orbital (PO) scattering method [55]. Many-body perturbation theory calculations by the UNSW group of Dzuba et.al. also predicted binding, and further predicted the existence an excited e+Mg(2P o) bound state [58, 227, 228]. A later PO calculation also suggested the stability of positronic magnesium [60]. However, the uncertainties associated with both of these methods meant that the predictions of binding were not universally accepted.

The FCSVM has also predicted positron binding to magnesium [71, 35, 69]. Whilst the FCSVMpol calculations are not fully ab-initio, they have been widely accepted as giving convincing evidence for the stability of e+Mg, since the predictions were rigorous with respect to the underlying model Hamiltonian. It had also been found that the stability of e+Mg largely depends on the nature of the interaction between the valence electrons and the positron, and the core potential can be varied quite markedly without aﬀecting the existence of the bound state. For example, in the work of Ryzhikh et.al. [35], complete removal of core-polarisation resulted in the binding energy increasing from 0.013906 Hartree to 0.014803 Hartree. However, the FCSVMpol calculation only gave a lower bound to the binding energy. Since this calculation relies on a stochastic search, there was no guarantee that the calculation might not severely underestimate the binding energy.

94 In particular, the difference between the FCSVMpol and MBPT calculations war- rants careful scrutiny since they both involve large scale calculations to treat electron- electron and electron-positron correlations. Mitroy and Ryzhikh [69] have previously stated that the MBPT calculation overestimated the strength of the polarisation- correlation potential energy due to the inclusion of two distinctly different manifolds + of states in the MBPT expansion. Here, independent CIpol calculations of the e Mg binding energy are performed which give a binding energy close to the latest FCSVMpol binding energy. This further reinforces the critique given by Mitroy and Ryzhikh [69] of the MBPT results [58, 227, 228]. One of the more interesting recent results in negative ion physics was the discovery that the alkaline-earth atoms, calcium, strontium and barium, could form a stable negative ion. Both the theoretical and experimental evidence for the existence of the Ca− 4s24p ground state were first presented in 1987 [229, 230]. This resulted in further theoretical activity demonstrating that both strontium and barium could also form a stable negative ion [231, 232, 233, 234]. Calculations of positron binding to a model alkali atom predicted positron binding to atoms as long as the ionisation potential exceeded 0.176 Hartree [81]. Since the group II elements (Ca, Sr, Ba, Ra) all exceed this threshold, this raised the possibility that positrons could bind to them. In this section calculations are presented that demonstrate the stability of both e+Ca and e+Sr. Interestingly, it will be seen that the positron affinity of both calcium and strontium is stronger than their electron affinity.

5.1.1 e+Mg results

There have been a number of calculations of positron binding to magnesium. The polarised orbital (PO) calculation by Szmytkowski [55] only included the dipole component of the polarisation potential. More recently, an improved PO calculation by McEachran and Stauﬀer [60] included higher multipoles of the polarisation potential. The energy of the both PO calculations, 0.00055 Hartree and 0.00459 Hartree respectively, were derived from the scattering length using the identity ε 1/(2A2) [235]. ≈ The UNSW group applied MBPT to the e+Mg system, initially giving a binding energy 3 of 0.0320 Hartree (and a dipole polarisability of the parent Mg atom of 68 a0) [58]. This was increased in a later calculation [227] to 0.0362 Hartree (1.06 eV) which also

95 predicted that e+Mg had a 2P o excited state with a binding energy of 0.0058 Hartree. The ground state e+Mg MBPT results are twice as large as the energy given by the FCSVMpol series of calculations [71, 35, 73]. Through visual examination of the

FCSVMpol convergence pattern the most recent FCSVMpol binding energy of 0.015612 Hartree is reported to remain about 10-15% below the true variational limit [73]. How- ever, the possibility remained that the FCSVMpol wavefunction is poorly converged and is underestimating the true binding energy. While it would be desirable to perform a larger FCSVMpol calculation for positronic magnesium, the existing calculation probably represents the best that can be performed without an improvement in the FCSVM algorithm or in the computing hardware [73]. The FCSVMpol core-polarisation potentials were also limited to using a single cutoff for all partial-waves. The present CIpol model Hamiltonian used cutoffs that vary with ℓ. The improvement in the model can be seen in the neutral magnesium atom energies in Table 2.7. + The e Mg CIpol calculations are reported in Table 5.1 as a function of Lmax using a fixed Lint = 3. The largest explicit calculation gives a binding energy of 0.014509 Hartree. Extrapolation to the L limit again using Equations (3.5), (3.6) and max → ∞ (3.7) gave 0.016756 Hartree for the binding energy, which is largely consistent with + the latest FCSVMpol energy of 0.015612 Hartree [73]. One salient feature of the e Mg calculation is that the positron is already bound at Lmax = 2. The CI calculations of other e+X systems have not established binding until L 3 [77, 78, 79]. max ≥ The e+Mg binding energy is increased by only 15% with the extrapolation and so even if the correction is in error by 20% it would not significantly increase the uncertainty in the final binding energy. Since the exponents of the LTOs (see Appendix Table F.1) were optimised to give the lowest possible energy, and there are at least 8 LTOs for every ℓ, the enlargement of the dimension of the basis would not have a great impact on the binding energy. The subdivision of the annihilation rate into core and valence components in Table

5.1 reveals that each has completely diﬀerent behaviour with Lmax. The core annihilation rate is calculated simply as the overlap between the positron and core electron densities (see Appendix D.4). Since the mean positron radius r decreases as L h pi max increases, it is not surprising that Γc increases at the same time. The Lmax = 10 value of Γc should be close to converged.

96 + + Table 5.1: Results of Lint = 3 CIpol calculations for e Mg up to a given Lmax. The three-body energy of the e Mg system is given by E(e+Mg), while the column E(Mg) gives the two-body energy of the model neutral magnesium atom. Both E(e+Mg) and E(Mg) are given relative to the energy of the Mg2+ core. The binding energy (ε) is the energy against dissociation into e++Mg. All energies are given in units of Hartree. The organisation of the rest of the table is otherwise the same as Table 3.1. The L extrapolation results in the rows max → ∞ ∞ and p are from the 8, 9, 10 series.

L N N N E(e+Mg) E(Mg) ε r r Γ Γ max e p CI h ei h pi c v 0 15 12 1440 -0.80098006 -0.80370263 -0.0027226 3.21096 23.0372 0.000250 0.000407 1 29 21 4590 -0.82996246 -0.83160419 -0.0016417 3.17108 19.4165 0.000690 0.005736 2 38 29 8544 -0.83386770 -0.83266453 0.00120317 3.19791 12.6464 0.003567 0.053068 97 3 46 37 13352 -0.83761027 -0.83285003 0.00476024 3.24029 9.42272 0.006820 0.141648 4 54 45 18992 -0.84059698 -0.83285003 0.00774695 3.27743 8.25555 0.008700 0.230906 5 62 53 25008 -0.84278927 -0.83285003 0.00993925 3.30701 7.73764 0.009686 0.308643 6 70 61 31312 -0.84436273 -0.83285003 0.01151271 3.33026 7.46584 0.010216 0.374010 7 78 69 37840 -0.84549299 -0.83285003 0.01264296 3.34842 7.30773 0.010512 0.428412 8 86 77 44368 -0.84631193 -0.83285003 0.01346190 3.36258 7.20922 0.010682 0.473618 9 94 85 50896 -0.84691270 -0.83285003 0.01406267 3.37362 7.14468 0.010784 0.511296 10 102 93 57424 -0.84735925 -0.83285003 0.01450922 3.38225 7.10069 0.010846 0.542867 p 2.82 - - 2.34 3.64 4.64 1.68 -0.8496056 -0.8328500 0.0167556 3.4425 6.9547 0.01099 0.98038 ∞ FCSVMpol [69] -0.847684 -0.832072 0.015612 3.437 7.018 0.0121 0.943 The behaviour of Γv with Lmax is completely diﬀerent. It converges very slowly, and the extrapolation correction adds about 80% to the annihilation rate. With such a large correction the obvious question is whether the extrapolation is reliable? A more detailed discussion of extrapolation issues surrounding e+Mg, e+Ca and e+Sr as well as the convergence of the system with respect to Lint is postponed till later.

The overall comparison in Table 5.2 between the extrapolated CIpol and the FCSVMpol binding energies suggests that these calculations agree when the associated uncertainties are taken into consideration. The best estimated result of 0.016151 Hartree is also compatible with a recently reported diﬀusion Monte Carlo (DMC) calculation with a binding energy of (0.0168 0.0028) Hartree [117]. The DMC calculation was ± fully ab-initio and did not use the frozen-core approximation. Taken in conjunction, the FCSVMpol, CIpol and DMC methods suggest a binding energy in the vicinity of (0.016 0.002) Hartree and provide conclusive evidence that the existing MBPT cal- ± culations [58, 227, 228] overestimated the positron binding energy.

Table 5.2: Binding energies (in Hartree) of positronic magnesium (as well as positronic calcium and positronic strontium). Three CIpol results are are shown. Firstly the explicit Lmax = 10 results, secondly the estimate resulting from extrapolating the explicit Lmax = 8, 9, 10 series and thirdly the best guess extrapolation estimates using a revised pbg which are discussed later. Only the latest calculations of a given type by a particular group are listed in this table.

e+Mg e+Ca e+Sr

CIpol Lmax = 10 0.014509 0.012358 0.004869 CI (with p ) 0.016756 0.018331 0.012528 pol ∞ 8,9,10 CI (with p ) 0.016151 0.016500 0.010050 pol ∞ bg POdipole [55] 0.00055 MBPT [227] 0.0362

POmultipole [60] 0.00459

FCSVMpol [69] 0.015612 DMC [117] 0.0168(28) a

a The uncertainty in the last digits includes both the statistical error and the estimated error in deﬁning the nodal surfaces due to single conﬁguration state wavefunction [117].

98 5.1.2 e+Ca results

The condition for positron binding to calcium is that the energy of the e+Ca state be lower than the energy of the Ca+ + Ps dissociation channel.

ε = E(Ca+) Ps(1s) E(e+Ca) , (5.1) − −

Given that the one electron energy E(Ca+) = 0.43628654 Hartree for the present − model Hamiltonian, binding occurs when E(e+Ca) < 0.68628654 Hartree. The de- − + tails of the basis and the results of the e Ca calculations as a function of Lmax are given in Table 5.3. The binding energies indicate that e+Ca is one of the most tightly bound positronic atoms with a binding energy comparable in magnitude to e+Mg. The partial-wave series is more slowly convergent for e+Ca than for e+Mg. This is expected since calcium has a smaller ionisation potential and thus it is easier for the positron to attract the electron. The stronger pileup of electron density around the positron requires a larger partial-wave expansion to represent correctly. The extrapolation of the e+Ca binding energy yields a 50% correction to the binding energy. The uncertainties associated with such a large correction are discussed in more detail later. Table 5.3 also shows the annihilation rate for e+Ca is larger than that of e+Mg. Previous calculations have shown that parent atoms with smaller ionisation potentials generally have a larger annihilation rate [75, 81]. Again, this is caused by the stronger pileup of electron density in the vicinity of the positron. The extrapolation corrections for both r and Γ listed in Table 5.3 are obviously h pi c not reliable. The e+Ca system at large distances consists of Ca+ + Ps. In other calculations of positron binding systems (see e+Li in Table 3.1) it has been noticed that systems that decay asymptotically into Ps+X+ do not have an r that changes h pi monotonically with Lmax. Initially, the positron becomes more tightly bound to the system as L increases, resulting in a decrease in r . However, r then tends to max h pi h pi increase at the largest values of Lmax as the description of the Ps clustering improves. The net result of all this is that r (and by implication Γ ) approach their asymptotic h pi c forms very slowly. The variations in r and Γ are relatively small and the best policy h pi c is to simply not to give any credence to the extrapolation for either of these operators for the present calculations of e+Ca (and as will be seen also e+Sr).

99 + + Table 5.3: Results of Lint = 3 CIpol calculations for e Ca up to a given Lmax. The three-body energy of the e Ca system is given by E(e+Ca), and is given relative to the energy of the Ca2+ core. The binding energy (ε) is the energy against dissociation into Ps + Ca+. All energies are given in units of Hartree. The organisation of the rest of the table is otherwise the same as Table 3.1. The L max → ∞ extrapolation results in the rows and p are from the 8, 9, 10 series. ∞ L N N N E(e+Ca) ε r r Γ Γ max e p CI h ei h pi c v 0 15 12 1440 -0.62968005 -0.0566065 4.06555 15.0877 0.001507 0.001754 1 26 21 3717 -0.65492270 -0.0313638 4.07231 12.4531 0.002813 0.015892 2 36 29 7535 -0.66654447 -0.0197421 4.11995 9.06910 0.008024 0.090071

100 3 44 37 12159 -0.67589424 -0.0103923 4.18778 7.73557 0.011811 0.200433 4 52 45 17623 -0.68291413 -0.0033724 4.24854 7.23477 0.013436 0.310805 5 60 53 23447 -0.68795200 0.00166546 4.30062 7.02092 0.013981 0.409795 6 68 61 29559 -0.69156879 0.00528225 4.34453 6.92215 0.014075 0.495772 7 76 69 35895 -0.69419400 0.00790746 4.38129 6.87751 0.013983 0.569390 8 84 77 42231 -0.69612282 0.00983628 4.41147 6.85672 0.013843 0.632608 9 92 85 48567 -0.69755979 0.01127325 4.43622 6.84842 0.013695 0.686919 10 100 93 54903 -0.69864433 0.01235780 4.45646 6.84777 0.013552 0.733532 p 2.67 - 1.91 24.2 0.33 1.45 -0.7046170 0.0183308 4.6679 6.8477 -0.05963 1.65042 ∞ 5.1.3 e+Sr results

The strontium atom has an (experimental) ionisation potential of 0.20925 Hartree [157], smaller than that of both magnesium and calcium. Therefore, the changes that occurred when going from e+Mg to e+Ca are also evident, but even more marked when going from e+Mg to e+Sr. It is expected that the e+Sr wavefunction and binding energy converge even more slowly with Lmax and the annihilation rate to be larger than that of e+Ca. Both of these features can be seen in Table 5.4. In the present model, the one electron Sr+ model energy was -0.40534976 Hartree, thus the threshold for positron binding of -0.65534976 Hartree was only passed at Lmax = 8. The binding energy increases by about 30% when Lmax is increased from 9 to 10 and the annihilation rate is larger than that of e+Ca. As was the case for e+Ca, the e+Sr extrapolations for r and Γ are nonsensical. h pi c There remains some uncertainty in the precise determination of the binding energy due to the large contribution from the extrapolation correction (157%), which is discussed later.

5.1.4 L 3 calculations of e+Mg, e+Ca and e+Sr int ≤

To further examine the convergence of each positronic system, the series of Lint < 3 calculations was performed to examine whether each system is stable against enlargement of Lint. The results of which are shown in Table 5.5. + For the e Mg system, there was only a 1% change in the binding energy when Lint was increased from 2 to 3, suggesting that this series has reached adequate convergence. + One interesting feature is the result that the e Mg binding energy for the Lint = 0 calculation was roughly twice as large as the energies of the Lint = 1, 2, 3 calculations. A similar result occurred for positronic beryllium (see Table 4.4). The reason for this derives from the mechanism for binding. Positronic beryllium, and to a lesser extent positronic magnesium consist of a positron bound to the system by the polarisation of the parent atom and in both cases the positron is predominantly found outside the electron charge distribution of the parent atom. The polarisabilities for both Be and

Mg are too large for the Lint = 0 calculation, leading to an anomalously large positron binding energy.

101 + + + Table 5.4: Results of Lint =3CIpol calculations for e Sr up to a given Lmax. The three-body energy of the e Sr system is given by E(e Sr), and is given relative to the energy of the Sr2+ core. The binding energy (ε) is the energy against dissociation into Ps + Sr+. All energies are given in units of Hartree. The organisation of the rest of the table is otherwise the same as Table 3.1. The L extrapolation results max → ∞ in the rows and p are from the 8, 9, 10 series. ∞ L N N N E(e+Sr) ε r r Γ Γ max e p CI h ei h pi c v 0 16 12 1632 -0.58217182 -0.0731779 4.37904 12.8625 0.003830 0.003218 1 28 23 4680 -0.60668919 -0.0486606 4.41995 10.9673 0.004959 0.020768 2 40 33 10680 -0.62159214 -0.0337576 4.47036 8.73491 0.010178 0.092608

102 3 52 41 19260 -0.63277466 -0.0225751 4.54409 7.72902 0.013970 0.198827 4 60 49 27004 -0.64114395 -0.0142058 4.61129 7.32303 0.015515 0.308144 5 68 57 34748 -0.64718350 -0.0081663 4.66976 7.15141 0.015913 0.408438 6 76 65 42540 -0.65155032 -0.0037994 4.71979 7.07848 0.015847 0.496994 7 84 73 50476 -0.65474366 -0.0006061 4.76192 7.05064 0.015618 0.574136 8 92 81 58412 -0.65710513 0.00175536 4.79708 7.04437 0.015350 0.640838 9 100 89 66348 -0.65887523 0.00352547 4.82617 7.04812 0.015094 0.698612 10 108 97 74284 -0.66021865 0.00486888 4.85010 7.05629 0.014867 0.748752 p 2.62 - 1.86 -7.40 1.14 1.35 -0.6678778 0.0125280 5.1146 0.00652 1.94361 ∞ ∞ + + The sequence of Lint calculations for e Ca and e Sr listed in Table 5.5 reveal a + diﬀerent convergence pattern than that of e Mg. The Lint = 0 binding energies for + + e Ca and e Sr are substantially smaller than their Lint = 3 binding energy, and in + fact the e Sr Lint = 0 calculation is not even bound. Positronic calcium and positronic strontium are best described as Ps bound to Ca+ and Sr+. Therefore, the overestima- tion of the neutral atom dipole polarisability does not result in an anomalously large binding energy. Instead, it is the steady buildup in the description of a polarised Ps atom with respect to Lint that drives the positron binding to these two species. The e+Sr binding energy changed by about 10% in going from L = 2 3, suggesting int → that it would have been worthwhile to increase L 4 for this system. int →

5.1.5 e+Mg, e+Ca and e+Sr extrapolation issues

The successive increments to the energy and valence annihilation rate for the e+Mg, e+Ca and e+Sr systems show a slow rate of convergence towards an asymptotic form. This can be seen in Figures 5.1 and 5.2, in which the binding energy and valence annihilation rates are plotted as a function of Lmax.

Figure 5.1: The binding energy (in units of Hartree) of e+Mg (dots), e+Ca (squares) + and e Sr (triangles) as a function of Lmax (with Lint = 3).

0.02 e+Mg 0.01

0 e+Sr -0.01 e+Ca (Hartree) ε -0.02

-0.03

-0.04 0 2 4 6 8 10 Lmax

103 Table 5.5: Results of CI calculations for e+Mg, e+Ca and e+Sr with the full L = 10 orbital basis for the series L 3. The three-body pol max int ≤ + + energy of each e X system is given by E(e X), while the column Edissoc is the energy (in Hartree) of the lowest dissociation channel of the + 2+ + three-body system. Both E(e X) and Ebind are given relative to the energy of the X core. The binding energy (ε) is the energy of the e X system against dissociation. All energies are given in units of Hartree. The organisation of the table is otherwise the same as Table 3.1, with 3 2 the addition of αd (in units of a0) of each model ns ground state neutral atom (see Tables 2.7, 2.9 and 2.15).

L N E(e+X) E α ε r r Γ Γ int CI dissoc d h ei h pi c v e+Mg 0 12090 -0.83038303 -0.80370263 98.417 0.02668040 3.52140 6.45376 0.012430 0.697872 1 29772 -0.84548912 -0.83160419 70.232 0.01388492 3.38828 7.18675 0.010441 0.530966

104 2 43776 -0.84702622 -0.83266453 71.542 0.01436169 3.38275 7.12158 0.010751 0.539079 3 57424 -0.84735925 -0.83285003 71.639 0.01450922 3.38225 7.10069 0.010846 0.542867 e+Ca 0 11805 -0.68911005 -0.68628654 203.31 0.00282351 4.56976 6.82218 0.013113 0.774553 1 25555 -0.69600424 -0.68628654 157.30 0.00971770 4.48375 6.90507 0.013032 0.726975 2 41255 -0.69816199 -0.68628654 161.67 0.01187545 4.45902 6.86573 0.013418 0.728588 3 54903 -0.69864433 -0.68628654 161.73 0.01235780 4.45646 6.84777 0.013552 0.733532 e+Sr 0 14368 -0.65223326 -0.65534976 245.94 -0.00311651 4.94719 7.09636 0.014127 0.765951 1 31396 -0.65742053 -0.65534976 199.31 0.00207077 4.88372 7.11383 0.014297 0.743373 2 52408 -0.65967128 -0.65534976 204.29 0.00432152 4.85382 7.07576 0.014708 0.743783 3 74284 -0.66021865 -0.65534976 204.31 0.00486888 4.85010 7.05629 0.014867 0.748752 Figure 5.2: The 2γ spin-averaged annihilation rate (in units of 109 sec−1) of e+Mg, + + e Ca and e Sr as a function of Lmax (with Lint = 3) for the valence electrons only.

0.8 e+Sr e+Ca 0.6 ) -1

s +

9 e Mg 0.4 (10 2γ Γ 0.2

0 0 2 4 6 8 10 Lmax

The binding energies for e+Ca, e+Sr and the annihilation rates for e+Mg, e+Ca and e+Sr are all subject to quite large extrapolation corrections, raising questions about their overall reliability. Fortunately, the analysis of Gribakin and Ludlow [214], which suggested that the asymptotic form for the energy increments was pE = 4 while the annihilation rate was described by pΓ = 2, can be utilised to assess the accuracy and furthermore help devise an improved extrapolation scheme. Figure 5.3 shows the energy exponents derived from the (L 2, L 1, L ) max − max − max + + + e Mg, e Ca and e Sr calculations where Lmax = 6, 7, 8, 9 and 10. It is evident that pE is not constant and that it increases as Lmax increases. This follows the same behaviour observed for e+Cu and e+Li (see Figure 3.4).

Given this variation in pE it is likely that the extrapolations using the Lmax = 10 values of pE = 2.82, 2.67 and 2.62 for each positronic atom overstates the contribution from ℓ>Lmax. In addition, an extrapolation with pE = 4 would tend to underestimate the magnitude of the extrapolation correction. Since the extrapolation with pE derived from the last three energy increments will tend to overestimate the extrapolation correction it is clear that upper and lower bounds can be placed on the extrapolation correction. A third best guess extrapolation with pbg chosen halfway between 4 and the Lmax = 10 exponent was also done. This perhaps gives a more reliable estimate of the binding energy than either of the other methods.

105 Figure 5.3: The exponent relating two separate energy increments using Equation + + + (3.7) as a function of Lmax (with Lint = 3) for e Mg, e Ca and e Sr. The analysis of Gribakin and Ludlow [214] suggests a limiting value of p =4 as L . E max → ∞ 3 2.8 2.6 2.4 2.2

E 2 p 1.8 + 1.6 e Mg 1.4 e+Ca 1.2 e+Sr 1 4 5 6 7 8 9 10 Lmax

The binding energies for all systems, using these three methods of determining pE are given in Table 5.6. The variations in the binding energy are 6% for e+Mg, 20% for e+Ca and about 40% for e+Sr. The actual uncertainty in the correction is about the same for all 3 systems, the smaller overall uncertainty for e+Mg occurs because the actual magnitude of the correction when compared with the rest of the binding energy is much smaller. For neutral calcium, this gives a best guess estimate of the positron aﬃnity of 0.04317 Hartree, which far exceeds the most recent estimates of the electron aﬃnity of between 0.00064 and 0.00090 Hartree [236, 237].

The variations in the diﬀerent estimates of the annihilation rate are larger than the binding energy, but they are not excessively large considering that only about 50% of the annihilation rate comes from the explicit calculation. The actual diﬀerence between the value of pΓ derived from comparison of the increments to the annihilation rate, and the asymptotic value (pΓ = 2) given by Gribakin and Ludlow [214], are 0.32 + + + for e Mg, 0.55 for e Ca and 0.65 e Sr. The annihilation rates for the middle pbg value of pΓ are also taken as the preferred estimate, giving total annihilation rates of 0.91 109 sec−1 for e+Mg, 1.36 109 sec−1 for e+Ca and 1.47 109 sec−1 for e+Sr. × × × In all three cases, the e+Sr binding energy is smaller than that of e+Ca, which is consistent with a previous analysis that investigated positron binding to a model alkali atom [81]. The binding energy of the model e+Alkali system decreased as the

106 Table 5.6: Sensitivity of the binding energy and annihilation rate to the use of diﬀerent exponents in the power series extrapolation. The exponents for energy are in the column pE and the exponents for the 2γ annihilation are rate in pΓv . The top estimates are the extrapolated values from Tables 5.1, 5.3 and 5.4. The bottom extrapolations use pE = 4 and pΓv = 2. The middle (best guess) estimates (in bold face) which use an intermediate pbg are the preferred values.

pE ε pΓv Γv e+Mg 2.82 0.016756 1.68 0.98038 3.41 0.016151 1.84 0.89905 4 0.015789 2 0.84175 e+Ca 2.67 0.018331 1.45 1.65042 3.33 0.016500 1.72 1.34408 4 0.015467 2 1.17482 e+Sr 2.62 0.012528 1.35 1.94361 3.31 0.010050 1.67 1.45131 4 0.008720 2 1.22342

ionisation energy of the parent atom decreased (provided that I < 0.250 Hartree). The model e+Alkali atoms with parent atom ionisation potentials equal to that of Ca and Sr (0.22465 and 0.20925 Hartree [180]), had positron binding energies of about 0.0105 + + and 0.0047 Hartree. The CIpol binding energy estimates of e Ca and e Sr (0.016500 and 0.010050 Hartree) suggest that two-electron atoms with I < 0.250 Hartree bind a positron more strongly than a one-electron atom with the same ionisation potential.

Recent SVM calculations of the (m2+,e−,e−,e+) system conﬁrmed this trend for the two-electron-positron system [238]. These SVM calculations used a double- positively charged particle m2+ with a variable mass. Changing m2+ systematically changes the ionisation potential of the two-electron (m2+,e−,e−) subsystem. Adding a positron to each two-electron system allowed positron binding to be examined as a function of the ionisation potential of two-electron systems. As the ionisation potential

107 of the two-electron system decreases below that of 0.250 Hartree, the system increasingly forms into m2+ +Ps−. This implies that as I 0, the binding energy reaches a → plateau of 0.0120 Hartree, the Ps− binding energy [238]. + This can already be seen in Table 5.5, in that, e Sr exhibits a slower Lint convergence than both e+Mg and e+Ca. This is consistent with the notion that an increasing presence of Ps−-type clustering demands more electron-electron correlation terms. An L 14 with L = 4 calculation would probably be needed to give an estimate of max ≥ int the e+Sr binding energy accurate at the 5% level. It remains to be seen if the binding energy for e+Sr does indeed exceed 0.0120 Hartree.

Positronic barium

Although an explicit calculation has not been done, there is a mounting body of evidence that positronic barium would also be stable:

An analysis of positron binding to a model alkali atom showed that binding • was expected for atoms with an ionisation potential larger than 0.1767 Hartree [81]. The ionisation energy of barium is 0.1915 Hartree [202] which exceeds this threshold. The model alkali atom with an ionisation potential of 0.1915 Hartree had a positron binding energy of around 0.0010 Hartree [81].

The CI e+Ca and e+Sr results further suggest that positronic systems with • pol two valence electrons (and an ionisation potential less than 0.250 Hartree) have binding energies that exceed those predicted by the model alkali calculation with the same ionisation potential.

The (m2+,e−,e−,e+) study [238] revealed that as the mass of particle m2+ 0, • → the (m2+,e−,e−,e+) system dissociates into (m2++Ps−). This suggests that the e+Ba binding energy will be approximately equal to the Ps− binding energy of 0.0120 Hartree.

As barium has an ionisation potential less than that of both calcium and strontium, it would be expected that the binding energy of e+Ba will converge very slowly with + Lmax. Since formal binding for e Sr was only established at Lmax = 8, one should + anticipate going beyond Lmax = 10 for e Ba.

108 5.2 Positron binding to group IIB atoms

Some of the ﬁrst suggestions that positrons could bind to a neutral atom were made for the group IIB elements of the periodic table. The initial predictions of positron binding were made in a (dipole only) polarised orbital (PO) calculation of positron scattering from zinc and cadmium [56] as well as mercury [57]. Many-Body Perturbation Theory (MBPT) has also been used to predict the positron binding energy for the Zn, Cd and Hg atoms [58]. Another PO calculation by McEachran and Stauﬀer [60] included the higher-order moments of the polarisation potential in predicting positron binding to zinc. The application of the same perturbative methods to the e+Mg system resulted in predictions of the e+Mg binding energy that were not reliable, and the same is expected to be true for the equivalent set of e+Zn and e+Cd calculations.

+ Although a FCSVMpol calculation of e Zn has already been reported [74], this calculation gave binding energies that were far from converged. These FCSVMpol calculations upon e+Zn were extremely tedious [74] and to a certain extent provided the initial stimulus to adapt the CI method to study positron binding to atoms. Since positronic zinc and positronic cadmium are expected to possess structures similar to that of positronic beryllium, i.e. the positron orbits the polarised neutral atom at a large distance, both these systems are relatively amenable to the CI treatment. The + + extrapolated CIpol binding energies of e Zn and e Cd are probably accurate to within 10%.

5.2.1 e+Zn Results

There is a rather large scatter amongst the previous calculations of positronic zinc. The polarised orbital (PO) calculations that predicted positron binding to zinc [56] included both non-relativistic and relativistic treatments of the atomic structure and scattering dynamics. The estimates obtained for the binding energy were ε = 0.00000038 Hartree (non-relativistic) and ε = 0.00001868 Hartree (relativistic). One limitation of these calculations was that only the dipole component of the polarisation potential was used even though it is known that the higher moments of the polarisation potential are important in positron scattering calculations [114, 29, 151]. McEachran and Stauﬀer [60] did include the high-order moments of the polarisation potential in a later non-relativistic PO calculation giving a binding energy of 0.00194

109 3 Hartree. However, their model zinc atom had a dipole polarisability of 54 a0 that exceeds the experimental polarisability of (38.8 0.8) a3 [197] by 40%, and could ± 0 therefore overestimate the strength of the polarisation potential. The binding energies in both the PO calculations were not calculated explicitly, rather they were deduced from the eﬀective range relation ε 1/(2A2) [235] that is often used to relate binding ≈ energy to the scattering length A. Dzuba et.al.’s MBPT calculations predicted the positron binding energy of zinc 3 to be 0.00845 Hartree (with a model Zn αd of 42 a0) [58]. However, the FCSVMpol calculation for positronic zinc [74] gave a much reduced binding energy of 0.001425

Hartree. Although the FCSVMpol energy was far from converged, the prediction of binding within the constraints of the underlying model potential was rigorous since the determination of the binding energy was performed variationally. + The results of the sequence of CIpol calculations for e Zn with a ﬁxed Lint = 3, and increasing Lmax are reported in Table 5.7 (the one-electron energy in this model is -0.6603009 Hartree). Positron binding is established by Lmax = 3 and the binding energy at Lmax = 10 is 0.003039 Hartree. When the extrapolation procedure is applied, the binding energy increases by about 25% to 0.003731 Hartree. Positronic zinc is in many ways similar to positron beryllium. Neutral zinc and neutral beryllium both have a valence ns2 shell, with roughly the same binding energies and dipole polarisabilities that are within 5% of each other (see Tables 2.5 and 2.13). Since zinc has the slightly larger polarisability it is expected that e+Zn would have a slightly larger binding energy and this is the case. The mean positron distance from + the nucleus for e Zn is 9.9 a0, this is about the same distance from the nucleus as the + positron in e Be. The ability to perform an accurate CIpol calculation that reproduces + + the best FCSVMpol calculation upon e Be, suggests that the e Zn calculations should be similarly reliable. The MBPT binding energy of 0.00845 Hartree [58] is about twice as large as the + estimated CIpol binding energy. As was seen from the e Mg results, the MBPT calculations of the UNSW group overestimated the strength of the polarisation-correlation potentials. Therefore, a MBPT binding energy of e+Zn that is twice as large as the current CIpol binding energy is not an anomaly since it is consistent with the pattern established for e+Mg.

110 + + Table 5.7: Results of Lint = 3 CIpol calculations for e Zn up to a given Lmax. The three-body energy of the e Zn system is given by E(e+Zn), while the column E(Zn) gives the two-body energy of the model neutral zinc atom. Both E(e+Zn) and E(Zn) are given relative to the energy of the Zn2+ core. The binding energy (ε) is the energy against dissociation into e++Zn. All energies are given in units of Hartree. The organisation of the rest of the table is otherwise the same as Table 3.1. The L extrapolation results in the rows and p are max → ∞ ∞ from the 8, 9, 10 series.

L N N N E(e+Zn) E(Zn) ε r r Γ Γ max e p CI h ei h pi c v 0 16 12 1632 -0.97189957 -0.97368919 -0.0017896 2.76518 27.3148 0.000378 0.000282 1 28 23 4680 -0.99220765 -0.99352357 -0.0013159 2.75449 24.2922 0.001185 0.002847 2 40 33 10680 -0.99430964 -0.99494216 -0.0006325 2.75962 20.0015 0.004140 0.016001 111 3 48 41 16432 -0.99556155 -0.99536093 0.00020061 2.77219 16.0616 0.009031 0.046893 4 56 49 22848 -0.99635571 -0.99536093 0.00099477 2.78700 13.5932 0.013526 0.086532 5 64 57 29376 -0.99699033 -0.99536093 0.00162940 2.79946 12.2296 0.016700 0.124487 6 72 65 36064 -0.99746724 -0.99536093 0.00210630 2.80914 11.4475 0.018814 0.157292 7 80 73 42976 -0.99781702 -0.99536093 0.00245608 2.81647 10.9712 0.020222 0.184565 8 88 81 49888 -0.99807251 -0.99536093 0.00271158 2.82195 10.6711 0.021166 0.206850 9 96 89 56800 -0.99826021 -0.99536093 0.00289927 2.82614 10.4626 0.021831 0.225234 10 104 97 63712 -0.99839946 -0.99536093 0.00303852 2.82927 10.3246 0.022292 0.240232 p 2.83 - - 2.80 3.91 3.49 1.93 -0.9990924 -0.9953609 0.0037314 2.8451 9.9139 0.02393 0.39270 ∞ FCSVMpol [74] -0.988586 -0.990011 0.001425 2.85 12.36 0.0140 0.234 Table 5.8 shows the convergence of the energy and other properties when Lint increases from 0 3. The tabulations suggest convergence in most properties at the → 1% level when Lint = 3. The PO calculations which included higher order multipoles of the polarisation potential [60], can be best regarded as an approximation to the Lint = 0, Lmax = 10 calculation. The PO binding energy was 0.00195 Hartree while the Lint=0, Lmax=10

CIpol calculation had a binding energy of 0.007173 Hartree. The PO calculation for positronic zinc seems to underestimate the strength of the electron-positron correlations. Although the PO method as applied by McEachran and co-workers gives a good description of positron-rare gas scattering [114, 239, 240], its inherent limitations are exposed for the present systems with their stronger electron-positron correlations. 3 The present Lint = 3 model Hamiltonian gives a dipole polarisability of 41.232 a0 which is slightly larger than the latest experimental estimate of (38.8 0.8) a3 [197]. ± 0 The specification of the core polarisation potential is one of the largest sources of uncertainty in these calculations. Since zinc (and similarly cadmium) is a moderately heavy atom, it is likely that the core-polarisation potential is also compensating for small relativistic energy shifts. However, the core-polarisation potential does not have much of a direct influence upon the positron, rather the core-polarisation potential influences the motion of the valence electrons which in turn influences the motion of the positron. For example, adding a core polarisation potential for a parent atom with I > 6.8 eV causes the positron binding energy to decrease [35, 66]. This was tested in a calculation in which the Zn2+ core dipole polarisability was 3 3 increased from 2.294 a0 to 3.9 a0, while everything else stayed the same. When this is done, the two-body energy of Zn dropped to -1.04427988 Hartree. The total dipole 3 polarisability of neutral zinc in this model potential decreased to 35.841 a0, which is less than the experimental polarisability. However, the positron still remains bound with an extrapolated binding energy of 0.002232 Hartree and a valence annihilation rate of 0.30424 109 s−1. The explicit and extrapolated results from these α 36 a3 × d ≈ 0 calculations are shown for comparison in the last two rows in Table 5.8.

112 Table 5.8: Results of CI calculations for e+Zn with the L = 10 orbital basis of Table 5.7 for the series L 3. The organisation pol max int ≤ of the table is the same as Table 5.5, except the energy of each model zinc atom is given in the E(Zn) column. The dipole polarisability of each model zinc 4s2 ground state, α , is also included from Table 2.13 (in units of a3). The L = 3 calculation had the Zn2+ core dipole d 0 int † polarisability increased to 3.9 a3 such that the model neutral zinc atom had a dipole polarisability of 36 a3. The L row is the 0 ≈ 0 max → ∞† extrapolated e+Zn results calculated using the same α 36 a3 model zinc atom. d ≈ 0

113 L N E(e+Zn) E(Zn) α ε r r Γ Γ int CI d h ei h pi c v 0 13856 -0.98086194 -0.97368919 50.457 0.00717276 2.89878 8.11867 0.032533 0.375857 1 30020 -0.99641033 -0.99352357 40.658 0.00288676 2.83243 10.4783 0.021542 0.234512 2 49736 -0.99793751 -0.99494216 41.182 0.00299534 2.83016 10.3660 0.022058 0.238290 3 63712 -0.99839946 -0.99536093 41.232 0.00303852 2.82927 10.3246 0.022292 0.240232 3 63712 -1.04603984 -1.04427988 35.841 0.00175996 2.72586 11.8527 0.021059 0.183209 † L -1.0465123 -1.0442799 35.841 0.0022324 2.7380 11.242 0.02309 0.30424 max → ∞† 5.2.2 e+Cd Results

The stability of the ground state of positronic zinc implies that positronic cadmium is stable, since the ionisation potential of cadmium, I = 0.330511 Hartree [202], is slightly smaller than that of zinc, I = 0.345219 Hartree [202]. A recent experimental determination of the dipole polarisability of cadmium giving (49.65 1.65) a3 [206], ± 0 and hence it should be easier to bind a positron to cadmium than it was for zinc. Previously there have been two calculations that had made a prediction of positron binding to Cd. A relativistic polarised orbital calculation gave a binding energy of 0.0000553 Hartree while the non-relativistic calculation gave a binding energy of 0.000276 Hartree [56]. The dipole polarisability derived from the Cd I Hartree-Fock 3 wavefunction (76.02 a0) is larger than the polarisability derived the Dirac-Fock wave- 3 function (63.68 a0) [56]. Therefore the smaller binding energy in the relativistic model is expected. Binding was also predicted by the MBPT calculations of Dzuba et.al. [58] with a binding energy of 0.0129 Hartree (with the Cd dipole polarisability of 53 3 a0). Although the FCSVM had been able to give positive evidence of positron binding to zinc, this calculation took almost one year to complete [74]. Therefore an FCSVM calculation on cadmium was not started by Ryzhikh and Mitroy [241], even though it seemed obvious that cadmium would bind a positron. The present e+Cd ground state was again treated as a system with two active electrons and a positron. The 46 electrons in the 1s 4d core orbitals were taken → from a HF calculation of the neutral Cd 5s2 1Se ground state. The properties of e+Cd are given in Table 5.9. The 5s single electron energy using the electron sub-basis was -0.62111175 Hartree. Thus, the L = 3,L 3 calculations have an ionisation int max ≥ potential of -0.31807853 Hartree, which is slightly smaller than the zinc model atom. 3 Consequently the Lint = 3 cadmium dipole polarisability of 50.073 a0 is 20% larger 3 than the equivalent zinc model atom 41.232 a0 (see Tables 2.17 and 2.13). Therefore, one expects that the binding energy for cadmium will be somewhat larger than that for zinc, and this is the case. The extrapolated e+Cd binding energy of 0.006100 Hartree is around 50% larger than that of e+Zn.

114 + + Table 5.9: Results of Lint = 3 CIpol calculations for e Cd up to a given Lmax. The three-body energy of the e Cd system is given by E(e+Cd), while the column E(Cd) gives the two-body energy of the model neutral cadmium atom. Both E(e+Cd) and E(Cd) are given relative to the energy of the Cd2+ core. The binding energy (ε) is the energy against dissociation into e++Cd. All energies are given in units of Hartree. The organisation of the rest of the table is otherwise the same as Table 3.1. The L extrapolation results in the rows max → ∞ ∞ and p are from the 8, 9, 10 series.

L N N N E(e+Cd) E(Cd) ε r r Γ Γ max e p CI h ei h pi c v 0 15 12 1440 -0.92236479 -0.92406240 -0.0016976 2.98690 27.6952 0.000505 0.000311 1 29 23 5010 -0.93592019 -0.93708058 -0.0011604 2.99166 24.1689 0.001781 0.003105 2 42 33 12014 -0.93823897 -0.93857673 -0.0003378 3.00016 19.1185 0.006607 0.017786 115 3 51 42 19457 -0.94001664 -0.93919028 0.00082636 3.01892 14.4648 0.015165 0.055511 4 59 50 26833 -0.94117600 -0.93919028 0.00198572 3.04051 11.9726 0.022512 0.103414 5 67 58 34161 -0.94211083 -0.93919028 0.00292055 3.05823 10.7451 0.027366 0.148939 6 75 66 41569 -0.94281714 -0.93919028 0.00362686 3.07199 10.0761 0.030481 0.188612 7 83 74 49121 -0.94333796 -0.93919028 0.00414768 3.08242 9.68952 0.032476 0.221840 8 91 82 56673 -0.94372045 -0.93919028 0.00453017 3.09038 9.43931 0.033818 0.249567 9 99 90 64225 -0.94400290 -0.93919028 0.00481262 3.09648 9.26897 0.034746 0.272639 10 107 98 71777 -0.94421346 -0.93919028 0.00502318 3.10113 9.15370 0.035391 0.291788 p 2.79 - - 2.58 3.71 3.44 1.77 -0.9452907 -0.9391903 0.0061004 3.1284 8.7820 0.03773 0.52731 ∞ Despite the larger binding energy, there are obvious qualitative similarities between the structures of e+Cd and e+Zn. The positron charge distribution for e+Cd is diﬀuse with a mean positron radius of 8.8 a0. This is about 10% more compact than the charge distribution e+Zn. The annihilation rate of 0.52731 109 sec−1 is about 40% larger than × the annihilation rate of e+Zn. Previous investigations have shown that parent atoms with smaller ionisation potentials generally have larger annihilation rates [75, 81, 73]. The larger annihilation rate occurs because the positron can more strongly attract an electron when the ionisation potential (which is a measure of the interaction strength between the electron and the atomic core) is smaller. A reasonable summary of the structure of e+Cd is that it is an analogue of e+Zn with a slightly more tightly bound positron. The e+Cd diagonalisations were computationally very demanding in terms of time taken. The iterative Davidson algorithm [217, 218] for reasons which remain unclear was extremely slow to converge for e+Cd (this point is illustrated further in Appendix C.2.1). As such, there was minimal optimisation of the Laguerre type orbitals for the e+Cd system beyond choosing exponents that were consistent with the e+Zn, e+Be + and e Mg trends (see Appendix F). A series of calculations for smaller Lint were not performed, however it is expected that the convergence of the energy and annihilation + rate with Lint would be similar to that of e Zn. The comparisons shown in Table 5.10 with other predictions of the e+Cd binding energy are broadly consistent with the pattern seen for positronic magnesium and positronic zinc. The dipole-only PO calculation only just predicts binding with a binding energy, 0.000056 Hartree, about 100 times smaller than the present CIpol estimate. The MBPT binding energy, 0.0129 Hartree, is about two times larger than the present CIpol binding energy.

5.2.3 e+Zn and e+Cd extrapolation issues

The behaviour of successive increments to the energy and valence annihilation rate for e+Zn and e+Cd systems shows a slow rate of convergence towards an asymptotic form. This can be seen in Figures 5.4 and 5.5, in which the binding energy and valence + annihilation rates are shown as a function of Lmax. The results of e Be are also plotted to emphasise the structural similarities between e+Be and e+Zn against their respective model Hamiltonians.

116 Figure 5.4: The binding energy (in units of Hartree) of e+Be, e+Zn and e+Cd as a function of Lmax (with Lint = 3).

0.006 0.005 0.004 e+Cd e+Zn 0.003 0.002 e+Be (Hartree) 0.001 ε 0 -0.001 -0.002 0 2 4 6 8 10 Lmax

Figure 5.5: The 2γ spin-averaged annihilation rate (in units of 109 sec−1) of e+Be, + + e Zn and e Cd as a function of Lmax (with Lint = 3) for the valence electrons only.

0.3

0.25

) 0.2 -1 s 9 0.15 (10 2γ

Γ 0.1 e+Cd 0.05 e+Zn e+Be 0 0 2 4 6 8 10 Lmax

117 + + The CIpol calculations for e Zn and e Cd gave extrapolation exponents for pE of 2.83 and 2.79 that are somewhat smaller than the Gribakin and Ludlow pE = 4

[214], presumably because the asymptotic region is not reached until Lmax > 10. In + + the e Ca and e Sr calculations pE = 4 and pΓ = 2 were used to develop improved + estimates for pE and pΓ. This more involved procedure was not adopted for the e Zn and e+Cd systems. There are two aspects where convergence can be incomplete. Besides the size of

Lmax, the number of LTOs for each value of ℓ could be increased. Using the value of pE taken from the Lmax = 8, 9 and 10 calculations tends to yield a extrapolation correction that is an overestimate, while using only 8 LTOs per partial-wave tends to give a binding energy that is slightly too small. Fortunately, these eﬀects do tend to cancel each other out. The application of exactly the same extrapolation procedure to the very similar e+Be system, with a similar sized LTO basis, gave a binding energy

2% lower than than a FCSVMpol calculation with almost identical physical content. The current procedure was retained for both e+Zn and e+Cd for the pragmatic reason that it was a good predictor for e+Be.

Table 5.10: Binding energies (in Hartree) of positronic zinc and positronic cadmium. Only the latest calculations of a given type by a particular group are listed in this table.

System PresentCIpol MBPT[58] Other L = 10 max ∞ e+Zn 0.003039 0.003731 0.00845 0.001425a, 0.00195 b, 3.8 10−7 c × e+Cd 0.005023 0.006100 0.0129 0.000056c

a FCSVMpol calculation, not converged [74] b Polarised orbital calculation, including multipoles [60] c Relativistic polarised orbital calculations, dipole polarisation only [56].

+ + For both e Zn and e Cd, the extrapolated CIpol binding energies given in Table 5.10 should be taken as the present best estimates for these systems. As mentioned throughout this chapter, the MBPT and polarised orbital results have signiﬁcant uncertainties associated with their perturbative treatments, whilst the FCSVM e+Zn binding energy was simply not converged.

118 5.3 Experimental veriﬁcation of positron-atom binding

To date, there is no experimental evidence of positron binding to a neutral atom. Mitroy and Ryhzikh proposed [242] that positronic atoms could be formed and iden- tiﬁed by a charge exchange reaction of the form

A− + e+ e+A + e− A+ + e− + 2γ (or 3γ) . (5.2) → → For atoms with ionisation potentials > 0.250 Hartree, the Ps formation channel is not available at threshold. Depending on the energy of the incident (low-energy) positron, there will be an onset energy where the charge exchange can occur, and the positronic atom will be formed. Detection of positive ions downstream would give a suitable signature of the formation of the positronic atom. Both Cu and Ag have stable negative ions, making them ideal for such a study [242]. Mitroy and Ivanov [243] further elaborated upon using the same charge exchange reaction of Equation (5.2) for atoms with an ionisation potential of less than 0.250 Hartree. Measurement of the kinetic energy of the ejected electron as a function of the incident positron energy would both verify binding and provide a measurement of the binding energy. The suggested candidates for such an experimental approach included Li, Na, Ca, Sr and Ba [243]. However, these experiments present fundamental diﬃculties (i.e. crossing a beam of these negative ions which is typically dirty, with a beam of energetically well-deﬁned slow positrons) which would have to be overcome. In this section an alternative protocol to verifying e+ binding is presented. The group IIB atoms, zinc and cadmium, have low vapour pressures and therefore it is possible that positron scattering experiments from these neutral atoms could be done [122]. Besides the determination of the low-energy cross-sections, it is desirable to analyse the cross-sections for features that could be used as indicators of positron binding in an experimental situation. The positron-atom interaction consists of a short-range repulsive interaction and 4 a long-range interaction due to the attractive αd/(2r ) polarisation potential. The polarisation potential ensures that the phase shifts for the ℓ 1 partial-waves are ≥ positive close to threshold. The large size of the polarisability for the group IIB atoms under consideration, means that the scattering lengths for these atoms will generally be large, i.e. the positron will be weakly bound or just fail to be bound. The scattering length will be positive when the positron can be bound to the atom and negative when

119 the system does not support the bound state. Before performing any calculations, it is instructive to apply eﬀective range theory to this problem. Using the relationship between the real part of the scattering length A and the binding energy ε as [235] 1 A = , (5.3) 2 ε | | + + where the binding energies of e Zn andpe Cd are taken from the extrapolated CIpol results. This gives an estimate of the positron scattering length as +11.58 a0 and

+9.05 a0 for zinc and cadmium respectively.

5.3.1 e+-Zn and e+-Cd model potential construction

Reasonable estimates of the elastic scattering cross-section can be obtained from the e+Zn and e+Cd binding energies using a model potential technique [244, 124]. The justification of the specific details of the model potential have been given elsewhere [124]. The underlying philosophy is purely semi-empirical, with no attempt at determining the specific form of the polarisation potential by any ab initio technique being made. The effective Hamiltonian for the single positron moving in the field of the atom is approximated by the model potential,

1 2 Z Hm = 0 + Vdir(r0)+ Vpol(r0) . (5.4) −2∇ r0 −

The attractive direct potential, Vdir is computed from the HF wavefunction of the (completely frozen) neutral atom. The polarisation potential is again given the form α V (r )= d (1 exp r6/ρ6 ) . (5.5) pol 0 −2r4 − − ± 3 3 where zinc has αd = 41.23 a0 and cadmium has αd = 50.07 a0. The adjustable cut-off parameter, ρ± is fixed by two different conditions for each atom. The first requirement is that the model potential supports a bound state with + a binding energy equal to that of the extrapolated CIpol calculations. For e Zn to + have a binding energy of 0.00373 Hartree, requires ρ+ = 2.593 a0. For e Cd to have

ε = 0.00610 Hartree requires ρ+ = 2.585 a0. Since the important issue that an experiment will have to resolve is whether the scattering length is positive or negative, an additional model potential has been constructed for each atom. A second cut-oﬀ parameter ρ− was detuned to give a binding

120 energy that was negative but had the same magnitude. The values of ρ− were 4.216 a0 and 5.00 a0 for Zn and Cd respectively. With these values of ρ−, the model positron- atom interaction now only supports a virtual state.

5.3.2 e+-Zn and e+-Cd model potential scattering

The model potential scattering calculations described in this section were performed using the program developed by Mitroy and Ivanov [124] for their semi-empirical analysis of low-energy positron-atom scattering. s-wave phase shifts

When the positron is bound and has a large positive scattering length, the zero-energy s-wave phase shift will be π. This will then decrease and eventually will pass through π/2 as the energy increases. When the scattering length is negative, the s-wave phase increases from zero until it reaches a maximum in the phase shift (the maxima in the phase shift cannot exceed π/2 [245]), then begins to decrease. The s-wave positron scattering phase shifts [241] are plotted in Figure 5.6. They show the typical behaviour for the zinc and cadmium cases when the potential either supports a bound state (ρ+, solid lines) or when the potential supports a virtual state

(ρ−, dashed lines). The positron scattering lengths calculated for the model Zn and Cd ρ potentials were 14.5 a and 11.6 a respectively [241]. The effective range ± ± 0 ± 0 theory estimates from the previous section are about 20% smaller than these model potential values. When the s-wave phase shifts are in different quadrants they will interfere with the higher ℓ phase shifts and have completely different differential cross-sections. This behaviour of the s-wave phase shifts close to threshold in these two cases can be exploited to verify experimentally whether the positron can bind to either atom. The kinematic region where the phase shifts are in different quadrants can be roughly estimated as k π/(2A). ≈

Elastic cross-section

In the ﬁrst instance, the energy dependence of the elastic cross-section is examined for a signature of positron-atom binding. One of the more interesting features of the energy dependence of the elastic cross-section is that it increases very close to threshold. This

121 −1 + Figure 5.6: The s-wave phase shifts as a function of k (in units of a0 ) for e -Zn and e+-Cd elastic scattering [241]. The phase shifts corresponding to the potential tuned to support a physical state (ρ+) are shown as solid lines, while the phase shifts for the virtual state (ρ−) are shown as dashed lines. The horizontal dashed line marks

δ0 = π/2.

occurs because of the interaction of the scattering length term and polarisability term of the eﬀective range expansion [246],

πα k2 tan(δ)= Ak d + ... (5.6) − − 3 close to threshold. The resulting cross-section can be approximated as

4πA2 σel = (5.7) (1 παdk )2 + A2k2 − 3 A negative scattering length leads to a cross-section that decreases from threshold. When the scattering length is positive the cross-section increases because of the linear term in the denominator. King [228] has suggested that the rate at which the cross-section changes, dσ/dk, can be used to identify positron binding. However, this can only give a unique signature of binding at energies very close to threshold since the elastic cross section will increase 2 2 2 for k < (3παd)/(π αd + 18A ). For magnesium, zinc and cadmium this occurs for −1 k < 0.020 a0 (i.e. E < 0.0055 eV), and therefore would be diﬃcult to measure with existing positron beam technology.

122 This can be seen in the low-energy elastic cross-sections (σ+) computed using the bound state (ρ+) Zn and Cd model potentials which are given for momenta up to 0.5 a−1 in Table 5.11. The slight increase in σ (k) from threshold to k 0.02 a−1 0 + ≈ 0 is barely visible for both atoms. The cross-sections derived from the virtual state

(ρ−) model potential calculations are not given in Table 5.11. However, it should be emphasised that at higher energies the diﬀerence between the bound and virtual state cross-sections is not so obvious.

2 Table 5.11: The elastic cross-section, σ+ (in units of π a0) and the annihilation −1 parameter Zeﬀ (discussed in the following chapter) as a function of k (in a0 ) for the bound state ρ+ model Zn and Cd potentials [241]. The cross-sections at k = 0 −1 a0 were obtained by extrapolation. The cross-sections are unlikely to be reliable at −1 energies above the Ps-formation threshold (k = 0.436 and 0.401 a0 for Zn and Cd respectively).

−1 k (a0 ) Zn Cd

σ+ Zeﬀ σ+ Zeﬀ 0.00 843.9 110.7 535.8 80.14 0.01 857.3 108.8 557.9 79.04 0.02 838.7 104.4 563.0 76.75 0.03 794.1 98.19 552.1 73.75 0.04 733.1 91.00 529.6 70.28 0.05 664.2 83.45 499.7 66.60 0.06 593.9 76.04 465.5 62.91 0.08 464.7 62.91 394.4 56.18 0.10 361.0 52.78 330.1 51.02 0.12 283.4 45.56 278.2 47.72 0.15 205.9 39.04 223.7 45.81 0.20 138.8 34.89 172.9 46.30 0.25 105.5 33.33 138.3 45.00 0.30 82.77 31.36 107.8 41.03 0.40 51.46 26.63 64.93 33.12 0.50 34.11 23.51 42.97 28.98

123 Diﬀerential cross-section

The differential cross-section (DCS) is particularly interesting to determine since sig- natures of a positive scattering length are much more apparent in the differential cross-section than in the integrated cross-section. Comments about using the differential cross-section to identify positron binding have been made previously but without presenting the results of any numerical calculations [247]. The differential cross-section is written

σ(θ)= f(θ) 2 , (5.8) | | where the scattering amplitude is

1 f(θ)= (2ℓ + 1)(exp(2iδ ) 1)P (cos θ) . (5.9) 2ik ℓ − ℓ Xℓ The shape of the DCS changes dramatically when the scattering length changes sign from negative to positive. The phase shifts for ℓ 1 are positive and to a ﬁrst ≥ approximation are given by

α πk2 tan(δ ) d (5.10) ℓ ≈ (2ℓ + 3)(2ℓ + 1)(2ℓ 1) − A positive scattering length, giving a phase shift between π/2 and π results in a differential cross-section that is larger at backward angles than at forward angles. −1 This is shown in Figure 5.7 where the differential cross-sections for k = 0.09 a0 (E = 0.1102 eV) positron scattering from the zinc and cadmium model potentials are plotted. The potential supporting the bound state has a differential cross-section (solid lines) that increases slowly as the scattering angle is increased. The potential with a negative scattering length, has a differential cross-section (dashed lines) that is strongly peaked in the forward direction. There is an order of magnitude difference between the two sets of cross-sections at a backward angle such as 135o. The differences between a positive and negative scattering length are most no- ticeable in Figure 5.8 where the ratio of differential cross-sections σ(135o)/σ(30o) are plotted as a function of k. The bound state potential gives a ratio that initially increases from threshold. The virtual state potentials give a ratio that decreases steadily from threshold. Measurement of the differential cross-section ratio at incident positron −1 energies from 0.060 eV to 0.300 eV, (k = 0.0664 to 0.1485 a0 ) should provide experimental evidence of the stability of both e+Zn and e+Cd.

124 2 Figure 5.7: The diﬀerential cross-sections (in a0/sr) for positron scattering from zinc −1 and cadmium model potentials at k = 0.09 a0 (E = 0.1102 eV) [241]. The DCS for the bound state (ρ+) potentials are plotted as the solid lines while the DCS for the virtual state (ρ−) potentials are plotted as the dashed lines.

o o Figure 5.8: The ratio of the diﬀerential cross-sections σ+(135 )/σ+(30 ) are plot- −1 ted as a function of k (in a0 ) for positron scattering from zinc and cadmium (solid lines) [241]. The cross-section ratios computed from the virtual state potentials o o σ−(135 )/σ−(30 ) are shown as the dashed lines.

125 e+Hg and e+Mg detection

Differential cross-section measurements of this type could conceivably be used to determine whether mercury will bind a positron. Positron binding to mercury was predicted by Dzuba et.al. [58] who estimated a binding energy of 0.00165 Hartree. However, as we have seen for magnesium, zinc and cadmium, the MBPT calculations tend to overestimate the attraction of the positron to the atom. The polarisability of mercury, namely (33.91 0.34) a3 [248] is smaller than that of both Be and Zn which bind a ± 0 positron weakly. With the available theoretical information, it is not possible to definitely determine whether positron binding to Hg will occur. Deciding this question theoretically will be difficult since a fully relativistic calculation would be necessary. It is quite likely that the best way to determine the stability of the positron-mercury system will be a differential cross-section experiment. The stability of positronic magnesium could also be investigated in a differential cross-section experiment. The positron-magnesium system has a smaller scattering length and a larger dipole polarisability [124] so the peaking of the differential cross- section toward backward angles should be even larger than in cadmium or zinc. The disadvantage of magnesium is that production of an atomic vapour requires a cell capable of higher temperatures.

126 5.4 CuPs

The neutral Ps atom is already known to bind to a number of one-electron atoms. Apart from the heavily calculated positronium hydride (PsH) [66, 89, 111], rigorous variational evidence of positronium binding only exists for lithium positride (LiPs) [109, 106, 105, 69]. In addition, the FCSVM has demonstrated the stability of positronium binding to the other alkali atoms; sodium [71, 35, 69] and potassium [110]. It is a feature of all of these systems that the positronium binding energy increased as the ionisation potential of the parent atom increased. Therefore, the copper atom with a larger ionisation potential than lithium, sodium and potassium was a promis- ing candidate for Ps binding. Previously, Karl et.al. [53] had predicted positronium binding to copper using spectral analysis of CuH and model potentials to estimate a binding energy of 0.3 eV (0.011 Hartree). In this section, the model Hamiltonian previously used for the e+Cu study was also used in two-electron-positron CIpol calculations of copper positride (CuPs). The CuPs system is amenable to treatment by the CI method as the Ps cluster was localised relatively close to the nucleus. The calculation for CuPs used the set of core potentials + and core orbitals identical to those used previously for the CIpol calculations on e Cu, however the number of LTO orbitals per partial-wave was necessarily reduced. The Cu(4s) energy of the CuPs set of single electron orbitals was -0.28394153 Hartree (relative to the Cu+ core). The condition for binding is that the energy of the CuPs state be lower than the energy of the Ps(1s) + Cu(4s) dissociation channel, and thus binding occurs when E(CuPs) < 0.53394153 Hartree. − The optimal exponents of the orthogonal Laguerre orbitals for CuPs were diﬀerent to those of the e+Cu system (see Appendix Table F.1). Table 5.12 gives energies and expectation values for a series of CIpol calculations with a ﬁxed Lint = 3 and Lmax up to 10, which gave a binding energy of 0.010732 Hartree. Using the Lmax = 8, 9 and 10 calculations to extrapolate to gave a binding energy of 0.015575 Hartree. ∞

127 Table 5.12: Results of Lint = 3 CIpol calculations for CuPs up to a given Lmax. E(CuPs) is the 3-body energy (in Hartree) of the CuPs system, and is given relative to the energy of the Cu+ core. ε gives the binding energy (in Hartree) against dissociation into Ps + Cu. The organisation of the rest of the table is otherwise the same as Table 3.1. The L extrapolation results in the rows and p are from max → ∞ ∞ the 8, 9, 10 series.

L N N N E(CuPs) ε r r Γ Γ max e p CI h ei h pi c v 0 15 12 1440 -0.43980554 -0.0941360 3.49439 5.36908 0.165785 0.102236 1 28 23 4677 -0.48234262 -0.0515989 3.54979 5.04620 0.173993 0.224655 2 40 32 10470 -0.50564194 -0.0282996 3.61446 4.93645 0.171024 0.372719

128 3 48 40 16194 -0.52000474 -0.0139368 3.67395 4.92097 0.163992 0.527867 4 56 48 22578 -0.52866812 -0.0052734 3.73059 4.95370 0.155521 0.661990 5 64 56 29106 -0.53421689 0.00027536 3.77965 5.00079 0.148159 0.774680 6 72 64 35858 -0.53793919 0.00399766 3.82104 5.04924 0.142191 0.868844 7 80 72 42834 -0.54051928 0.00657774 3.85546 5.09396 0.137464 0.947731 8 88 80 49810 -0.54235361 0.00841208 3.88374 5.13311 0.133753 1.014171 9 96 88 56786 -0.54368679 0.00974526 3.90674 5.16623 0.130848 1.070588 10 104 96 63762 -0.54467345 0.01073191 3.92528 5.19365 0.128574 1.118878 p 2.86 - 2.04 1.79 2.33 1.48 -0.5495170 0.0155754 4.0934 5.5205 0.11253 2.02813 ∞ Figure 5.9: The binding energy (in units of Hartree) of PsH and CuPs as a function of Lmax (with Lint = 3).

0.04 0.03 0.02 PsH 0.01 0

(Hartree) -0.01 ε CuPs -0.02 -0.03 -0.04 0 2 4 6 8 10 Lmax

The behaviour of the energy and annihilation rate for CuPs demonstrates a slower rate of convergence with Lmax towards an asymptotic form than was the case for PsH. This can be seen in Figures 5.9 and 5.10, showing the binding energy and valence annihilation rates as a function of Lmax for both PsH and CuPs. One notable feature about the annihilation rate was the relatively large component arising from the core. The extrapolated value of the Γ (0.11253 109 s−1), is about c × one order of magnitude larger than the Γc so far reported for any other positron (or positronium) binding atom. The large size of Γc is a consequence of the relatively compact size of the CuPs ground state. The CuPs r was only 5.52 a , not much h pi 0 larger than PsH (3.66 a0 [66]) and somewhat smaller than LiPs (6.35 a0 [69]). The close proximity of the positron to the ten electrons in the 3d-shell results in a relatively large Γc. However, the present CuPs wavefunction is still a way from convergence and it is still an open question as to whether a core annihilation rate larger than 0.1 109 × s−1, would be maintained in a more accurate calculation. With a large correction to the binding energy of almost 50%, some estimate of the uncertainty in the extrapolated results is warranted. The exponent pE was 2.86, somewhat smaller than the expected value of pE = 4 [214]. In keeping with the same treatment as that for e+Mg, e+Ca and e+Sr (see Table 5.6), two further extrapolation estimates were made. An estimate of the minimum binding energy simply uses pE = 4

129 Figure 5.10: The 2γ spin-averaged valence annihilation rate (in units of 109 sec−1) of

PsH and CuPs a function of Lmax (with Lint = 3).

2 1.8 1.6 PsH

) 1.4 -1 1.2 s 9 1

(10 0.8 2γ

Γ CuPs 0.6 0.4 0.2 0 0 2 4 6 8 10 Lmax which gave a binding energy of 0.013560 Hartree. Choosing an intermediate value of pE = 3.43, gave a binding energy of 0.014327 Hartree, which probably gives the most reasonable estimate of the CuPs binding energy. The explicit calculation gives about half of the valence annihilation rate of 2.0328 × 9 −1 10 s . The value of pΓ derived from the Lmax = 8, 9, and 10 calculations was 1.48. Again using the Gribakin and Ludlow p = 2 [214] gave Γ = 1.5760 109 s−1 while the Γ v × in-between p of 1.74 gave Γ = 1.7351 109 s−1. The annihilation rates for PsH [66], Γ v × LiPs [69], and NaPs [69] suggest that CuPs should also have a reasonably well-formed Ps cluster, with an annihilation rate that should be slightly greater than 2.0 109 s−1. × The CIpol annihilation rate is consistent with this when the uncertainties with respect to the inadequate convergence of the wavefunction are taken into consideration.

Table 5.13 reports the sequence of calculations for Lint = 0, 1, 2 and 3 with Lmax set to 10. These calculations retained all the electron and positron orbitals of the

Lmax = 10 basis and the dimension of the largest calculation was 63762 conﬁgurations.

One notable feature of Table 5.13 is that CuPs is stable for all values of Lint. The variations in most of the quantities, with the exception of the binding energy, was relatively slow as L increased from 0 3. The convergence pattern suggests that int → ε is converged to better than 5% with respect to further enlargement of Lint. Other quantities would appear to be converged at the 1% level.

130 Table 5.13: Results of CI calculations for CuPs with the full L = 10 orbital basis of Table 5.12 for the series L 3. The organisation pol max int ≤ 3 of the table is the same as Table 3.1, with the addition of αd the core + valence dipole polarisability (in units of a0) of the model copper negative ion (see Table 2.11).

131 L N E(CuPs) α ε r r Γ Γ int CI d h ei h pi c v 0 12885 -0.53821466 400.07 0.00427313 3.98210 5.28450 0.122930 1.105187 1 30344 -0.54250938 281.04 0.00856785 3.94580 5.23042 0.126119 1.111030 2 49886 -0.54408501 277.15 0.01014348 3.93086 5.20467 0.127830 1.115487 3 63762 -0.54467345 275.64 0.01073191 3.92528 5.19365 0.128574 1.118878 5.5 Chapter summary

The CI method has been used to compute the wavefunctions and energies for the group II positronic atoms (e+Mg, e+Ca and e+Sr), the group IIB positronic atoms (e+Zn and e+Cd), as well as demonstrating the stability of CuPs. Table 5.14 shows the best

CIpol estimates for the binding energy and annihilation rate for each system. There are two primary sources of uncertainty to these results, the core polarisation potentials and the extrapolation corrections. The extrapolations corrections are most signiﬁcant for the e+Ca, e+Sr and CuPs systems. The uncertainties associated with the core-polarisation potential are largest for e+Zn, e+Cd and CuPs.

Table 5.14: CIpol binding energies (ε in Hartree) and valence annihilation rates (Γv 9 −1 in units of 10 sec ) of the two electron-positron systems. Firstly the explicit Lmax CI results, and secondly the best extrapolation L estimates. pol max → ∞ L = 10 L max max → ∞ ε Γv ε Γv e+Be 0.002533 0.23912 0.003083 0.37120 e+Zn 0.003039 0.24023 0.003731 0.39270 e+Cd 0.005023 0.29179 0.006100 0.52731 e+Mg 0.014509 0.54287 0.016151 0.89905 e+Ca 0.012358 0.73353 0.016500 1.3441 e+Sr 0.004869 0.74875 0.010050 1.4513 CuPs 0.010732 1.1189 0.014327 1.7351

The e+Zn and e+Cd systems, together with e+Be form a set of very similar exotic atoms. The positron is found at moderately large distances from a largely undisturbed atom, and is weakly bound to the atom by polarisation forces. Examination of the e+Be convergence pattern suggests that for e+Zn and e+Cd, the error associated with the binding energy extrapolations is of the order of a few percent. However, this uncertainty is not large enough to invalidate the most signiﬁcant result, namely that both zinc and cadmium are likely to bind a positron. The low-energy elastic positron scattering calculations of Zn and Cd indicate that measurement of the diﬀerential cross-sections will verify whether they can bind a positron. It will be necessary to take measurements at positron beam energies below 0.150 eV, which is achievable with current positron sources [122, 247, 249].

132 Chapter 6

Positron Scattering Using the CI-Kohn Method

“I took my dogma for a walk”

- Russell Kilbey via The Crystal Set, from the Almost Pure album 1991

133 In one respect, the calculation of positron-atom scattering is simpler than that of electron-atom scattering; there is no exchange interaction between the positron and target electrons. However, every other aspect of the theoretical treatment of positron- atom scattering presents a more difficult proposition than electron-atom scattering, due to the attractive nature of the electron-positron interaction. In this chapter, the CI bound state program is extended to perform single-centred single-channel Kohn variational calculations of low-energy positron-atom scattering. Firstly, the methodology is outlined with the uncertainties surrounding Schwartz singularities investigated by running test calculations of model potential scattering. Cal- culations of positron scattering upon two atomic systems, hydrogen and copper, are then reported in this work. The calculations of the positron-atomic hydrogen scattering system were used to validate the computer programs, as both the s-wave phase shifts and annihilation parameter, namely Zeff, for this system have been calculated by many authors to a high degree of accuracy [250, 48, 251, 252, 253, 121]. The calculations of positron scattering from copper were done for a number of reasons. First, this system supports a positron bound state so it is worthwhile to check whether Zeff is abnormaly large for the system. Next, copper is a system with an ionisation potential (7.726 eV) not much larger than the Ps ionisation potential (6.80 eV). As such, it provides an exacting test of the computational issues to achieve convergence for such scattering systems.

6.1 Heuristics of positron-atom scattering

One of the simplest (and most instructive) examples of a charge transfer reaction in many-body physics is the formation of positronium during positron-atom collisions. For systems with an I < 6.80 eV, positronium formation is possible at all incident positron energies. For atoms with I > 6.80 eV only elastic channels are open at low- energy, with positronium formation only possible if the energy of the incident positron exceeds a threshold of E = I E = I 6.80 eV . (6.1) − Ps − The processes possible during low-energy positron-hydrogen and positron-copper scattering are listed in Table 6.1. Whilst always possible, the cross-sections for in-

134 Table 6.1: The diﬀerent processes that are possible during positron-hydrogen and positron-copper scattering. Each channel is said to be open if it is energetically possible for that process to occur, i.e. Eopen denotes the required energy of the incident positron for each channel.

InitialState FinalState Process Eopen (eV) e+ + H(1s) e+ + H(1s) (Elasticscattering) 0 → 2γ (or 3γ)+p (Annihilation) 0 → Ps(1s)+p (Psformation) 6.80 → e+ + H∗(2ℓ) (Excitation) 10.2 → e+ + e− +p (Ionisation) 13.6 → e+ + Cu(4s) e+ + Cu(4s) (Elasticscattering) 0 → 2γ (or 3γ) + Cu+ (Annihilation) 0 → Ps(1s) + Cu+(3d10 1Se) (Ps Formation) 0.923 → e+ + Cu∗(3d94s2 2De) (Excitation) 1.490 → e+ + Cu∗(3d104p 2P o) (Excitation) 3.806 → e+ + e− + Cu+(3d10 1Se) (Ionisation) 7.726 →

ﬂight electron-positron annihilation are negligibly small compared to the other cross- sections [24, 254], and this process is treated using perturbation theory. Hence, the basic multichannel problem in positron scattering occurs when only the elastic and positronium formation channels are open. This is the Ore gap, which for positron- hydrogen scattering occurs for 6.80 E 10.2 eV, a range in which excitation of ≤ ≤ neither the H or Ps atoms is energetically possible.

6.1.1 Methods

There have been a number of theoretical methods employed to describe positron-atom scattering, these are thoroughly reviewed in a number of articles and do not need further discussion here [255, 256, 257, 258].

Based on the type of wavefunction employed, these can be loosely divided into three categories either following a close-coupling (CC) methodology [259] (close-coupling approximation CCA; e.g. convergent close-coupling CCC, R-matrix, and Lippmann-

Schwinger variational methods), variational methods using rij coordinates [260], or

135 methods using various perturbation series expansions [261] (e.g. Born, eikonal, distorted- wave series and many-body theory).

For similar reasons to what has been seen for the positronic-atom bound state calculations, perturbative methods struggle to deal with strong correlations associated with low-energy positron-atom interactions. Their success has been primarily limited to the description of positron-noble gas interactions [262, 114, 239, 263, 264, 29]. These diﬃculties are exempliﬁed in the polarised orbital calculations of McEachran and co- workers [114], in which polarisation multipoles of up to order 12 are required to describe positron scattering from the noble gases.

The single-centre close-coupling methods can in principle be extended to convergence by systematically increasing the number of target states which describe both the physical eigenstates and pseudostates [265]. The electron-positron correlations mani- fest themselves in CC expansions of positron-hydrogen scattering that are notoriously slow to converge [113], when compared to the equivalent electron-atom CC expansions [266].

One way to avoid the slow CC convergence is to explicitly include positronium channels into the CC expansion. The inclusion of such channels aids the CC expansion to represent both virtual positronium formation (significant at low energies), as well as the positronium formation itself (when it is energetically allowed). Due to the two- centre nature of the problem, inclusion of Ps channels into the CC carries its own set of difficulties associated with the calculation of the matrix elements between states in the positron-atom and positronium-(residual ion) groups of channels. In the case of positron-hydrogen scattering, these difficulties have been solved [267, 268, 269] and large-scale calculations can now be routinely performed [270, 271, 253, 121, 272].

The generalisation of such techniques to treat scattering from the alkali atoms is non-trivial. One area of diﬃculty lying in the Ps-residual ion channels is the treatment of the exchange interaction between the positronium atom and the residual ion. Ex- isting calculations on positron-alkali atom scattering have largely ignored these issues [273, 274, 275, 276, 277].

These diﬃculties in the applicability of the CC with Ps channels method is a cause for great concern. Apart from hydrogen (-2.1036 a0 [48]) and helium (-0.48 a0 [278]), it is diﬃcult to name one other system for which it could be asserted that the positron scattering length is known to within an accuracy of 5%! ±

136 Indeed, some of the more reliable estimates of positron atom scattering lengths are from simple model potential analyses of the group II and IIB elements [244, 124] (see Chapter 5.3.1). These calculations are believed to be reliable since the model potentials were tuned to reproduce the positron aﬃnities obtained from the bound state FCSVM or CI positronic atom calculations. Put succinctly, the ability to calculate the scattering observables for target systems with ionisation potentials greater than the Ps binding energy (6.80 eV) will lead to an improved understanding of the dynamics of positron annihilation at thermal energies. The advantage of sticking with a single-centre basis is that the matrix elements of the scattering model Hamiltonian can be evaluated without any approximations. Therefore, the following Kohn variational calculations of copper are not subject to the same degree of uncertainty as previous low-energy calculations on the alkali metals [276, 277].

6.2 Kohn variational method

In this work, the Kohn variational method is used to study positron-atom scattering using a CI type basis with the electron and positron orbitals centred on the nucleus.

The drawbacks of this approach is the slow convergence of the phase shifts with Lmax, and the restriction that the method can only sensibly be applied at energies below the Ps-formation threshold. For a system such as the positron-hydrogen system it is necessary to explicitly include orbitals with ℓ 15 for s-wave phase shifts which are ≤ converged to the 1% level [112, 113]. In the present approach, the difficulties with slow convergence are handled by simply accepting that the trial wavefunctions will have a basis of very large dimension and developing procedures to perform the necessary calculations as accurately and efficiently as possible. This turned out to be not too difficult since the existing CI program developed to study positronic atoms in a single-centre basis could be readily adapted to perform the necessary calculations.

6.2.1 Theory

The closely related Hulth`en and Kohn variational methods [279, 280, 281, 282] were among the ﬁrst variational methods applied to the quantum scattering problem. The

137 Kohn method is the continuum analog of the Rayleigh-Ritz variational method so often used for bound state problems, and is based on the differential form of the Schrödinger equation. Here, the Kohn variational method is presented using the formalism outlined in the monograph of Burke and Joachain [283] (with recourse to [284, 259, 260]). Two classes of trial wavefunctions are considered here. The first describes single particle scattering (i.e. a positron or an electron scattering off a potential), the second describes positron scattering off a single active electron in an initial state (both particles moving in the field of a potential). In addition, each class has two different functional forms, depending on the whether the scattering particle has zero- or finite-energy. Thus there are four different trial wavefunctions under consideration here, all of which are described by the generic form

NSR Ψ ; LS = α Φ ; LS + α Φ ; LS + c Φ ; LS . (6.2) | t i 0| s i 1| c i n| n i n=1 X The ﬁrst two terms Φ ; LS and Φ ; LS contain the non-square integrable (long-range | s i | c i sinusoidal) continuum functions. The set of Φ ; LS functions describe the shorter- | n i ranged correlations of the system. The factors α0 and α1 are the Kohn normalisation factors, which are discussed later.

Potential scattering trial wavefunction

For potential scattering, the trial wavefunction of Equation (6.2) is simply a combination of single particle orbitals given by

NSR Ψ ; LS = ψ (r)= α θ (r)+ α θ (r)+ c φ (r) , (6.3) | t i t 0 s 1 c i i Xi=1 where φi(r), θs(r) and θc(r) are the same single particle orbitals that are decomposed into radial and angular components as per Equation (2.6). The radial dependence of the set of NSR φi(r) orbitals is described by the same square-integrable functions used for the CI calculations (see Chapter 2.3).

The single particle orbitals θs(r) and θc(r) are composed of continuum radial functions corresponding to the regular and irregular solutions of the free-particle Schr¨odinger equation at large distances from the nucleus

1 ℓπ lim rχkℓ(r)= sin kr + δℓ(k) . (6.4) r→∞ √k − 2 138 They are chosen to be

χs,kℓ(r) = √k jℓ(kr) , (6.5)

χ (r) = √k (1 exp( βr))2ℓ+1 n (kr) . (6.6) c,kℓ − − − ℓ

The momentum, k, of the scattering particle is simply given by k2 = 2 E , where | T | ET is the energy of the system. The details of the Bessel jℓ(kr) and Neumann nℓ(kr) functional forms and other useful relationships are given in Appendix A.4. The (1 − exp( βr)) cut-oﬀ factor ensures that the irregular solution n (kr) goes to zero as − l r 0. It will be seen that with a large enough basis, the calculations are insensitive → to the precise value chosen for β.

For calculations at exactly zero-energy, one is only interested in the ℓ = 0 partial- wave which requires a diﬀerent normalisation of the continuum functions (to absorb the √k dependence) such that

rχ (r) s = r (6.7) √k rχ (r) c = (1 exp( βr)) . (6.8) √k − −

All of the basis functions so far except θs(r) and θc(r) are identical in functional form to the basis functions used in earlier CI calculations. Therefore, the amount of work required to adapt the CI program to perform scattering calculations was minimal.

Besides the normalising conditions α0 and α1, there is one other area where there is ﬂexibility in the choice of the continuum functions. This concerns whether the radial 2 functions θs(r) and θc(r) are orthogonalised to the short-range L radial basis functions of φi(r). Either choice is permissible, but we always chose to orthogonalise since this simpliﬁed the evaluation of the matrix elements. Thus the θs(r) and θc(r) radial functions will also contain linear combinations of STOs and/or LTOs basis functions

(i.e. the functions used to describe each φi(r)).

139 Positron-electron scattering trial wavefunction

The trial wavefunction for positron scattering on a single electron in a ground-state denoted by φgs(r1)s(µgs) is composed of LS-coupled electron-positron conﬁgurations

In this expression θs(r0) and θc(r0) are the same continuum functions of Equations

(6.5) and (6.7), while φgs(r1) is the ground-state orbital of the (single electron) target atom. The remaining summation of the trial electron-positron wavefunction explicitly 2 includes the ground-state orbital, φgs(r1), as well as the L set of φi(r1) electron 2 orbitals. These are coupled to the L set of φj(r0) positron orbitals as:

NSR Ne (Np−2) c Φ ; LS = c φ φ ; LS , (6.10) n| n i i,j| i j i n=1 X Xi Xj (Np−2) Ne = c φ φ ; LS + c φ φ ; LS . (6.11) gs,j| gs j i i,j| i j i Xj h iX6=gs i The L2 basis was constructed by populating all the possible configurations that could be formed by letting the electron and positron populate all the orbitals subject to the ℓ L . In keeping with the notation for the bound state calculations, N = ≤ max SR (N 2), denoting that Φ ; LS and Φ ; LS are treated within the CI-Kohn program CI − | s i | c i as configurations, for a total number of configurations of N . Each φ φ ; LS (and CI | i j i φ φ ; LS ) were of the form | gs j i

Φ ; LS = ℓ m ℓ m LM 1 µ 1 µ SM φ (r )s(µ )φ (r )s(µ ) . (6.12) | ij i h i i j j| Lih 2 i 2 j| Si i 1 i j 0 j m ,m µ ,µ Xi j Xi j

Choosing the target orbital φgs(r)

There are two one-electron atoms considered here, hydrogen and copper. Both of these require the inclusion of a single electron orbital of the radial form

gs φgs(r)= dα χα,ℓgs (r) . (6.13) α X 140 Firstly, the only atom that has an exact ground state wavefunction, i.e. H(1s), has a single electron orbital described by the inclusion of only one STO (or LTO) basis function with nα = 1,λ = 1.0. Secondly, for copper, the ground state (4s) orbital is expanded in the same ℓ = 0 mixed STO and LTO electron basis as that going to be used for the positron scatter- gs ing calculation. The coeﬃcients, dα , are determined by diagonalisation of the single particle Hamiltonian in this mixed basis. Additional ℓ = 0 electron orbitals, φ (r) , { i } spanning the same mixed STO and LTO space are used to describe the short-range electron-positron correlations in the scattering calculation.

Given that the energy of the one-electron target state is Egs, and that the total energy of the scattering system is ET , then the momentum of the incident scattering positron is given by: k2 = E E . (6.14) 2 | T − gs| where Egs is exactly -0.50 Hartree for hydrogen, and Egs for copper is the energy that comes from the diagonalisation of the one-electron target state basis (in the calculations reported here E = 0.283942174246 Hartree). Note that if extra ℓ = 0 electron basis gs − functions are added to the CI-Kohn calculation of positron-copper scattering, then the gs target coeﬃcients dα and E must be recomputed. { } gs As was the case for the potential scattering, the scattering particle (the positron) 2 has the two continuum orbitals θs(r) and θc(r) orthogonalised against the set-of L - type orbitals φj(r). The same orthogonalisation routines used for the bound state CI calculations ensures all of the electron orbitals φi(r) are orthogonalised against the single target-state orbital φgs(r).

6.2.2 Kohn linear equations

The asymptotic form of the scattering wavefunctions can be written with a number of diﬀerent normalisations depending of the form adopted for α0 and α1 [285]. These conditions can be most generally written as

α = cos τ α sin τ (6.15) 0 − t

α1 = sin τ + αt cos τ

α = tan(δ τ) t t −

141 where δ is the phase-shift of the trial wavefunction and τ [0, π ]. When τ = 0, α t ∈ 2 t reduces to tan(δt) which is just the K-matrix element. For the inverse Kohn normali- π sation τ = 2 [286], αt = cot(δt) which is the reciprocal of the K-matrix element. The generalised Kohn functional α (= tan(δ τ)) is given by [260] v v −

α = α 2 Ψ ; LS H E Ψ ; LS . (6.16) v t − h t | − T | t i where it is useful to deﬁne the trial correction term as

I = Ψ ; LS H E Ψ ; LS . (6.17) tt h t | − T | t i

If the trial wavefunction was exact, then Itt = 0. Applying the Kohn condition that the Kohn functional is stationary with respect to the linear variational parameters α and c ( n = 1,...,N ) in the trial wave- t n ∀ SR function, leads to the linear equations

∂α ∂α v =0 and v = 0 . (6.18) ∂αt ∂cn

In either case, solving the resulting set of (NSR + 1) linear equations determines the parameters αt and cn of the trial wavefunction. The details of the construction of the linear equations are left to the Appendix C.3. It should be noted that neither the use of the (Itt = 0) Hulth`en constraint [279], nor the Harris variant [287] was considered. Having obtained the trial parameters α and c , calculation of the trial correction t { n} term Itt and the use of Equation (6.16) gives αv (and hence δv). These variationally corrected quantities have an error of second order compared with the error in the trial wavefunction itself.

To calculate the variationally corrected scattering length, Av, requires Equation (6.16) to be considered in the form

tan(δ ) tan(δ ) Ψ Ψ A = v = t + 2 t ; LS H E t ; LS , (6.19) v − k − k √ − gs √ k k

i.e. using continuum functions of the forms given by Equatio n (6.7) removes the k dependence from the Kohn linear equations, and ensures that at zero-energy α t ≡ tan(δ )/k = A in Equation (6.16). t − t The scattering wavefunction and model potential components of the CI-Kohn program were validated by repeating a number of calculations described in Burke and Joachain [283]. The results of these validations are collated in the Appendix E.2.

142 Whilst the Kohn method is variational, it merely obtains the stationary parameters and does not guarantee a rigorous bound on the phase shift [283]. There are two exceptions to this, both occuring at the zero-energy threshold. Firstly, it has been shown that at zero-energy the Kohn method does provide a rigorous upper bound to the scattering length provided that there are no bound states of the system [288]. And secondly, if there are bound states, a rigorous upper bound is also given when the diagonalisation of the L2 basis produces the correct number of bound states [289]. The ﬁrst case is applicable to the positron-hydrogen system, the second to the positron- copper system. Although for positron-copper, the results are only an upper bound for the model Hamiltonian employed.

6.2.3 Annihilation coeﬃcient Zeﬀ

In examining the interactions of positrons with atoms, there is always the possibility of in-ﬂight annihilation. This is typically described by the annihilation parameter (Zeﬀ), which is related to the spin-averaged absorption cross section (σabs) as a function of the incident positron momentum (k) [23]

k σabs(k) Zeﬀ(k)= 2 , (6.20) πc r0 where c is the speed of light and r0 is the classical electron radius. The annihilation parameter is calculated from the scattering wavefunction by the identity [24, 114, 121],

2 Z = 4N d3r ...d3r d3r OˆsΨ(r ,..., r ; r ) δ(r r ) , (6.21) eff e 1 Ne 0 1 1 Ne 0 1 − 0 Z where Ψ(r1,..., rNe ; r0) is the wavefunction of the Ne electron and one positron system. Due to antisymmetry of the wavefunction, the δ(r r ) operator is only required to i − 0 operate on the first electron. In a sense, Zeff is the effective number of electrons that the positron sees as it annihilates with the atom. In the plane-wave Born approximation, the positron wavefunction is written as a plane-wave and the annihilation parameter is equal to the number of atomic electrons, i.e. Zeff = Ne. However, especially in the low-energy regime, the strong electron-positron correlations lead to an increased electron density in the vicinity of the positron, and Zeff is typically larger than Ne. Fortunately it was trivial to convert the code used to calculate the bound state

2γ core and valence annihilation rates to handle Zeﬀ (see Appendix Equations (D.18)

143 and (D.19)). Whilst the Kohn variational method does not provide a rigorous bound on Zeﬀ, the convergence with respect to increasing the ﬂexibility of the wavefunction

Ψ(r1, r0) is generally well-behaved [252, 251]. It has been noticed in the polarised orbital calculations of McEachran et.al. [114] that the phase shift converges faster than Zeﬀ.

6.3 Schwartz singularities

One of the problems of the Kohn variational method as originally formulated lies in the presence of catastrophic singularities when the K-matrix is plotted as a function of energy [290, 291]. A good deal of attention has been devoted developing procedures to eliminate or otherwise handle these singularities (refer to the extensive discussions in [284, 259]).

To be complex or not to be complex?

One of the more ingenious ways to avoid these singularities is to formulate the scattering problem with complex (i.e. S-matrix) rather than real (i.e. K-matrix) boundary conditions [292, 293]. Since the chances of these spurious singularities appearing in the complex-Kohn formulation can be minimised [294], it has been increasingly applied to a variety of scattering problems in the last decade [295, 296]. The complex-Kohn method does have two drawbacks. The ﬁrst is the annoyance of dealing with complex arithmetic and the second relates to the fact that the resulting S-matrix cannot be guaranteed to be unitary even though it can be expected to satisfy the unitary condition with increasing quality of the trial wavefunction [296]. The K-matrix version of the Kohn method was adopted in this work as it was found that any problems with singularities became increasingly minor as the size of the basis used to represent the scattering function was enlarged (Nesbet has previously commented on this point [259]). To illustrate this, we present the results of some test calculations using a model potential.

6.3.1 Model potential investigations

To investigate the so-called Schwartz singularities [290, 291], various model problems of single and multi-channel scattering have been employed [297, 294]. The present

144 investigations are based on the earlier research of Brownstein and McKinley [298] who investigated the behaviour of the Kohn variational phase shift for scattering from an attractive square well with a short range basis consisting of a small number of STOs. Whilst the test-case of [298] was also successfully reproduced, it was quickly found that the discontinuous nature of a square well potential is not described well by increasing the number the smoothly varying Slater basis functions included in the calculation. In addition, with such a discontinuity, the numerical integrations of the matrix elements were sensitive to the exact location of the Gaussian integration meshes. Since both of these eﬀects could inhibit the ability to observe in isolation the Schwartz singularities, it was not the best model for a computational exercise. Instead, the same square well is mimicked with a real Woods-Saxon optical model potential [299] (originally used for nucleon-nuclei scattering) which allows parametric control of the squareness of the square well

V V (r)= 0 . (6.22) − r−R0 1 + exp( a )

Choosing V0 = 2, R0 = 1 a0 and the surface diﬀuseness parameter, a = 0.05 a0, gives a squarish attractive potential well similar to that used in [298]. Note that a <

0.01 a0 produces a sharp enough potential gradient to still cause the same numerical issues as the square well. Whilst Brownstein and McKinley [298] investigated the variation of the K-matrix with respect to the continuum orbital cutoff β (at a fixed energy), the present model potential calculations investigated the variation in the phase shift with incident particle momenta k. This was done since with a large L2 basis, the K-matrix elements were found to be insensitive to the specific value chosen for β of Equation (6.5). This is exemplified in the static potential calculations shown in Appendix Table E.9. As the L2 STO basis was extended, the calculations with β [1, 4] became identical. The ∈ value of β = 2.0 was adopted for the remainder of calculations reported in this thesis. These conclusions are consistent with those made by Lucchese [294] for the complex Kohn method. The phase shifts will be insensitive to β as long as there is some degree 2 of overlap between the L orbitals and the continuum orbital θc(r). Calculations were performed with two sets of basis functions. The first was a set with 4 Laguerre type orbitals which had exactly the same exponents as the rn exp( λr), − (n = 1, 2, 3, 4; λ = 1.0) STO basis of Brownstein and McKinley [298]. Since the LTO

145 and STO basis sets span the same space they are eﬀectively equivalent. The second L2 basis with 28 LTOs was able to give phase shifts very close to convergence. Al- though the LTOs all have a common exponent and are thus mutually orthogonal, the two continuum orbitals were still subjected to a Gram-Schmidt orthogonalisation to ensure they were orthogonal to the LTO set. For both the N = 4 and N = 28 Laguerre basis sets, three calculations were π π performed corresponding to the Kohn normalisation conditions: τ = 0, 4 and 2 . The phase shifts for each basis set and normalisation are denoted by δN,τ . To demonstrate the variations amongst the 6 calculations, the phase shifts from the N = 28,τ = 0 calculation were taken as the reference phase shift. The relative deviation in each phase shift is calculated against the reference as

δ (k) δ (k) Relative deviation = 28,0 − N,τ . (6.23) δ28,0(k)

The three N = 4 calculations shown in Figure 6.1 clearly exhibit the catastrophic appearance of Schwartz singularities as the incident momenta is changed. Over the momentum range plotted, there are two singularities for each of the three values of τ. The basis deficiencies are most clearly exhibited in the fact that the phase shift plateaus are consistently 5% larger than the reference phase shift. Using four L2 basis functions struggles to generate an accurate wavefunction in the inner region. Figure 6.1 is very reminiscent of the figures previously published by Schwartz [290] and Brownstein and McKinley [298]. A completely different picture emerges when the L2 part of the basis is enlarged to include 28 LTO basis functions. The variations of the τ = π/4 and π/2 phase shifts relative to the normal Kohn formulation, τ = 0, are shown in Figure 6.2. The first thing to note is that the relative difference, has been multiplied by a factor of 10,000 in order to make the variations visible. Although there are one or two spikes where the relative difference reaches 3 10−4 there is no feature that could be unambiguously × identified as a Schwartz singularity. It is not possible to completely rule out the possibility that singularities may be present in the k [0.0, 1.0] a−1 range. Many ∈ 0 singularities of increasingly narrow width could very well still exist in this momentum range [298, 300].

146 Figure 6.1: Investigations of Schwartz singularities using single particle scattering from an attractive Woods-Saxon potential (deﬁned in text). The phase shifts of three N = 4 calculations with Kohn normalisations: τ = 0, π/4 and π/2, as a function of incident particle momentum k. The phase shifts are plotted relative to the N = 28,τ = 0 calculation. The exact k at which the singularities occurred for each τ were not mapped out, due to the ﬁnite set of k values for which calculations were performed.

0.15

0.1

28,0 0.05 δ ) / τ

4, 0 δ - -0.05 28,0 δ ( τ = 0 -0.1 τ = π/4 τ = π/2 -0.15 0 0.2 0.4 0.6 0.8 1 -1 k (units of a0 )

Figure 6.2: Investigations of Schwartz singularities using single particle scattering from an attractive Woods-Saxon Potential (deﬁned in text). The phase shifts of two N = 28 calculations with Kohn normalisations: τ = π/4 and π/2, plotted relative to the N = 28,τ = 0 calculation. as a function of incident particle momentum k.

28,0 0 δ ) / τ -1 28, δ -

28,0 -2 δ ( ×

4 -3 τ π 10 = /4 τ = π/2 -4 0 0.2 0.4 0.6 0.8 1 -1 k (units of a0 )

147 It is worth noting that spurious resonances above the ionisation threshold have long been a feature of close-coupling calculations of electron-hydrogen scattering that have used a pseudostate basis [301, 302]. However, when the pseudostate basis sets have been enlarged it was found that the impacts of these spurious resonances was less noticable [303, 265]. The pseudostate basis used for these CC calculation was a LTO basis identical in construction to the basis adopted for the present series of CI-Kohn calculations. The interesting thing amongst all of this is that the spurious features so prominent in calculations using a small ad-hoc pseudostate basis, seem to be of minor importance when a large Laguerre basis is used.

The philosophy underpinning the present thesis was to “Let sleeping dogs lie”. It was decided not to actively search for singularities as long as they did not mani- fest themselves in an overt manner and detract from the accuracy of the calculations. Hence, the application of the anomaly-free Kohn variational methods of Nesbet [259] was not investigated. The reliability of the Kohn and inverse Kohn variational methods for this model problem suggested that the standard Kohn method with real boundary conditions could be applied to calculations upon H and Cu. These subsequent π π calculations, which were performed for τ = 0, 4 and 2 , did not show any trace of a catastrophic singularity and furthermore the three Kohn variants gave phase shifts and Zeﬀ that agreed with each other to well within 0.1%.

6.4 s-wave elastic e+ scattering from hydrogen

The calculations upon atomic hydrogen were undertaken mainly to validate the analytic and numerical details of the CI-Kohn program used to perform the positron- copper calculations. They were also done to give information about the convergence of the phase shift and Zeﬀ with increasing Lmax. There have been many high-precision calculations of this system at low-energies (for a summary see Kuang and Gien [253] and Mitroy [271]). Brieﬂy, the pioneering calculations for this system used the Kohn variational method with Hylleraas type wavefunctions [291, 304, 250, 305]. Since then, investigations of low-energy positron- hydrogen scattering have used theoretical methods of increasing sophistication. Single- centre CC type methods applied to this system include the moment T -matrix method [306], the intermediate R-Matrix method [112], the Schwinger variational method [307]

148 and the convergent close coupling method [113]. CC calculations with explicit inclusion of Ps channels have also been highly successful [267, 268, 269, 270, 271, 253, 272]. It should be noted that there have been relatively fewer calculations that report Zeﬀ [48, 252, 308, 251, 309, 121]. The Hamiltonian used for the positron-hydrogen system is simply

1 2 1 2 Z Z 1 H = 1 0 + . (6.24) −2∇ − 2∇ − r1 r0 − r10

The present CI-Kohn approach is only applicable to the elastic scattering region, which −1 for hydrogen means in the energy range from 0 to 6.80 eV (k = 0.707 a0 ). The calculations of elastic positron scattering on a one-electron atom were initially validated by calculations upon the e+-hydrogen system using simple target states including both eigenstates H(1s) and H(1s2s2p), as well as pseudostates H(1s2s2p).

In terms of the Lmax Laguerre target basis states used here, Bray and Stelbovics applied the CCC method [265] using single particle orbitals with ℓ 15 [113] to obtain ≤ close to converged positron-hydrogen phase shifts. However, the interest here was not in obtaining 4+ significant figure variational accuracy, merely to demonstrate that reasonable accuracy of the low-energy phase shift and Zeff was achievable with medium sized (Lmax = 10) calculations.

The simple target state and Lmax calculations were performed at two-energies. The −1 ﬁrst was at a momenta of k = 0.5 a0 (where the exponents of the Lmax = 10 LTO basis were optimised). The second energy was explicitly at zero-energy. This ensured that the CI-Kohn program was working reliably at both zero- and ﬁnite-energy regimes for the purpose of the later positron-copper calculations.

−1 6.4.1 Validations and calculations at k =0.5 a0 (3.4 eV)

−1 The results of k = 0.5 a0 positron scattering from the three target hydrogen states; H(1s), H(1s2s2p) and H(1s2s2p) are collated in Appendix Table E.14. The s-wave phase shifts are compared against three previous calculations [310, 311, 241]. The

Zeff results are also in good agreement with a previous calculation which solved the Lippmann-Schwinger equations in momentum space [121]. The results in Appendix Ta- ble E.14 can usefully serve as benchmark values of Zeff for solutions of the Schrödinger −1 equation for these target states at k = 0.5 a0 .

The basis set for the Lmax calculation begins with the exact ground H(1s) target

149 state represented by a single LTO with ℓ = 0,n = 1,λ = 1.0. For the s-wave electron orbitals, 16 additional LTOs (n 2,λ = 1.6) were added to ensure convergence of the ≥ target states. For the s-wave positron orbitals, 30 LTOs with λ = 1.6 augmented the 2 continuum orbitals for ℓ = 0 convergence. A Gram-Schmidt orthogonalisation ensures that the set of electron λ = 1.6 LTOs are orthogonal to the single H(1s) orbital, and that the positron s-wave scattering orbitals are orthogonal to the 30 ℓ = 0 LTOs. The remainder of the basis uses a pure Laguerre basis with a minimum of 14 LTOs per ℓ, and, as was the case for most of the CI calculations, kept a common λ for both positron and electron orbitals.

With Lmax = 10, the positron-hydrogen scattering system at an incident momenta −1 k = 0.5 a0 was roughly optimised. The resultant LTO exponents, and the calculations using the three Kohn normalisations are shown in Table 6.2. Pleasingly, the k = 0.5 −1 a0 phase shifts for the three Kohn normalisations (τ = 0, π/4 and π/2) were identical to all the signiﬁcant ﬁgures given in Table 6.2. The Lmax = 7 and extrapolated phase shifts are also in agreement with the conceptually similar R-matrix calculations of Higgins et.al. [112]. The close to exact agreement of the extrapolated phase shift with 0.0624 [250] is a tad fortuitous, as the present system is not completely optimised at

Lmax = 10, and moreover, pδ0 = 3.41 probably has not reached its asymptotic form.

Whilst all three Kohn normalisations produce similar Zeﬀ results, the overall convergence in Zeﬀ is much slower than it was for the phase shift. At Lmax = 10

Zeff = 2.2101, which is only 80% of the variational result of Bhatia et.al. [251]. The extrapolation of L improves Z to 2.667 (i.e. 98% of [251]), suggesting that max → ∞ eff reliable values of Zeff can be obtained with the present CI-Kohn method.

A note on the trial correction term Itt

Given that the trial wavefunctions used here are never exact, the trial correction term

Itt of Equation (6.17) is never quite equal to zero. It was noticed that during the model potential calculations shown in Figure 6.1, the appearance of the singular points was reﬂected in Itt varying by orders-of-magnitude. Table 6.2 and Appendix Table E.14 also suggest that the abnormalities between the diﬀerent Kohn normalisations are also correlated against variations in Itt. The possibility is that Itt provides an integrity check of the trial wavefunctions against manifestations of Schwartz singularities.

150 −1 + Table 6.2: Results of CI-Kohn calculations for k = 0.5 a0 s-wave elastic e -H scattering up to a given Lmax. The total number of electron and positron orbitals are denoted by Ne and Np, with the LTO exponent for each partial-wave given by λ. The NCI column is the number π π of configurations included in each calculation. The results of three Kohn normalisations (τ = 0, 4 and 2 ) are reported. The s-wave phase shifts for each τ are identical to all significant figures given in the δ0 column. The s-wave annihilation parameter is denoted by Zeff, while the correction from the trial wavefunction given by I . The results in the row are from the L extrapolation, with the row p being tt ∞ max → ∞ the extrapolation exponent for each expectation value of the 8, 9, 10 series.

Lmax Ne Np λ NCI δ0 Zeff(τ0) Zeff(τπ/4) Zeff(τπ/2) Itt(τ0) Itt(τπ/4) Itt(τπ/2) 0 17 32 1.6 512 -0.23945031 0.442185 0.442185 0.442185 -8.08−11 -2.01−10 4.633−9 1 33 54 1.7 864 -0.05793051 0.916225 0.916225 0.916220 -2.155−7 -4.272−7 5.229−5 2 48 69 2.5 1089 0.00201135 1.274370 1.274369 1.274360 -2.646−7 -4.931−7 -1.851−3 3 62 83 3.0 1285 0.02825575 1.532561 1.532561 1.532548 -3.047−7 -5.533−7 -1.517−4 151 4 76 97 3.5 1481 0.04143965 1.720422 1.720421 1.720406 -3.193−7 -5.725−7 -1.084−4 5 90 111 4.0 1677 0.04870032 1.860103 1.860102 1.860086 -3.219−7 -5.732−7 -9.295−5 6 104 125 4.3 1873 0.05298441 1.966274 1.966272 1.966256 -3.246−7 -5.756−7 -8.613−5 7 118 139 4.6 2069 0.05565435 2.048754 2.048753 2.048735 -3.261−7 -5.767−7 -8.235−5 8 132 153 4.9 2265 0.05739438 2.114106 2.114105 2.114087 -3.262−7 -5.761−7 -7.989−5 9 146 167 5.3 2461 0.05857170 2.166878 2.166877 2.166858 -3.258−7 -5.746−7 -7.817−5 10 160 181 5.6 2657 0.05939360 2.210096 2.210095 2.210076 -3.254−7 -5.734−7 -7.699−5 p 3.41 1.90 1.90 1.90 0.0624148 2.66755 2.66755 2.66753 ∞ Lmax = 7 R-matrix [112] 0.055 L R-matrix [112] 0.062 max → ∞ CC(13,8) 0.0621 [271] 2.7527 [121] Kohn-Hy 0.0624 [250] 2.730 [251] 6.4.2 Validations and calculations at zero-energy

At zero-energy the (threshold) Zeff was validated with the H(1s), H(1s2s2p) and H(1s2s2p) close-coupling approximations. These zero-energy results are shown in Ta- ble 6.3, and are consistent with the (extrapolated to zero-energy) results from the momentum-space T -matrix calculations [121]. The program was also run at a very small but finite-energy (k 0.001 a−1) which ensured that the finite energy wavefunc- ≈ 0 tions converged towards the zero-energy wavefunction.

The scattering length and threshold Zeﬀ results of the Lmax zero-energy positron- hydrogen scattering calculations are also shown in Table 6.3. These calculations used −1 the identical Lmax = 10 basis to that used for the k = 0.5 a0 calculations of Table 6.2, with no attempt made to either extend the basis nor reoptimise the exponents. The L 10 scattering length of -1.88778 represents a true variational upper max ≤ limit, which is 90% of the -2.103 of Humberston and Wallace [252]. However, this percentage is only marginally improved to 91% with the L extrapolation. The max → ∞ convergence in Zeﬀ is again much slower than the scattering length. The Lmax = 10

Zeff is only 76% of the Zeff = 8.868 result of Houston and Drachman (cited in [251]). The extrapolated Z (L ) improves this to 98%. eff max → ∞ One aspect about all of these positron-hydrogen calculations that should be emphasised was the need to include a very large basis of L2 functions for the positron partial-wave that is coupled to the H(2p) excitation. The interaction between the H(1s) and H(2p) channels decays as 1/r2 at large r and to represent the virtual excitation to the H(2p) level requires a large L2 basis. For s-wave positron scattering one needs a large number of ℓ = 1 positron orbitals (i.e. 30). This requirement appears to ≥ be most demanding at energies close to threshold, which explains the inability for the −1 present Lmax = 10 positron-hydrogen basis (which was optimised at k = 0.5 a0 using only 22 ℓ = 1 positron orbitals), to give an accurate scattering length. There are three ways that the zero-energy positron-hydrogen wavefunction could be improved. Firstly, separate optimisation of the λe and λp for at least the s, p and d partial-waves (see λ trends of Appendix Table F.1). Secondly, increasing the number of long-ranged positron LTOs included for the s- and p-waves. Thirdly, increasing both

Lmax and the number LTOs per partial-wave beyond the current minimum of 14 is likely to be required for the Zeﬀ to reach an asymptotic form. However, the intent of the positron-hydrogen calculations was merely to give the

152 Table 6.3: Results of CI-Kohn calculations for zero-energy s-wave elastic e+-H scattering firstly for the model target hydrogen atoms (defined in Appendix E.2), and secondly for the Lmax series. The organisation is the same as Table 6.2, with the simplification that only the τ = 0 Kohn normalisation is sensible at zero-energy, and the difference being that the scattering length A is reported (in units of a0). The results in the row are from the L extrapolation, with the row p being the ∞ max → ∞ extrapolation exponent for each expectation value of the 8, 9, 10 series.

Lmax Ne Np λ NCI A Zeﬀ Itt 1s 1 35 1.0 35 0.58224284 0.405584 1.657−8 1s2s2p 3 65 1.0 98 -0.16180180 0.658954 -1.154−6 1s2s2p 3 65 1.0 98 -0.83540125 1.108296 -3.849−6 0 17 32 1.6 512 0.53685008 0.474487 5.362−9 1 33 54 1.7 864 -0.95498075 1.743346 -1.034−5 2 48 69 2.5 1089 -1.39497379 2.969801 -1.168−5 3 62 83 3.0 1285 -1.60398200 3.960397 -1.135−5 4 76 97 3.5 1481 -1.71703233 4.721065 -1.130−5 5 90 111 4.0 1677 -1.78291119 5.301126 -1.136−5 6 104 125 4.3 1873 -1.82350953 5.747222 -1.140−5 7 118 139 4.6 2069 -1.84966942 6.095367 -1.143−5 8 132 153 4.9 2265 -1.86716447 6.371458 -1.144−5 9 146 167 5.3 2461 -1.87922782 6.594003 -1.146−5 10 160 181 5.6 2657 -1.88778092 6.775886 -1.148−5 p 3.26 1.92 -1.9215178 8.66011 -1.154−5 ∞ Kohn-Hy [252, 260] -2.103 8.9 Kohn-Hy [48, 251] -2.1036 8.868

153 CI-Kohn method and numerics a solid test prior to performing the positron-copper scattering calculations. This aim has been achieved.

6.5 s-wave elastic e+ scattering from copper

The copper atom has an ionisation potential (7.726 eV) not much larger than the Ps ionisation potential (6.80 eV), and in addition supports a positron-atom bound state. As such, low-energy positron-copper scattering provides a more exacting test of the basis-set requirements to achieve convergence than was the case for the hydrogen atom. Presented here is a uniﬁed treatment of both the bound state positronic copper CI system and the CI-Kohn positron-copper scattering system. The positron-copper scattering model Hamiltonian is the same as Equation (3.2), with the details of the HF frozen-core orbitals and semi-empirical core-polarisation potentials being exactly the same as those used for the e+Cu ground state. The same short-range basis functions from the CI calculations were included, although positron orbitals were added to the basis to assist in representing the scattering positron at large distances from the nucleus.

The contributions to Zeff from the core and valence electrons were computed sepa- c v rately and denoted as Zeff and Zeff. As was seen in the core-annihilation rate (Γc) CI c calculations, the resultant Zeff generally provides an underestimate. This can be seen in exploratory frozen-core calculations of positron scattering from helium and neon in Appendix E.2.

6.5.1 Convergence with Lmax

To examine low-energy positron-copper scattering as a function of Lmax, two sets of −1 calculations were done. One at zero-energy and the other at k = 0.1 a0 (0.136 eV). + The scattering length and (threshold) core and valence Zeﬀ for e -Cu scattering is shown in Table 6.4 as a function of Lmax. At zero-energy only the Kohn normalisation −1 (τ = 0) is applicable. Table 6.5 gives the k = 0.1 a0 s-wave phase shifts and core and valence Zeﬀ as a function of Lmax for the τ = 0 and τ = π/2 (inverse) Kohn normalisations.

154 + Table 6.4: Results of CI-Kohn calculations for zero-energy s-wave elastic e -Cu scattering up to a given Lmax. The organisation is mostly c v the same as Table 6.3. The core and valence annihilation parameters are given separately as Zeff and Zeff. Two additional columns are added, v v v 2 −2 firstly ∆Zeff is the incremental contribution to Zeff with each Lmax. Secondly, the combined value of Zeff/A (in units of a0 ) is shown. The results in the row are from the L extrapolation, with the row p being the extrapolation exponent for each expectation value of ∞ max → ∞ the 16, 17, 18 series. The row Zv uses the L results of A2 and Zv /A2 to generate an improved estimate of Zv . eff† max → ∞ eff eff c v v v 2 Lmax Ne Np NCI A Zeff Zeff Itt ∆Zeff Zeff/A 0 22 30 618 0.1296219 0.934990 0.497298 -1.915−2 - 29.59783 1 42 50 1018 -16.26256 13.80431 16.52639 -2.289−2 16.02910 0.062489 2 60 66 1306 -372.1632 5421.634 10971.50 -1.191−1 10954.97 0.079213 3 75 81 1531 41.460977 59.66164 178.2353 -7.231−3 -10793.3 0.103685 4 89 95 1727 25.534955 20.94131 82.80159 -1.155−2 -95.4337 0.126990 5 103 109 1923 20.590112 12.86783 62.72724 -1.290−2 -20.0744 0.147958 −2

155 6 117 123 2119 18.245298 9.678288 55.39451 -1.354 -7.33273 0.166404 7 131 137 2315 16.912853 8.043046 52.20894 -1.391−2 -3.18557 0.182520 8 145 151 2511 16.074744 7.077523 50.79479 -1.415−2 -1.41416 0.196576 9 159 165 2707 15.511902 6.455043 50.25382 -1.431−2 -0.54097 0.208852 10 173 179 2903 15.116145 6.029074 50.17874 -1.442−2 -0.07508 0.219603 11 187 193 3099 14.828230 5.724819 50.36076 -1.450−2 0.182027 0.229041 12 201 207 3295 14.613151 5.500363 50.68491 -1.456−2 0.324149 0.237351 13 215 221 3491 14.449014 5.330538 51.08498 -1.461−2 0.400068 0.244690 14 229 235 3687 14.321541 5.199410 51.52055 -1.465−2 0.435573 0.251189 15 243 249 3883 14.221059 5.096445 51.96740 -1.468−2 0.446851 0.256961 16 257 263 4079 14.140841 5.014442 52.41032 -1.470−2 0.442914 0.262100 17 271 277 4275 14.076085 4.948335 52.84027 -1.472−2 0.429951 0.266687 18 285 291 4471 14.023293 4.894474 53.25195 -1.474−2 0.411683 0.270792 p 3.57 3.58 0.76 - - 1.94 13.67963 4.54533 118.088 - - 0.34630 ∞ Zv ֒ 64.8040 eff† → −1 + Table 6.5: Results of CI-Kohn calculations for s-wave elastic k = 0.1 a0 e -Cu scattering up to a given Lmax, using the same basis defined c v 6.4. The organisation is mostly the same as Table 6.2. The core and valence annihilation parameters are given separately as Zeff and Zeff. v v Two additional columns are added, firstly ∆Zeff is the incremental contribution to Zeff with each Lmax. Secondly, the combined value of v 2 −2 Zeff/X (in units of a0 ) where X = sin(δ)/k is shown. The results in the row are from the Lmax extrapolation, with the row p ∞ v → ∞ 2 being the extrapolation exponent for each expectation value of the 16, 17, 18 series. The row Zeff uses the Lmax results of X and v 2 v † → ∞ Zeff/X to generate an improved estimate of Zeff. c c v v v 2 v 2 Lmax δ0(τ0) δ0(τπ/2) Zeff(τ0) Zeff(τπ/2) Zeff(τ0) Zeff(τπ/2) Itt(τ0) Itt(τπ/2) Zeff/X (τ0) Zeff/X (τπ/2) 0 -0.0537163 -0.0537162 0.867343 0.867632 0.451188 0.451336 1.098−4 -3.161−3 1.565180 1.565694 1 0.5602140 0.5602140 3.119401 3.121575 3.661005 3.663539 2.217−4 5.468−4 0.129661 0.129751 2 1.0500912 1.0500913 4.836386 4.840069 9.594584 9.601845 4.644−4 4.121−4 0.127502 0.127599 3 1.3941351 1.3941352 4.922114 4.924992 14.42071 14.42908 1.313−3 3.628−4 0.148803 0.148889 4 -1.5457535 -1.5457534 4.409587 4.411448 17.10034 17.10748 -8.676−3 3.362−4 0.171111 0.171182 5 -1.4257073 -1.4257072 3.926940 3.928186 18.77483 18.78070 -1.414−3 3.196−4 0.191757 0.191817 156 6 -1.3505040 -1.3505040 3.565854 3.566754 20.01703 20.02199 -8.896−4 3.082−4 0.210207 0.210260 7 -1.3009393 -1.3009392 3.305284 3.305980 21.04308 21.04742 -7.013−4 3.001−4 0.226531 0.226578 8 -1.2668645 -1.2668645 3.116000 3.116569 21.93480 21.93871 -6.061−4 2.941−4 0.240927 0.240969 9 -1.2426243 -1.2426243 2.976171 2.976656 22.72793 22.73153 -5.498−4 2.896−4 0.253628 0.253668 10 -1.2248942 -1.2248941 2.870990 2.871419 23.44091 23.44430 -5.132−4 2.862−4 0.264855 0.264893 11 -1.2116271 -1.2116270 2.790530 2.790918 24.08446 24.08770 -4.879−4 2.834−4 0.274796 0.274833 12 -1.2015084 -1.2015084 2.728046 2.728404 24.66640 24.66952 -4.697−4 2.813−4 0.283615 0.283651 13 -1.1936644 -1.1936644 2.678869 2.679205 25.19336 25.19641 -4.562−4 2.796−4 0.291459 0.291494 14 -1.1874984 -1.1874983 2.639703 2.640022 25.67066 25.67364 -4.459−4 2.782−4 0.298448 0.298483 15 -1.1825915 -1.1825915 2.608180 2.608485 26.10333 26.10627 -4.379−4 2.771−4 0.304690 0.304725 16 -1.1786446 -1.1786446 2.582566 2.582860 26.49568 26.49858 -4.315−4 2.761−4 0.310276 0.310310 17 -1.1754389 -1.1754389 2.561573 2.561859 26.85174 26.85462 -4.265−4 2.754−4 0.315284 0.315318 18 -1.1728127 -1.1728126 2.544232 2.544512 27.17516 27.17803 -4.224−4 2.747−4 0.319784 0.319818 p 3.49 3.49 3.34 3.34 1.68 1.68 - - 1.87 1.87 -1.1550791 -1.1550791 2.41946 2.41970 35.2057 35.2102 0.40901 0.40906 ∞ Zv ֒ ֒ 34.2306 34.2343 eff† → → v Also given in Tables 6.4 and 6.5 are the incremental contribution to Zeff with each v 2 −1 Lmax and the value of Zeff/X , where for zero-energy X = A, while at k = 0.1 a0 v X = sin(δ)/k. These are used to generate an improved estimate of Zeff, the reasoning behind which is discussed later.

The regular asymptotic behaviour of the scattering length as well as the Zeff with −1 Lmax, and the agreement between the different Kohn normalisations at k = 0.1 a0 continues to suggest no appearance of catastrophic Schwartz singularities. The small variations of expectation values such as δ0 and Zeff for different Kohn normalisations are insignificant when compared to the incomplete convergence of the basis and the error resultant from the extrapolation processes.

Convergence of scattering length

The scattering length in Table 6.4 changes sign at Lmax = 3 since this is the minimum

Lmax for positron binding (see positronic copper results in Table 3.2). The scattering length decreases monotonically for Lmax > 3 and was A = 14.02 a0 at Lmax = 18.

The power series extrapolation using pA = 3.57 gave an estimated scattering length of

+13.68 a0. This scattering length is about 15% larger than a copper model potential scattering length of +11.8 a0 by Mitroy and Ivanov [32]. The model potential [32] was tuned to the FCSVM e+Cu binding energy of 0.005597 Hartree, which accounts for about 5% of the discrepancy between the model potential and Kohn scattering lengths. −1 The convergence of both the scattering length and the k = 0.1 a0 phase shift is well behaved, requiring only minor extrapolation corrections. Both the phase shift and scattering length partial-wave convergence patterns are consistent with the convergence of the e+Cu CI calculations.

c Convergence of Zeﬀ

c Figure 6.4 shows that the core annihilation parameter Zeff in the zero- and finite-energy cases has quite different behaviour with Lmax. When the scattering length is at its c most negative (at Lmax = 2), the zero-energy Zeff is massive, exceeding 5400!

This Zeff behaviour is consistent with effective range analyses showing that the threshold Z A2 for large values of the scattering length [312, 30, 124]. Essen- eff ∝ tially the normalisation condition, which relates the asymptotic wavefunction to the wavefunction in the interaction region, leads to a large Zeff whenever A is large.

157 + Figure 6.3: The scattering length (A in a0) for e -Cu scattering as a function of

Lmax.

25 ) 0 20 A (a

10 0 5 10 15 20 Lmax

−1 c The k = 0.1 a0 Zeff has only a slightly pronounced peak at Lmax = 2, with the contribution from Z A2 becoming less significant with increasing energy. This eff ∝ suggests that the strong dependence of Z A2 at threshold diminishes as the energy eff ∝ of the positron increases.

c At both energies, Zeff appears to be well converged by Lmax = 18. The extrapo- + lation exponents pZc = 3.58 and 3.34 are consistent with the CI calculations of e Cu which resulted in pΓc = 4.04 (see Table 3.2). Interestingly, there are opposing convergence trends between the bound state and scattering systems. The Γc convergence is closely related to the average location of the positron r . As the positron slowly h pi c drifts in with Lmax there is a gradual increase in Γc. The opposite is true for Zeff at −1 c both zero-energy and k = 0.1 a0 , with a systematic decrease in Zeff. The decrease c in Zeff with Lmax is related to the decrease in A which has been previously seen in a semi-empirical study of positron scattering from a model argon atom [124].

v Convergence of Zeﬀ

v The variation of Zeﬀ versus Lmax shown in Figure 6.5 reveals a more complicated pattern. The k = 0 annihilation parameter reaches a maximum near Lmax = 3, decreases steadily until Lmax = 10 and then starts to increase for Lmax > 10. This behaviour is caused by two opposing trends:

158 −1 −1 c Figure 6.4: The threshold k = 0 a0 (solid line) and k = 0.1 a0 (dashed line) Zeﬀ + c for s-wave e -Cu scattering as a function of Lmax. At Lmax = 2 the Zeﬀ(k = 0) was over 5,400.

8 -1 k = 0 a0 6 c eff Z 4

2 -1 k = 0.10 a0 0 0 5 10 15 20 Lmax

Firstly, that the threshold Z A2 for large A [312, 30, 124]. Therefore, Zv • eff ∝ eff v should decrease along with A as Lmax increases. The tendency for Zeff to decrease as L increases from 3 10 is essentially a consequence of the normalisation max → of the asymptotic wavefunction.

The increase in Zv for L > 10 occurs for the same reason that the valence • eff max + annihilation rate Γv of the e Cu ground state increases as Lmax increases. The inclusion of orbitals with increasingly larger ℓ into the calculation permits an improved localisation of the positron close to the electron leading to an increase in the annihilation matrix element. In effect, the ability of the basis to describe electron pileup around the positron outweighs the effect of a decreasing A.

The Zv behaviour at k = 0.1 a−1 exhibits a gradual increase in Zv for L 0, eﬀ 0 eﬀ max ≥ as the electron-positron pileup description improves.

159 −1 −1 v Figure 6.5: The threshold k = 0 a0 (solid line) and k = 0.1 a0 (dashed line) Zeﬀ + v for s-wave e -Cu scattering versus Lmax. At Lmax = 2 the k = 0 Zeﬀ was 11,000.

70 -1 k = 0 a0 60

40 v eff Z 30

20 -1 k = 0.10 a0 10

0 0 5 10 15 20 Lmax

v Behaviour of ∆Zeﬀ

v v To unravel the Zeff trends with Lmax, the incremental contributions to Zeff are shown as a function of Lmax for the zero-energy case in Figure 6.6. Also shown (for later v 2 comparison) are the incremental contributions to the factor Zeff/A . v v The increments to Zeff for successive values of Lmax > 10 show clearly that Zeff v is nowhere near an asymptotic form at Lmax = 18. Whilst the increase in Zeff begins v at Lmax = 11, the maximum value of ∆Zeff is at Lmax = 15 and by Lmax = 18, v ∆Zeff has only just begun to head back down towards zero. Due to the competing processes of Zv A2 and the gradual improvement in the description of electron- eff ∝ v positron clustering, the low-energy extrapolations of Zeff are fraught with uncertainty. v For example, using the zero-energy Zeff Lmax = 16, 17 and 18 results, one obtains v pZV = 0.75 which yields an extrapolated Zeff of 118.1. Such a large contribution to Zv from the L = 19 is simply an artifact of the extrapolation procedure. eff max → ∞ 2 The competition between the Zeff vs A effect and the clustering effect is less promi- −1 nent at k = 0.1 a0 . This is shown in Figure 6.7, which shows that the incremental v v contributions, ∆Zeff, are positive for all Lmax. While there is a small peak in ∆Zeff at the e+Cu binding threshold of L = 2, once L 10 both ∆Zv and ∆(Zv /X2) max max ≥ eff eff have settled into steady asymptotic forms.

160 v Figure 6.6: The incremental contributions to the threshold Zeﬀ (solid line) for s-wave + e -Cu scattering as a function of Lmax. Also shown are incremental contributions to v 2 the ratio Zeﬀ/A (see Table 6.4).

2.5 -1 [k = 0 a0 ] 2

1.5 × ∆ v 2 100 (Zeff/A ) 1

0.5

0 ∆ v Zeff -0.5

-1 0 5 10 15 20 Lmax

v Figure 6.7: The incremental contributions to the threshold Zeﬀ (solid line) for s-wave + e -Cu scattering as a function of Lmax. Also shown are incremental contributions to v 2 the ratio Zeﬀ/A .

6 -1 [k = 0.10 a0 ] 5

3 ∆ v Zeff 2

0 × ∆ v 2 100 (Zeff/X ) -1 0 5 10 15 20 Lmax

161 v Improved Zeﬀ extrapolation procedure

v Taking the view that the slow convergence of Zeff is largely due to the simultaneous v variation of A with Lmax suggested that some combination of Zeff and A was more 2 amenable to analysis. At zero energies the combination Zeff/A was examined, while 2 Zeff/X where X = sin(δ)/k is the obvious generalisation at finite energies. The ratio, v 2 Zeff/X is plotted as a function of Lmax in Figure 6.8 at both k = 0 and k = 0.1 −1 a0 . Both ratios show a remarkably steady increase for Lmax > 2, especially given the 2 variations in both Zeff and X over this range.

v 2 Figure 6.8: The ratios Zeﬀ/X where X = A for k = 0 and X = sin(δ)/k for k = 0.1 −1 + a0 , plotted as a function of Lmax for s-wave e -Cu scattering.

0.35 -1 k = 0.10 a0 0.3

0.25 -1 k = 0 a0 2 0.2 /X v eff

Z 0.15

0.1

0.05

0 0 5 10 15 20 Lmax

v 2 Furthermore, the incremental changes to Zeff/X for successive Lmax that were plotted in the previous Figures 6.6 and 6.7 show a steady decrease for L 5 and max ≥ L 10 respectively. The regular behaviour of Zv /X2 with L suggested that the max ≥ eff max L = 19 contribution could be obtained by extrapolating the quantity Zv /X2 max → ∞ eff to . Then the extrapolated Zv /X2 could be multiplied by the independently ∞ h eff i∞ extrapolated X2 to give an improved estimate, denoted by Zv , i.e. eff† Zv = Zv /X2 X2 . (6.25) eff† h eff i∞ × h i∞ v 2 At Lmax = 18, the decay of the zero-energy ratio Zeff/A resulted in an exponent, pZv/A2 = 1.94, significantly larger than the original pZv = 0.76. Multiplication of the v 2 −2 extrapolated Zeff/A (0.3436 a0 ) and the extrapolated scattering length (+13.68 a0)

162 gave an estimated threshold Zv = 64.80. This 22% increase in Zv from 19 is eff† eff → ∞ + far more compatible with the 28% change in Γv during the e Cu extrapolation (see Table 3.2). −1 At k = 0.1 a0 the Lmax = 16, 17 and 18 results gave an exponent of pZ/X2 = 1.87. Using the extrapolated phase shift (-1.155) in X = sin(δ)/k, and multiplying by v 2 −2 the independently extrapolated Zeff/X = 0.409 a0 , gave the preferred estimate of Zv = 34.23. This is close to the extrapolated Zv = 35.21, obtained by extrapolating eff† eff v Z with p v = 1.68. That both extrapolation methods give similar results at eff → ∞ Z −1 v v 2 k = 0.1 a0 can be seen in Figure 6.7, where the ∆Zeff and Zeff/X curves tracked alongside each other for Lmax > 10.

6.5.2 Positron-copper scattering for 0 < k 0.2 a−1 (0 0.544 eV) ≤ 0 → Having found stable CI-Kohn convergence patterns for both zero-energy and k = 0.1 −1 a0 positron-copper scattering, further calculations were performed out to a maximum −1 incident positron energy of 0.544 eV (k = 0.2 a0 ). These calculations remaining well below the energy at which the Ps formation channel opens, i.e. 0.923 eV (k = 0.260 −1 a0 ). c v The results of the s-wave phase shifts, Zeﬀ and Zeﬀ are shown in Tables 6.6 and 6.7. All of the e+-Cu calculations were run with three Kohn normalisations (τ = 0, π/4 and π/2), but since there were still no signs of Schwartz singularities, only the standard Kohn normalisation τ = 0 is reported.

163 Table 6.6: Results of CI-Kohn calculations for s-wave elastic k = 0.02 0.08 a−1 → 0 + e -Cu scattering up to a given Lmax, using the same basis defined in Table 6.4. The organisation is mostly the same as Table 6.5 with only the τ = 0 Kohn calculations are reported. Two columns are added; firstly momentum k (in a−1). Secondly, Zv 0 eff† which uses the L results of δ and Zv /X2 (where X = sin(δ )/k) to gen- max → ∞ 0 eff 0 erate improved estimates of Zv . The results in the row are from the L eff ∞ max → ∞ extrapolation, with the row p being the extrapolation exponent for each expectation value of the 16, 17, 18 series.

k L δ Zc Zv Zv /X2 Zv max 0 eff eff eff eff† 0.02 16 -0.28066908 4.817756 50.31421 0.262298 0.02 17 -0.27942671 4.756121 50.74815 0.266856 0.02 18 -0.27841349 4.705936 51.16082 0.270936 0.02 p 3.57 3.57 0.88 1.94 0.02 -0.2717992 4.37829 98.1830 0.34646 62.4262 ∞ 0.04 16 -0.54566895 4.325440 45.06850 0.267716 0.04 17 -0.54344859 4.274738 45.50582 0.272304 0.04 18 -0.54163598 4.233281 45.91562 0.276413 0.04 p 3.55 3.52 1.14 1.92 0.04 -0.5297172 3.95743 71.3497 0.35341 56.3948 ∞ 0.06 16 -0.78439297 3.710071 38.50967 0.277828 0.06 17 -0.78156033 3.671413 38.93497 0.282500 0.06 18 -0.77924500 3.639697 39.32798 0.286689 0.06 p 3.53 3.46 1.38 1.91 0.06 -0.7638799 3.42326 54.6009 0.36653 48.7161 ∞ 0.08 16 -0.99452100 3.106298 32.07530 0.291976 0.08 17 -0.99139422 3.077802 32.47101 0.296785 0.08 18 -0.98883534 3.054342 32.83293 0.301103 0.08 p 3.51 3.40 1.56 1.89 0.08 -0.9716981 2.88987 43.4336 0.38499 41.0267 ∞

164 Table 6.7: Table 6.6 continued. Results of CI-Kohn calculations for s-wave elastic k = 0.12 0.20 a−1 e+-Cu scattering up to a given L . → 0 max k L δ Zc Zv Zv /X2 Zv max 0 eff eff eff eff† 0.12 16 -1.34087941 2.157460 21.96757 0.333662 0.12 17 -1.33771718 2.141661 22.28212 0.338945 0.12 18 -1.33512397 2.128571 22.56653 0.343697 0.12 p 3.47 3.29 1.76 1.85 0.12 -1.3174869 2.03206 28.9646 0.43990 28.6298 ∞ 0.14 16 -1.48504116 1.824572 18.42236 0.363747 0.14 17 -1.48198128 1.812261 18.69866 0.369400 0.14 18 -1.47946980 1.802032 18.94777 0.374491 0.14 p 3.46 3.24 1.81 1.83 0.14 -1.4622747 1.72483 24.2247 0.47990 24.1975 ∞ 0.16 16 1.52738874 1.564008 15.65048 0.401408 0.16 17 1.53032418 1.554038 15.89346 0.407540 0.16 18 1.53273564 1.545731 16.11212 0.413068 0.16 p 3.44 3.19 1.84 1.81 0.16 1.5493572 1.48157 20.5790 0.53045 20.7113 ∞ 0.18 16 1.41069622 1.351265 13.40129 0.445524 0.18 17 1.41350492 1.342931 13.61528 0.452232 0.18 18 1.41581459 1.335966 13.80766 0.458288 0.18 p 3.42 3.14 1.86 1.79 0.18 1.4318585 1.28081 17.6596 0.59068 17.8810 ∞ 0.20 16 1.30428253 1.198408 11.78938 0.506723 0.20 17 1.30697397 1.191001 11.98181 0.514242 0.20 18 1.30919013 1.184791 12.15477 0.521044 0.20 p 3.40 3.08 1.87 1.75 0.20 1.3247425 1.13419 15.6063 0.67577 15.8919 ∞

165 The extrapolated s-wave phase shifts from Tables 6.5, 6.6 and 6.7 are plotted as a function of k in Figure 6.9. These are in reasonable agreement with the phase shifts of positron-copper scattering that were obtained using the model potential approach of Mitroy and Ivanov [124, 241]. As mentioned earlier in regard to the scattering length, some of the diﬀerence between the two curves can be attributed to the model potential having been tuned to the slightly stronger FCSVM e+Cu binding energy of 0.005597 Hartree. This indicates that the model potential approximation gives the s-wave phase shifts to an accuracy of about 10%.

Figure 6.9: The s-wave phase shift as a function of k for positron scattering from copper. Included are the extrapolated results from the CI-Kohn calculations, as well as the results of the model potential calculations of Mitroy and Ivanov [124, 241].

3.5

2.5

(k) Model potential 0 δ 2 CI-Kohn∞ 1.5

1 0 0.05 0.1 0.15 0.2 -1 k (a0 )

v The variation of Zeff with k is shown in Figure 6.10. For comparison the explicit v Lmax = 18 results are plotted, secondly the extrapolated Zeff results, and thirdly the improved (and preferred) Zv extrapolated results using the Zv /X2 procedure. eff† eff −1 v Since the two extrapolation curves merge for k > 0.1 a0 , convergence in Zeff is achieved relatively easily for such energies. For the very-low-energy regime (k < 0.1 a−1), the Zv convergence is impacted by the A2 relation. The use of the improved 0 eff ∝ c extrapolation procedure appears to solve this problem. Combining the core Zeff and v valence Zeff yields a value of 58.15 at Lmax = 18. The improved extrapolation to the L = limit gave a total core and valence Z = 69.35, which should be regarded max ∞ eff as the best CI-Kohn estimate of the threshold positron-copper annihilation parameter.

166 v Figure 6.10: The valence annihilation parameter Zeﬀ is plotted as a function of k for + s-wave e -Cu scattering. Firstly for the explicit Lmax = 18 series, secondly for the L = 19 extrapolated series, and thirdly the improved extrapolation results max → ∞ (Zv ). eﬀ† 120

100 → ∞ Lmax 80 v Zeff²

v eff 60 Z

40 Lmax=18 20

0 0 0.05 0.1 0.15 0.2 -1 k (a0 )

6.6 Zeﬀ for metal vapours

These estimates of Zeff are relevant to the existing debate about the mechanisms responsible for positron annihilation in collisions with gases which has attracted much discussion [313, 29, 30, 122, 123, 124]. Analysis of the results of Zeff from a large number of annihilation experiments for noble gases and molecules with single bonds by Murphy and Surko [313] resulted in Figure 6.11. This suggested a simple scaling with a semi-empirical formula of [313, 28, 122] B ln(Z ) + A, (6.26) eff ≈ I E | − Ps| where B 33 when I is given in eV. Using Equation (6.26) as a guide, there have ≈ been speculations that metal vapours such as Zn and Cd (and by implication Cu), with 1/ I E of 0.39, 0.46 and 1.08 eV−1 respectively, could have a threshold Z | − Ps| eff of order 106 [122].

The present Zeﬀ results for positron-copper scattering seen in Figure 6.10 stand in stark contradiction to those predicted from the semi-empirical formula of Equation (6.26). This is now examined further in the context of eﬀective range theory and model potential scattering.

167 Figure 6.11: Experimental measurements of the annihilation parameter Zeff for noble gases (filled circles), alkanes (hollow circles), and the F,Cl, and Br substituted alkanes (filled squares, hollow squares, filled triangle) as a function of I E −1, according | − Ps| to Murphy and Surko [313]. Note that the CI-Kohn estimate of Zeff = 69.35 for Cu (which has 1/ I E = 1.08 eV−1) lies well beyond the bottom-right of the graph. | − Ps|

6.6.1 Eﬀective range theory of positron annihilation

The tendency to associate large values of Zeﬀ with bound states [25, 26, 27, 29, 30, 31], immediately raises the question of whether the converse is true. Is the threshold Zeﬀ always large when the scattering system supports a bound state?

An initial answer to this question can be determined by applying effective range theory to the problem [124]. The real part of the scattering length A relates to the binding energy ε as A = 1/ 2 ε (Equation (5.3)), while at zero-energy Z becomes | | eff p Γ Zeff(k = 0) = 0.0440153 . (6.27) ε 3 | | p In this equation ε is expressed in Hartree while the annihilation rate Γ is given in units of 109 sec−1.

Equations (5.3) and (6.27) were applied to the positron binding atoms with ionisation potentials > 6.80 eV. The eﬀective range theory estimates based on the CIpol calculations of the previous chapters are shown in Table 6.8. Also shown are the estimates from the latest FCSVM results [74, 69, 241].

168 Table 6.8: Estimates of scattering length and threshold Zeﬀ for the positron binding atoms; Be, Mg, Cu, Zn and Cd. These estimates were calculated using Equations (5.3) and (6.27) [124]. The CIpol data for each positronic atom ﬁrstly used the explicit Lmax results and secondly the L extrapolated estimates. The FCSVM values use max → ∞ the latest FCSVM estimates [74, 69, 241]. It should be noted that the total (core and valence) annihilation rates were used.

CI (L ) CI (L ) FCSVM pol max pol max → ∞ pol A Zeff A Zeff A Zeff e+Mg 5.87 13.9 5.46 20.1 5.66 21.3 e+Cd 9.98 40.5 9.05 52.2 e+Cu 10.24 51.2 9.91 59.5 9.45 60.8 e+Zn 12.83 69.0 11.58 80.5 18.73 203 e+Be 14.15 84.3 12.82 97.5 12.60 104

6.6.2 Model potential estimates of positron annihilation

More reﬁned estimates of the threshold Zeﬀ for copper have been determined by Mitroy and Invanov [124] using the model potential scattering approach described in Chapter

5.3.1. The annihilation parameter Zeﬀ for a model potential wavefunction can be written as [124]

Z = d3r G ρ (r)+ G ρ (r) Φ(r) 2 , (6.28) eﬀ v v c c | | Z where ρc(r) and ρv(r) are the electron densities associated with the core and valence electrons of the target atom, and Φ(r) is the positron scattering function.

The enhancement factors G are introduced to take into consideration the impact that electron-positron correlations will have upon the annihilation rate. The attractive nature of the electron-positron interaction leads to electron-positron correlations that increase the electron density at the position of the positron, and consequently increases the annihilation rate. Such enhancement factors are routinely used in the calculation of the annihilation rate of positrons in condensed matter systems [314, 315, 316].

The enhancement factor for valence and core electrons is treated diﬀerently. For core orbitals, Gc is simply set to 2.5 due to reasons outlined in [124]. The valence

169 enhancement factor Gv is computed by the simple identity

bound Γv Gv = model , (6.29) Γv

bound where Γv is the annihilation rate of the positron with the valence orbitals as given model by the bound state calculation, and Γv is the valence annihilation rate predicted by the model potential calculation with Gv = 1.

An estimate of the positron-copper Zeﬀ(k) was obtained by Mitroy and Ivanov + [124] using the FCSVMpol e Cu results. This model potential resulted in a scattering length of +11.8 a0, the phase shifts seen in Figure 6.9, and gave a threshold Zeﬀ of

96.4. The dependence of Zeﬀ with k can be seen in Fig. 6.12, which compares the total

(core and valence) extrapolated CI-Kohn Zeff against the model potential Zeff results [124, 241]. As was the case for the scattering length and phase shifts, part of the difference between the CI-Kohn Zeff(k) and model potential Zeff(k) can be attributed to the copper model potential having been tuned to the FCSVMpol results. The present

CI-Kohn positron-copper phase shifts and Zeﬀ lend further justiﬁcation to the model potential approach of Mitroy and Ivanov [124].

Figure 6.12: The sum of the core Zc and valence Zv plotted as a function of k for eff eff† + s-wave e -Cu scattering. Also shown is the model potential s-wave Zeff calculated by Mitroy and Ivanov [124, 241].

100

eff Model potential Z 40 CI-Kohn² 20

0 0 0.05 0.1 0.15 0.2 -1 k (a0 )

Similarly, the application of the model potential based on the present e+Zn and + e Cd CIpol results gave a threshold Zeﬀ of 110.7 and 80.14 for Zn and Cd respectively.

170 The variation of Zeff with k for the zinc and cadmium model potentials are shown in Table 5.11. These calculations show a similar magnitude and variation with k as the copper calculations shown in Figure 6.12. This adds further weight to the accuracy of the Zn and Cd model potential scattering differential cross-section calculations seen in Chapter 5.3.1, which were used to suggest a method of experimental verification of positron-atom bound states.

Summary

The effective range and model potential estimates of Mitroy and Ivanov [124] had suggested that for a metal vapour such as copper a large value of the threshold Zeff was very unlikely. However, it is the present CI-Kohn calculations of positron-copper scattering that unequivocally show that the existence of a positron-atom bound state does not lead to a very large Zeff at threshold. These positron-copper scattering calculations were not performed in the expectation that they would motivate an experimental investigation. Due to its high melting temperature and the existence of a low-lying metastable state, a gas of neutral copper atoms in the ground state is rather hard to make. Instead these calculations were performed to improve understanding about the dynamics of the positron annihilation process.

171 172 Chapter 7

Conclusion

He payed and waited for his change. When the hotdog seller failed to return his money, the Zen Buddhist said “where’s my change?”

The hotdog seller said “Ahh, change should come from within”.

- anonymous via spam email

173 The conﬁguration interaction method has been applied to the study of positron interactions with atoms possessing one or two valence electrons. One of the signiﬁcant features of this research has been the treatment of the bound state and scattering systems within the same theoretical framework.

To describe the various model atoms, Hartree-Fock core electron orbitals were utilised, with semi-empirical core polarisation model potentials added and tuned to reproduce the single-electron experimental energy levels. For the group I and II atoms of the periodic table, this ansatz provided atomic structure descriptions that are comparable in accuracy to the best calculations. The level of accuracy was not so good for the group IB and IIB atoms. The resultant uncertainties in the positron binding energies associated with the choice of the core potential are probably of the order of 1-2% for the group I and II atoms, and are most likely of order 10-20% for the group IB and IIB atoms. The soft nd10 core electron shell and relativistic eﬀects are probably responsible for the reduced level of accuracy in the group IB and IIB atoms.

It is possible to obtain CI convergence for the one-electron-positron systems such as positronic copper and positron-hydrogen scattering. For the two-electron-positron systems compromises must be made. Preferentially treating the electron-positron correlations over the electron-electron correlations leads to a sensible CI scheme that halves the amount of work required. It was found that the inclusion of conﬁgurations involving the simultaneous excitations of both electrons to states with ℓ > 3 is not needed for a description of the positronic atom (or ion) that is accurate at the 1% to 2% level.

The results of this thesis are consistent with two general trends seen in previous positron-neutral atom binding research. FCSVM calculations of a model alkali atom showed the positron binding energy should be largest when the ionisation potential of the parent atom is closest to 0.250 Hartree [81]. The CIpol binding energy estimates for Ca, Sr, Zn and Cd plotted in Figure 7.1 are compatible with this principle and other positronic atom calculations. Furthermore, an eﬀective range analysis shows that the quantity Γ/√ε should be roughly constant for atoms with an ionisation potential greater than 0.25 Hartree [124]. The CIpol datapoints for Zn and Cd shown in Figure

7.2 are clearly compatible with the values of Γ/√ε derived from SVM and FCSVMpol calculations of other positronic atoms.

174 Figure 7.1: The binding energy (E in Hartree) of the e+Atom systems as a function of the model atom ionisation potential (also in Hartree). The source for each atom is: 3 e He( S ) [66], Li [67], Be [69], Na [67], Mg [69] are from the latest SVM or FCSVMpol rel estimates of Mitroy et.al.. The Cu and Ag results are from CIpol-MBPT calculations of Dzuba et.al. [78, 79]. The Ca, Sr, Zn and Cd points are from the CIpol estimates. The dashed line is from a study of positron binding to a model alkali atom [81].

Figure 7.2: The ratio Γ/√ε (in units of 109 s−1 Hartree−1/2) versus √ε (ε in Hartree) for a number of positron-atom bound states. The CIpol results of Zn and Cd are shown along with the FCSVMpol Be [69], Mg [69], Cu [68] and Ag [68] results. The unlabelled points correspond to diﬀerent values of m+ for the (m+, e−, e+) system [317].

175 7.1 Speciﬁc results

Some of the speciﬁc accomplishments of the thesis were:

The static dipole polarisabilities of the neutral atoms were computed as a by- • product of the positronic atom studies, giving estimates of 163.9, 37.7, 71.7, 162, 3 41.6, 41.3, 204 and 50.1 a0 for Li, Be, Mg, Ca, Cu, Zn, Sr, Cd respectively.

The ﬁrst CI calculation of PsH giving a realistic energy has been completed. The • largest calculation explicitly gave 94% of the binding energy, which improves to 99% upon extrapolation.

The CI calculation of positron binding to lithium achieved binding, although • pol the largest calculation remained far from converged (mostly just to see what ℓ was required for binding).

The calculation of e+Be conﬁrms the earlier rigorous prediction of binding by • the SVM calculations [35]. The estimated binding energy and annihilation rates

are consistent with the most recent FCSVMpol calculation [69].

The present CI calculations of the e+Cu ground state complement the previous • pol rel FCSVMpol and CIpol-MBPT calculations of this system [72, 78]. Variations in the details of the core-polarisation potential contribute to three diﬀerent binding energies ranging from 0.0051 to 0.0063 Hartree.

The computed binding energy for e+Mg is consistent with a previous FCSVM • pol calculation and a Quantum Monte Carlo calculation [69, 117]. The best CIpol estimate of the binding energy is 0.016151 Hartree with an overall uncertainty due to extrapolation of about 4%. ±

The existence of positron binding to the group II atoms, calcium and strontium, • has been demonstrated by explicit calculation. The best estimate of the e+Ca binding energy was about 0.016500 Hartree, while the e+Sr binding energy was 0.010050 Hartree.

The stability of CuPs has been demonstrated with an estimated binding energy • of 0.014327 Hartree.

176 A greatly improved e+Zn binding energy, 0.003731 Hartree, has been obtained. • The convergence of the CIpol calculation is vastly superior to the FCSVMpol calculation of this system [74].

Positronic cadmium has been predicted to be stable with a binding energy of • 0.006100 Hartree. This prediction is not hedged by the same uncertainties of earlier calculations [56, 58].

The CI calculations upon e+Mg, e+Zn and e+Cd provide ironclad evidence that • the earlier MBPT calculations of the same systems [58, 227, 228] signiﬁcantly overestimated the positron binding energy. This highlights the diﬃculty of describing the physics of a positron binding system using an approach based on perturbation theory.

Based on the CI results of e+Zn and e+Cd, the experimental possibility of • pol measuring their diﬀerential cross-sections to conﬁrm positron binding to these atoms was discussed. The experiment is within the capabilities of current low- energy positron beam technology [318].

The eﬀects of ’catastrophic’ Schwartz singularities were barely perceptible in • large basis CI-Kohn calculations of both model potential and positron-hydrogen scattering.

The conﬁguration basis used for the e+Cu bound state calculation was used as • part of the trial wavefunction for low-energy positron-copper scattering, giving

a threshold Zeff of about 69.34. This is the first explicit calculation of Zeff for one of these metal vapour atoms. This has demonstrated in a convincing manner that atoms that have an ionisation potential not much larger than 6.80 eV, and

that can bind a positron, do not necessarily have a very large Zeﬀ at threshold.

177 7.2 Perspectives for the future

The universal theme running through all the calculations reported here is the very slow convergence of the wavefunction with respect to the orbital angular momentum, ℓ, of the orbitals used to construct the wavefunctions. By taking due care, and using numerical procedures that are fast and accurate, it has been possible in a number of cases to perform calculations that are superior to the earlier FCSVM calculations of Ryzhikh and Mitroy. However, the CI method cannot be regarded as the solution to all positron binding problems. The mathematical basis simply does not correspond to the underlying physical representation of a system containing a positronium cluster. In the case of e+Li it is likely that the single particle basis would have to be increased to around 1000 electron and 1000 positron orbitals for the CI wavefunction to be reasonably close to converged. This should be contrasted with the FCSVM calculation upon this system which can achieve a binding energy of comparable accuracy in about 1-2 hours CPU time [241]. Positronic sodium, with the positron and electron cluster located even further from the nucleus, would represent an even more arduous calculation for the CI method, probably requiring the inclusion of orbitals with ℓ = 100. While the present calculations are usefully accurate, it would be possible to reduce the uncertainties associated with the extrapolations by performing even larger calculations. One of the main problems with doing larger CI calculations is that an orbital basis of 100 single electron and 100 single positron orbitals results in a very large number of electron-electron and electron-positron 1/r12 Slater integrals. For the two-electron-positron valence systems, these are currently stored in random access memory (RAM) and even a modest increase in the size of the calculation would result in a list of Coulomb integrals and orbital indices that took more than 1 GByte of RAM to store.

7.2.1 Improvements

Two improvements to the computer program that could be done without too much sweat are

Segmenting the list of Slater integrals would probably lead to calculations that • could use an orbital list at least 50% larger than the present series of calculations.

The most natural way to move to larger calculations is to use strategies based •

178 on parallel computation. It is certainly possible to generate parts of the Slater integral list on diﬀerent CPUs. Furthermore the next most expensive aspect of the calculation, the matrix diagonalisation is done using sparse-matrix techniques which also admit easy parallelisation.

On a more fundamental level, improving the accuracy of systems heavier than Cu and Zn would mandate the use of a relativistic framework. Development of a relativistic treatment would also mean that systems such as e+Ag as well as e+Ba could be investigated.

7.2.2 Natural springboard

This work provides the foundations for research into many e+ topics. The procedures developed during the course of this thesis could readily be extended to a number of interesting problems such as:

p- and d-wave positron scattering and annihilation of copper • There has been no Z calculations of positron scattering from hydrogenic ions • eﬀ (only low-energy phase shifts have been reported [319, 320, 321]). Calculations of these systems would be rather easy to do and could be used to probe the dynamics of positron annihilation in a strong Coulomb ﬁeld.

The issue of positron binding to non-zero angular momentum atoms (perhaps • aluminium or Be(2s2p 3P o)) is yet to be tackled. This could have implications in the understanding of resonances in positron-atom scattering.

Positron-rare gases scattering could be treated with polarisation potentials re- • lying on one-electron excitation from the core. This could provide an improved representation of the dynamical polarisation potentials beyond the PO approach of McEachran, Stauﬀer et.al. [114].

To ﬁnalise, these orthodox CI investigations have demonstrated the feasibility of using single particle orbitals centred on the nucleus to accurately describe certain classes of positronic systems with one or two valence electrons. Combining the CI and Kohn variational methods presents a uniﬁed approach, which has led to an improved understanding of positron-atom interactions.

179 180 Appendix A

Radial basis wavefunctions

The spatial part of the single particle orbitals that make up the many-particle wavefunctions are written as a product of a radial function and a spherical harmonic,

P (r) φ (r)= iℓi Y (ˆr) , (A.1) i r ℓimi

where each Piℓi (r) is normalised according to the relation

∞ 2 [Piℓi (r)] dr = 1 . (A.2) Z0 The radial functions are further expanded in a discrete set of radial basis functions which can be diﬀerent for both electron and positron orbitals.

i Piℓi (r)= dα rχα,ℓi (r) , (A.3) α X with each radial basis function normalised such that

∞ χ χ = χ (r) χ (r) r2dr = 1 . (A.4) h α| αi α,ℓi α,ℓi Z0 The r2 term here is required for the transformation within spherical co-ordinates from dr3 ∞ r2dr. The radial basis functions satisfy the boundary condition → 0 R R lim rχα,ℓ (r)=0 (A.5) r→0 i

To expediate matrix-element calculations (see Appendix B), the analytic form of the radial basis functions used is thus [rχα,ℓi (r)]. d2 The analytic form dr2 [rχα,ℓi (r)] for each radial basis function is also given here as

181 it is required for the evaluation of the one-body kinetic energy operator ( 2): ∇ ∞ ∞ 1 ∂ ∂ χ 2χ r2dr = χ r2 χ r2dr ∇ r2 ∂r ∂r Z0 Z0 ∞ 1 ∂2 = χ rχ r2dr (A.6) r ∂2r Z0 ∞ ∂2 = rχ rχ dr ∂2r Z0 this is further discussed in Appendix B. There are three types of discrete square-integrable (L2) basis types used here to represent both the real and continuous spectrum of energies ; Slater type orbitals (STO), Laguerre type orbitals (LTO) and Gaussian type orbitals (GTO). For an in- depth discussion on the analytic properties of such L2 functions see [141] and references therein. Also deﬁned here are the two s-wave scattering radial basis functions used in the CI-Kohn scattering calculations which have sinusoidal asymptotic forms for large r. These are not square-integrable, and are subject to diﬀerent normalisation conditions to that of Equation (A.4).

A.1 Slater type orbitals

The Slater type orbital (STO) is the most common type of radial basis functions employed in atomic structure calculations. It is deﬁned as:

χ (r)= N rnα−1 exp( λ r) , (A.7) α,ℓ α − α where n ℓ +1. The normalisation condition on rχ (r) (Equation (A.4)) requires: α ≥ α,ℓ

2n +1 (2λα) α Nα = . (A.8) s (2nα)!

The analytic form for the second derivative of rχα,ℓ(r) is:

d2 [rχ (r)] = N r(nα−2) exp( λ r)[(λ r)2 2λ n r + n (n 1)]. (A.9) dr2 α,ℓ α − α α − α α α α −

182 A.2 Laguerre type orbitals

The orthogonal Laguerre basis for a speciﬁc partial-wave (ℓ) can be characterised in terms of a single parameter, the exponent (λα) in the exponential. The LTO radial basis functions are deﬁned as

ℓ (2ℓ+2) χ (r)= Nαr exp( λαr)L (2λαr) , (A.10) α,ℓ − nα−ℓ−1

The function L(2ℓ+2) (2λ r) is an associated Laguerre polynomial that can be deﬁned nα−ℓ−1 α in terms of a conﬂuent hypergeometric function [322] as

(2ℓ+2) (nα + ℓ + 1)! L (2λαr)= M( (nα ℓ 1), 2ℓ + 2, 2λαr) . (A.11) nα−ℓ−1 (n ℓ 1)!(2ℓ + 2)! − − − α − − The Laguerre functions were not computed using the power series expression for the conﬂuent hypergeometric function. Rather, the recursion relation

(n + 1)Lα (x)=(2n + α + 1 x)Lα(x)+(n + α)Lα (x) , (A.12) n+1 − n n−1 was used with L2ℓ+2(x) = 1 and L2ℓ+2(x) = 2ℓ + 3 x as starting values. The 0 1 − normalisation condition (Equation (A.4)) results in:

3 (2λα) (nα ℓ 1)! Nα = − − . (A.13) s (ℓ + nα + 1)!

The analytic form for the second derivative of rχα,ℓ(r) is:

d2 [rχ (r)] = N (2λ)2 exp( λ r) dr2 α,ℓ α − α × d2L dL (2λr)ℓ+1 2(ℓ + 1)(2λr)ℓ (2λr)ℓ+1 (A.14) dr2 − − dr 1 + ℓ(ℓ + 1)(2λr)ℓ−1 (ℓ + 1)(2λr)ℓ + (2λr)ℓ+1 L . − 4 One advantage that the LTOs have over the STOs (and GTOs) is that for a given ℓ, all LTOs with a common λ are orthogonal to each other, i.e.

χ χ = δ . (A.15) h α,ℓ| β,ℓi αβ

183 A.3 Gaussian type orbitals

The Gaussian type orbitals (GTO) were implemented to enable exact comparison with the previous CI calculations of PsH by Strasburger and Chojnacki [115]. GTOs are also used extensively in atomic and molecular structure calculations [323, 144]. The GTO radial functional form is

χ (r)= N rnα−1 exp( λ r2) , (A.16) α,ℓ α − α

The normalisation condition on rχα,ℓ(r) (Equation (A.4)) results in:

3 1 (2nα+ ) (nα+ 2 ) 2 2 λα Nα = v . (A.17) u √π(2nα 1)!! u − t The analytic form for the second derivative of rχα,ℓ(r) is:

d2 [rχ (r)] = N rnα−2 exp( λ r2) (2λ )2r4 2λ (2n +1)r2+n (n 1) . (A.18) dr2 α,ℓ α − α α − α α α α− A.4 s-wave scattering basis functions

Scattering wavefunctions are also required that have the long-range form of Equation

(6.4). For scattering oﬀ neutral targets, spherical Bessel (jl) and Neumann (nl) functions are solutions of the radial Schr¨odinger equation with V = 0. Restricting ourselves to ℓ = 0 (the s-wave), Equations (6.5) have the functional forms

sin(kr) χ (k,r)= √kj (kr)= √k , (A.19) s,ℓ=0 0 kr cos(kr) χ (k,r)= √kn (kr)= √k 1 exp−βr . (A.20) c,ℓ=0 − 0 − kr The cutoﬀ-parameter β also ensures that √kn0(kr) is regular at the origin (Equation A.5). The analytic form for the second derivatives of Equation (A.19) is:

d2 [rχ (r)] = k√k sin(kr) , (A.21) dr2 s,k0 − d2 1 [rχ (r)] = (k2 β2)exp−βr k2 cos(kr) 2kβ exp−βr sin(kr) . dr2 c,k0 √ − − − k (A.22)

Also required is the zero-energy (k = 0) limit of the s-wave functions, which were d2 given by Equation (6.7), with correspondingly straightforward forms for dr2 [rχk0(r)].

184 Appendix B

Matrix elements

The atomic systems investigated in this thesis can include combinations of up to 2 valence electrons and/or a positron. Representing the diﬀerent one-body and two- body interactions between these active particles, as well with a frozen-core of inactive electrons, and all moving in the nuclear Coulomb potential, requires the evaluation of a myriad of matrix elements. The Slater, Laguerre and Gaussian radial functions all possess known analytic forms for the required matrix elements. However, in this thesis the decision was made to compute all of the matrix elements numerically. This approach allows for the use of single particle orbitals which contain a mixture of radial basis functions. Most of the calculations reported here use a mixture of STO and LTO functions (for reasons outlined in Chapter 2.3).

The radial component of each single particle orbital (r φi(r)) is evaluated, and stored on an r-space grid. All of the one- and two-body radial matrix elements were then computed using numerical quadrature.

B.1 One-body matrix elements

The matrix elements of the various scalar one-body operators (T ) describing the non- relativistic motion of a particle moving in a central potential ﬁeld of a nuclear charge Z can be considered as a linear combination of kinetic and potential interactions. For a single electron the one-body potentials are:

1 2 Z φi T1 φj = φi 1 + Vdir(r1)+ Vexc(r1)+ Vp1 (r1) φj . (B.1) h | | i h |− 2∇ − r1 | i

185 and for a single positron:

1 2 Z φi T0 φj = φi 0 + Vdir(r0)+ Vp1 (r0) φj . (B.2) h | | i h |− 2∇ r0 − | i

The direct potential (Vdir) which represents the interaction with the electron core is derived here from the HF wavefunction of the neutral atom ground state. The net [ Z/r + V (r)] static potential is attractive ( ) for valence electrons and repulsive ± − dir − (+) for positrons. The exchange potential (Vexc) is only between valence electrons and the HF core electrons. Whilst both Vdir and Vexc, are one-active body operators, they require the evaluation of two-body matrix elements, so further discussion of their computation is postponed to the following section (Sec. B.2.3). Here, the one-body polarisation potential of the core, (Vp1 ), is a semi-empirical polarisation potential (the form of which was given by Equation (2.18)). It is instructive to expand the simple one-body operators that do not involve core- valence interactions into radial integral form as

2 Z ∞ 1 d2 ℓ(ℓ + 1) Z φ ∇ φ = δ φ (r) + φ (r)r2dr h i|− 2 ± r | ji ℓi,ℓj i − 2 dr2 2r2 ± r j Z0 1 ∞ d2 = δ rφ (r) rφ (r) dr ℓi,ℓj − 2 i dr2 j Z0 (B.3) ℓ(ℓ + 1) ∞ rφ (r) rφ (r) + i j dr 2 r2 Z0 ∞ rφ (r) rφ (r) Z i j dr . ± r Z0 where we have used Equation (A.6), and the is for electrons, and + for positrons. ± − The discussion of the numerical procedures used for the radial integrations is left for Appendix Sec. B.4.2.

B.1.1 Scattering matrix element

For the Kohn scattering problem it is necessary to evaluate an additional matrix element according to the speciﬁed total energy of the scattering system (ET )

Given that the energy of the target state is Egs, and the momentum of the incident scattering particle is given by k, then

k2 E = E + . (B.5) T gs 2

186 2 For potential scattering, this reduces to ET = k /2. The only additional one-body matrix-elements required are of the form

∞ E φ φ = E rφ (r) rφ (r)dr , (B.6) − T h i| ji − T i j Z0 where for the CI-Kohn calculations, this is only included into the one-body (T ) Hamil- tonian for each orbital of the scattering particle only.

B.2 Two-body matrix elements

The matrix elements of the two-body operators (V ) in this section (and the next) use the conventions of angular momentum coupling of Brink and Satchler [324] (which is based on the Condon-Shortley phase convention).

B.2.1 Two-body integrals

The interparticle Coulomb interaction V12 = 1/r12, is handled with the usual multipole tensor expansion over spherical harmonics (Yk(ˆr)) to decompose the operator into radial and angular components:

∞ k ∞ k 1 r< 4π k k r< k k = k+1 Y (ˆr1) Y (ˆr2)= k+1 C (ˆr1) C (ˆr2) , (B.7) r12 r> 2k + 1 · r> · Xk=0 Xk=0 k where r< = min(r1,r2) and r> = max(r1,r2). The C (ˆr) are renormalised spherical harmonics. k k Since both C (ˆr1) and C (ˆr2) are tensors of the same rank, their dot product is simply a scalar. Application of the Wigner-Eckart theorem to the two-particle LS- coupled conﬁguration state functions leads to angular (rk) and radial (Rk) terms as:

1 ′ ′ φaφb; LS φcφd; L S h |r12 | i k (B.8) k k ′ ′ r< ′ ′ = φaφb; LS C (ˆr1) C (ˆr2) φcφd; L S φaφb; LS k+1 φcφd; L S . h k · k ih |r> | i Xk The two-body radial integral that has to be done is known as the Slater integral which can be written as a double integral:

k k r< R (a, b, c, d)= φa(r1)φb(r2) φc(r1)φd(r2) h |rk+1 | i > (B.9) ∞ ∞ k r< = r1 φa(r1) r2 φb(r2) r1 φc(r1) r2 φd(r2) dr1dr2 . rk+1 Z0 Z0 > 187 The details of how the Slater integrals were numerically integrated is postponed to later. To treat the reduced two-body angular integrals standard Racah algebra involving Wigner-3j and Wigner-6j symbols is used. The Wigner-Eckart theorem used here follows the deﬁnitions in Brink and Satchler [324]. The coupled tensor operator expands as

φ φ ; LS Ck(ˆr ) Ck(ˆr ) φ φ ; L′S′ h a b k 1 · 2 k c d i ℓ ℓ L ℓa+ℓc+L a b k k = δLL′ δSS′ ( 1) Lˆ φa C φc φb C φd (B.10) − ℓ ℓ k h k k ih k k i  d c  k = δLL′ δSS′ r (ℓa,ℓb,ℓc,ℓd,L ) . 

The short-hand notation ℓˆ= √2ℓ + 1 is used here. This is further simpliﬁed using

ℓi k ℓj k ℓi φi C φj =( 1) ℓˆi ℓˆj , (B.11) h k k i − 0 0 0    and thus the reduced two-body angular matrix element is given by:

ℓ k ℓ ℓ k ℓ ℓ ℓ L k ℓa+ℓc+L a c b d a b r (ℓa,ℓb,ℓc,ℓd,L)=( 1) Lˆ ℓˆa ℓˆb ℓˆc ℓˆd . −  0 0 0   0 0 0  ℓ ℓ k  d c      (B.12)  

B.2.2 Electron-electron anti-symmetrisation

Whilst Equation (B.8) can be directly used for the electron-positron system, due to anti-symmetrisation the two-body electron-electron operators require the evaluation of only a pair of two-body matrix elements, e.g.

φ φ ; LS V φ φ ; LS = N φ φ ; LS V φ φ ; LS h a b | 12| c d iA A h a b | 12| c d i (B.13) +( 1)ℓc+ℓd+L+S φ φ ; LS V φ φ ; LS . − h a b | 12| d c i The normalisation (NA) required to account for the possibility of the electrons occupying identical orbitals is 1 NA = . (B.14) (1 + δa,b)(1 + δc,d)

There are two matrix elements accountingp for the direct and exchange 1/r12 electron- electron interactions. Each of these can be evaluated using the angular and radial forms using the conﬁguration state function matrix elements (of Equation (B.8)).

188 B.2.3 Core matrix elements

Two-body matrix elements are also required to describe the direct interaction between valence particles and the core electrons, as well as the exchange interaction between the valence electrons and core electrons. Both of these are computed exactly (i.e. no local exchange approximations are made). Despite the 1/r12 being a two-body interaction, these are in eﬀect one-body matrix elements since they only involve one valence particle. For the one-body positron core matrix element, we have:

Ncore δℓ ,ℓ 1 φ V φ = i j (2L +1)(2S +1) φ φ ; L S φ φ ; L S , (B.15) h i| core| ji ˆ ˆ c c h c i c c|r | c j c ci 2(ℓi ℓj) c=1 10 X LXc,Sc where i and j are the positron orbitals, and c only runs over the closed shell electron orbitals which are coupled to the positron as LcSc. This equation can be simpliﬁed to

Ncore φ V φ = δ 2(2ℓ + 1)r0(ℓ ,ℓ ,ℓ ,ℓ ,L )R0(c,i,c,j) h i| dir| ji ℓi,ℓj c c i c j c c=1 (B.16) X∞

= δℓi,ℓj r0 φi(r0) Vcore(r0) r0 φj(r0)dr0 , Z0 k=0 where we have used r (ℓc,ℓi,ℓc,ℓj,Lc) = 1. The direct potential of every core electron has been incorporated into a radial potential, Vcore(r). This is given by

Ncore r0 ∞ r1 φc(r1) r1 φc(r1) dr1 r φ (r ) r φ (r ) V (r )= 2(2ℓ + 1) 0 + 1 c 1 1 c 1 dr core 0 c r r 1 c=1 R 0 Zr0 1 Xr0 ∞ ρcore(r1) dr1 ρ (r ) = 0 + core 1 dr . r r 1 R 0 Zr0 1 (B.17) where the core electron density, ρ (r)= Ncore 2(2ℓ + 1) rφ (r) 2, has been used. core c=1 c | c | For the valence-core electron interactionP the anti-symmetrised two-particle states (as per Equation (B.13)) can be expanded as

Ncore δℓ ,ℓ 1 φ V φ = i j (2L + 1)(2S + 1) φ φ ; L S φ φ ; L S h i| core| ji ˆ ˆ c c h c i c c|r | c j c ciA 2(ℓi ℓj) c=1 12 X LXc,Sc Ncore δℓ ,ℓ = i j (2L + 1)(2S + 1) rk(ℓ ,ℓ ,ℓ ,ℓ ,L ) Rk(c,i,c,j) ˆ ˆ c c c i c j c 2(ℓi ℓj) c=1 X LXc,Sc h +( 1)ℓc+ℓj +Lc+Sc rk(ℓ ,ℓ ,ℓ ,ℓ ,L ) Rk(c,i,j,c) . − c i j c c (B.18)i

189 These contributions are computed separately. Firstly the direct potential is similar to the positron case, i.e.

Ncore δℓi,ℓj 0 0 φi Vdir φj = (2Lc + 1)(2Sc + 1)r (ℓc,ℓi,ℓc,ℓj,Lc) R (c,i,c,j) h | | i 2(ℓˆ ℓˆ ) i j c=1 Lc,Sc ∞X X

= δℓi,ℓj r2 φi(r2) Vcore(r2) r2 φj(r2)dr2 , Z0 (B.19) where exactly the same core potential, Vcore(r), was used for both the positron and electron core-direct evaluations. The core-exchange contribution is slightly more time- consuming to evaluate since k can be non-zero, i.e.

Ncore δℓ ,ℓ φ V φ = i j (2L + 1)(2S + 1)( 1)ℓc+ℓj +Lc+Sc h i| exc| ji ˆ ˆ c c − 2(ℓi ℓj) c=1 X LXc,Sc k k r (ℓc,ℓi,ℓj,ℓc,Lc)R (c,i,j,c) (B.20) × 2 Ncore ℓc k ℓi = δ (2ℓ + 1) Rk(c,i,j,c) . − ℓi,ℓj c   c=1 0 0 0 X   where there is no further simpliﬁcation of Rk(c,i,j,c).

B.2.4 Vp2 matrix elements

The two-body polarisation potential (Vp2 ) is deﬁned as per [133, 132]

αd Vp2 (r1, r2)= 3 3 g(r1)g(r2)(r1 r2) , (B.21) ± ±r1r2 · where the is for the electron-electron, and + for the electron-positron interactions. ± − Separating radial and angular components, i.e. r r = r r C1(ˆr ) C1(ˆr ), the radial 1 · 2 1 2 1 · 2 matrix element for the two-particle conﬁguration state functions are calculated as:

1. If these are satisﬁed then the radial component of Vp2 can be added to the equivalent valence electron-electron or electron-positron (k = 1) Slater integral R1(a, b, c, d) of Equation (B.8).

190 B.3 Two-electron-positron system

The conﬁguration state functions for the two-electron-positron problem consist of an antisymmetric two-electron state function coupled to an intermediate LS and then coupled with the positron to give total quantum states of LT ST . For the two-electron-positron type conﬁguration state functions there are three interparticle two-body operators to evaluate: the repulsive electron-electron term 1/r12, and the two attractive electron-positron terms 1/r10 and 1/r20 (where the 0 denotes the positron co-ordinate):

T T 1 1 1 ′ ′ T T φaφb[LS]φp; L S φcφd[L S ]φq; L S . (B.23) h | r12 − r10 − r20 | i For each two-body operator, the third particle is purely a spectator. As an example, using Equation (B.7), the electron-positron 1/r10 interaction is

T T 1 ′ ′ T T φaφb[LS]φp; L S φcφd[L S ]φq; L S h |r10 | i = φ φ [LS]φ ; LT ST Ck Ck φ φ [L′S′]φ ; LT ST h a b p k 1 · 0k c d q i (B.24) Xk k T T r< ′ ′ T T φaφb[LS]φp; L S k+1 φcφd[L S ]φq; L S , ×h |r> | i where r< = min(r1,r0) and r> = max(r1,r0). Since the radial integrals of Equation k k+1 (B.24) are independent of LS, the r radial integrals corresponding to the 1/r10,

1/r20 and 1/r12 interactions are given by 1 for Rk(a,p,c,q) (B.25) r10 → 1 for Rk(b, p, d, q) (B.26) r20 → 1 for Rk(a, b, c, d) . (B.27) r12 →

B.3.1 Reduced two-body angular integrals

The two-electron-positron angular integrals are more complicated than the electron- electron or electron-positron cases due to the intermediate LS coupling of the two electrons. Using standard Racah algebra, the angular integral can be expanded over the intermediate coupling as

φ φ [LS]φ ; LT ST Ck Ck φ φ [L′S′]φ ; LT ST h a b p k 1 · 0k c d q i T (B.28) T ′ L ℓ L ℓq+L +L p k ′ ′ k =( 1) φaφbLS C1 φcφdL S φp C φq , − ℓ L′ k  h k k ih k k i  q 

  191 which is further composed of

′ ℓa L ℓb k ′ ′ ′ ℓa+ℓb+L +k ˆ ˆ′ k φaφbLS C1 φcφdL S = δS,S δb,d( 1) L L φa C φc . h k k i − L′ ℓ k  h k k i  c  (B.29)   Hence expanding the single-particle angular integrals using Equation (B.11) gives:

T T k k ′ ′ T T ℓ +ℓ +ℓ +LT φ φ [LS]φ ; L S C C φ φ [L S ]φ ; L S = δ ′ δ ( 1) b p q Lˆ Lˆ′ ℓˆ ℓˆ ℓˆ ℓˆ h a b p k 1 · 0k c d q i S,S b,d − a c p q T L ℓp L ℓa L ℓb ℓa k ℓc ℓp k ℓq . × ℓ L′ k  L′ ℓ k   0 0 0   0 0 0   q   c          (B.30)

Similarly, the 1/r20 angular component is given by

T T k k ′ ′ T T ℓ +ℓ +ℓ +LT φ φ [LS]φ ; L S C C φ φ [L S ]φ ; L S = δ ′ δ ( 1) a p q Lˆ Lˆ′ ℓˆ ℓˆ ℓˆ ℓˆ h a b p k 2 · 0k c d q i S,S a,c − b d p q T L ℓp L ℓb L ℓa ℓb k ℓd ℓp k ℓq . × ℓ L′ k  L′ ℓ k   0 0 0   0 0 0   q   d          (B.31)

The 1/r12 term involves no recoupling

φ φ [LS]φ ; LT ST Ck Ck φ φ [L′S′]φ ; LT ST h a b p k 1 · 2k c d q i = δ φ φ ; LS Ck Ck φ φ ; L′S′ (B.32) p,qh a b k 1 · 2k c d i k = δp,qδL,L′ δS,S′ r (ℓa,ℓb,ℓc,ℓd,L) .

B.4 Numerical integrations

The necessity to include single particle orbitals of relatively high angular momentum meant that some technical difficulties that do not normally occur in ordinary atomic structure calculations were identified and overcome. Difficulties arose in the evaluation of the electron-positron and electron-electron interaction matrix elements.

B.4.1 Grid transform

A transformed grid is used to ensure that there is a high density of grid points near the nucleus (where the potentials and wavefunctions vary the most), and a lower density of grid points at large r. The transform used here from the U-grid to the r-grid was performed according to: αU 2 r(U)= , (B.33) 1+ βU

192 such that U [0,U ]). The Jacobian for the transformation is thus ∈ max dr 2αU αβU 2 (U)= . (B.34) dU 1+ βU − (1 + βU)2

For all computations used in this thesis, the transform variables were simply set to α = 1 and β = 0. The beneﬁts of diﬀerent grid transformations (e.g. choosing β = 0) 6 was not investigated to any great extent since the transformation was adequate. A Gaussian integration based approach was adopted for evaluation of all of the one- and two-body radial matrix elements. Two grids are required, an outer grid used for the one-body integrals, and an inner grid which is also required for the two-body k k+1 r running integrals.

B.4.2 Two-body radial matrix elements (running integrations)

The two-body radial integral that has to be done is the Slater integral of Equation (B.9) which can also be written as:

∞ k k Z (r2) k k R (a, b, c, d)= r2 φb(r2) r2 φd(r2) + r2 Y (r2) dr2 , (B.35) rk+1 Z0 2 where r2 k k Z (r2)= r1 φa(r1) r1 φc(r1) r1 dr1 , (B.36) Z0 and ∞ k r1 φa(r1) r1 φc(r1) Y (r2)= dr1 . (B.37) rk+1 Zr2 1 It is necessary to ﬁrst form the so-called running integrals, Zk(r) and Y k(r), so that Equation (B.35) can then be evaluated. Thus the integrations used to construct Zk(r) and Y k(r) are also termed the inner integrations. This requires two diﬀerent grids for the evaluation of the two-body matrix elements, a U grid and a V grid, both of which use the same transformation to/from r of Equation (B.33). The outer U grid integration over the r [0,R ] range was done with a composite ∈ max 16-point Gauss rule. Typically the radial component of each orbital was evaluated and stored in memory at 512 radial grid points (i.e. 32 segments consisting of 16 points each) to be used in the all of the one-body integrations (e.g. Equation (B.3)), as well as the ”outer” two-body radial integration of Equation (B.9).

193 The inner integrations of Equations (B.36) and (B.37) require the propagation of k k k k Z (r) and Y (r) between outer grid points, i.e. Z (ri) to Z (ri+1), and similarly k k propagated inwards from Y (ri+1) to Y (ri). To evaluate between these outer grid points, a second ”inner” 6-point Gaussian grid (V ) was required. Shown in Figure B.1 and Table B.1 is a 6-point Gaussian inner-grid (V ) in-between each of the outer-grid (U) points. At each of the (typically 512) U grid points, the radial component of each orbital (φi(r)) is evaluated and stored in memory. For each V point, the orbital is continuously generated as required by the inner integrations (i.e. they are not permanently stored in memory since the number of V points would be 6 513). Fortunately, there is not a signiﬁcant overhead in generating the orbitals × at each V as needed for the inner integrations.

Figure B.1: Schematic diagram relating the outer (U) and inner (V ) radial grid points used for the Gaussian evaluation of the running integrals. U(0)=0 U(1) U(2) U(3) U(4)

Umax

V(1)...V(6) V(7)...V(12) V(13)...V(18) V(19)...V(24)

This algorithm was capable of generating Slater integrals with high values of k to close to machine accuracy, and in relatively quick time. For example, the k = 48 Slater integral with φ (r)= φ (r)= φ (r)= φ (r)= r24 exp( 4r) was done as a test. a b c d − Analytic evaluation gave R48 = 0.0129915361229981 while the Gaussian quadrature gave R48 = 0.0129915361229982. For the most extreme calculations in this thesis, i.e. e+Li, machine precision was achieved with a 1024-point outer grid extending out to Umax = 10 (rmax = 100 a0), whilst the inner was 6-point based. For the CI-Kohn scattering calculations, due to the slow radial decay of the continuum orbitals (especially at zero-energy), the grid had to be extended further outwards typically to Umax = 25 (rmax = 625 a0), with 2048 outer points required.

194 Table B.1: Table of example r values for two outer U(Iout) (16-point) Gaussian grid meshes over U = 10 (i.e. U [0, 10] r [0, 100]), as well as the ﬁrst two inner max ∈ → ∈ V (Iin) (6-point) Gaussian grids as seen in Figure B.1. The index of each grid is given by I.

Iout U(Iout) Iin V (Iin) r 0 0.00000 0.0000000 1 0.00089 0.0000008 2 0.00449 0.0000201 3 0.01009 0.0001018 4 0.01641 0.0002693 5 0.02201 0.0004844 6 0.02560 0.0006555 1 0.02650 0.0007021 7 0.03028 0.0009170 8 0.04548 0.0020685 9 0.06916 0.0047831 10 0.09590 0.0091969 11 0.11958 0.0142992 12 0.13478 0.0181653 2 0.13856 0.0191996 3 0.33592 0.1128436 4 0.61149 0.3739188 5 0.95531 0.9126160 6 1.35496 1.8359113 7 1.79599 3.2255841 8 2.26247 5.1187647 9 2.73753 7.4940775 10 3.20401 10.265673 11 3.64504 13.286331 12 4.04469 16.359522 13 4.38851 19.259029 14 4.66408 21.753624 15 4.86144 23.633575 16 4.97350 24.735726 17 5.02650 25.265679 ...... 32 9.97350 99.470749

195 Do not numerically calculate running integrations as a subtraction

k The running inner integral Y (r2) (of Equation (B.37)) is sometimes written as a subtraction, i.e.

∞ r2 k r1 φa(r1) r1 φc(r1) r1 φa(r1) r1 φc(r1) Y (r2)= dr1 dr1 . (B.38) rk+1 − rk+1 Z0 1 Z0 1 k in order to have this integration performed in the outward direction along with Z (r2) (of Equation (B.36)). However, this subtraction led to catastrophic errors for the large values of k encountered in this thesis. These occured because slightly diﬀerent

∞ xφa(x) xφc(x) numerical procedures are used to evaluate the definite integral, 0 xk+1 dx and r xφa(x) xφc(x) the running integral 0 xk+1 dx [125]. This means theR value of the running integral for large r canR be slightly different from the definite integral and that the limiting value of Y k(r) as r goes to is slightly different from zero. This can lead to ∞ large errors in the product rkY k(r) when both r and k are large since a small error in Y k(r) is multiplied by the rk factor. The form of Equation (B.37) for the evaluation of Y k(r) is definitely preferred.

Other running integration methods

These two-body running integrals were initially performed with a 5-point Bodes ”outer” integration grid (which requires grid points to be equally spaced in U-space). The 3- point Simpsons integration method was used to propagate Zk(r) and Y k(r) between each ”outer” point. The Simpsons approach only requires one extra ”inner” point, which is not stored in memory, to be evaluated midway between ri and ri+1. The outer grid U-space integrations required approximately 6000 radial points. However, when using the Bodes/Simpson approach, catastrophic numerical instabilities in the two-body running integrations began to appear once single particle orbitals with ℓ > 6 were included (the e+Li calculations required ℓ = 30). Thus, the decision was made to switch to a composite Gaussian based grid. A third ”outer” integration method was also brieﬂy investigated to integrate Equa- tion (B.35). Firstly, the Zk(r) and Y k(r) were calculated by either the ”inner” Gaus- sian or ”inner” Simpsons integrations. Then cubic splines were constructed along the U-grid from the kernel of Equation (B.35) using the inner Zk(r) and Y k(r) integrations, which were then analytically integrated. Generally the accuracy of this approach was found to be worse than that of the Bodes ”outer” integration method.

196 Appendix C

Eigenvalue problem and solving

This section describes the construction of the Hamiltonian Matrix for the single particle, the two-electron, the electron-positron and the two-electron-positron problems. For the CI bound state the Hamiltonian is diagonalised using one of two methods. For the present implementation of the CI-Kohn method, the Kohn linear equations were ﬁrst constructed from the Hamiltonian Matrix, and then solved.

C.1 Hamiltonian construction

In general, the Hamiltonian matrix construction requires the calculation of

H = Φ ; LS H Φ ; LS = Φ ; LS T + V Φ ; LS , (C.1) IJ h I | | J i h I | | J i for each conﬁguration, I,J = 1,...,NCI

For the one-body problem NCI is simply the number of single particle orbitals, and

H = φ H φ = φ T φ . (C.2) IJ h i| | ji h i| | ji where the one-body matrix elements of Equation (B.1) are used for an electron, and Equation (B.2) for a positron.

C.1.1 Two-body Hamiltonian matrix elements

Each two-body Hamiltonian matrix element requires one-body terms to be combined with the two-body matrix elements for each conﬁguration I,J = 1,...,NCI . For the

197 electron-positron system these are given by

H = φ φ ; LS H φ φ ; LS IJ h a b | | c d iA = φ φ ; LS V φ φ ; LS + N δ δ φ T φ + δ δ φ T φ h a b | 12| c d iA A b,d ℓa,ℓc h a| 1| ci a,c ℓb,ℓd h b| 1| di +( 1)ℓc+ℓd+L+S δ δ φ T φ + δ δ φ T φ , − b,c ℓa,ℓd h a| 1| di a,d ℓb,ℓc h b| 1| ci (C.4)

where NA is deﬁned as per Equation (B.14). These electron-electron matrix elements are further expanded using Equation (B.13) and V = 1/r V (r , r ). 12 12 − p2 1 2

C.1.2 Two-electron-positron Hamiltonian matrix elements

The Hamiltonian matrix elements for the LT = 0 two-electron positron system are

H = φ φ [LS] φ ; LT ST T + V + V + V φ φ [L′S′] φ ; LT ST IJ h a b A p | 12 10 20| c d A q i = δ δ δ ′ δ φ T φ a,c b,d L,L ℓp,ℓq h p| 0| qi + δ δ ′ δ ′ N δ δ φ T φ + δ δ φ T φ p,q L,L S,S A b,d ℓa,ℓc h a| 1| ci a,c ℓb,ℓd h b| 1| di ′ ′ +( 1)ℓc+ℓd+L +S δ δ φ T φ + δ δ φ T φ − b,c ℓa,ℓd h a| 1| di a,d ℓb,ℓc h b| 1| ci dir ℓ +ℓ +L′+S′ exc + δ δ ′ δ ′ N V +( 1) c d V p,q L,L S,S A 12 − 12 ′ ′ + N δ V dir +( 1)ℓc+ℓd+L +S δ V exc A b,d 10 − b,c 10 ′ ′ + N δ V dir +( 1)ℓc+ℓd+L +S δ V exc , A a,c 20 − a,d 20 (C.5)

198 k where assuming that each Vp2 (ri, rj) contribution has been absorbed into the R terms

dir k k V12 = r (ℓa,ℓb,ℓc,ℓd,L) R (a, b, c, d) Xk exc k k V12 = r (ℓa,ℓb,ℓd,ℓc,L) R (a, b, d, c) Xk V dir = φ φ [LS]φ ; LT ST Ck Ck φ φ [L′S′]φ ; LT ST Rk(a,p,c,q) 10 h a b p k 1 · 0k c d q i Xk (C.6) V exc = φ φ [LS]φ ; LT ST Ck Ck φ φ [L′S′]φ ; LT ST Rk(a,p,d,q) 10 h a b p k 1 · 0k d c q i Xk V dir = φ φ [LS]φ ; LT ST Ck Ck φ φ [L′S′]φ ; LT ST Rk(b, p, d, q) 20 h a b p k 2 · 0k c d q i Xk V exc = φ φ [LS]φ ; LT ST Ck Ck φ φ [L′S′]φ ; LT ST Rk(b, p, c, q) . 20 h a b p k 2 · 0k d c q i Xk C.2 Hamiltonian diagonalisation

Two off-the-shelf programs based on different algorithms were used to diagonalise the Hamiltonian Matrix; an EISPACK routine, and an iterative sparse-matrix algorithm. The EISPACK full-matrix diagonalisation routine was primarily used for the single particle and two active particle systems. The EISPACK routine returns the complete set of eigenvalues and eigenvectors. This is particularly important for the dipole polarisability sum rule of Equation (2.30), which requires wavefunctions of all final states (even the virtual states with positive eigenvalues). Since the EISPACK routine is stored completely in RAM, the electron-electron or electron-positron Slater integrals could first be calculated and stored in linear arrays on disk, and the Hamiltonian matrix constructed by sequentially reading the linear array. When this was done, the dimension of the full-matrix for the two-particle systems was limited to around NCI = 7500 on a workstation with 1GByte of RAM.

C.2.1 Davidson algorithm

The secular equations that arise with 3 active particles typically have dimensions exceeding 10,000 with a signiﬁcant number of the Hamiltonian elements = 0 (i.e. the matrix is sparse). Therefore the two-electron-positron matrix diagonalisation was performed with an iterative algorithm. The program of Stathopoulos and Froese Fischer [218], which utilises the Davidson algorithm [217] was used (and modiﬁed) to perform the diagonalisations.

199 There was one compiler/machine dependent bug found in their code (Note that the ftnchek program [325] also produced other benign warnings on this code). In ﬁle dvdson.f [218]:

CALL DSPEVX(’Vectors also’,’In a range’,’Upper triangular’, ACPZ0507 KPASS,TEMPS,-1.,-1.,1,NUME,0.D0, ACPZ0508 NFOUND,EIGVAL,SVEC,KPASS,SCRA1,ISCRA2,INCV,INFO) ACPZ0509

To avoid (compiler dependent) memory crossovers between 4-Byte reals and 8-Byte double-precision reals, this should be corrected as

KPASS,TEMPS,-1.0D0,-1.0D0,1,NUME,0.0D0, ACPZ0508

The Davidson algorithm is optimised to return the lowest (or highest) energy eigen- state/s, which for the two-electron-positron problem is exactly what is desired. The largest CI calculations performed had dimensionalities of over 100,000 (see Table 4.3), with the sparse-matrix either stored in memory, or for the larger calculations on disk. Due to the method of the Hamiltonian construction, the lists of both the electron- electron and electron-positron matrix elements were stored in memory. This meant that the number of single particle orbitals included in the two-electron-positron problem was much less than was possible for the electron-positron systems. Since the trends of the Laguerre type orbital exponents were not known at the start of these investigations, most of the basis optimisations involved repeating exactly the same calculation with only small variations of the Laguerre exponents to observe the eﬀect on the energy. The Davidson algorithm is an iterative algorithm, and it was possible to start diagonalisation with the previously obtained eigenvector (assuming the matrices are of the same dimension). This reduces the number of iterations required to achieve convergence in the next calculation, and hence the time taken. The extensive basis optimisations for each two-electron positronic atom were only possible due to this restarting ability of the Stathopoulos and Froese Fischer [218] program. The Davidson algorithm was chosen to terminate once the energy had converged to better than 10−10 Hartree. To achieve this level of convergence, typically 240 iterations were required to diagonalise the PsH Hamiltonian while about 800 iterations were required to diagonalise the e+Be Hamiltonian. However, the convergence of e+Cd was anomalously slow requiring up to 30,000 iterations before convergence was reached. Table C.1 shows the number of iterations required for the Davidson routine to reach

200 satisfactory convergence. The reason for the slow convergence of the e+Cd system remains unknown. Of related interest is the percentage of non-zero matrix elements in the matrix as the calculation size increases.

Table C.1: Number of iterations required for the e+Cd diagonalisations to reach convergence for each Lmax run. Also given is the number of configurations and the fullness of the matrix (as a percentage of the matrix that was non-zero). A denotes † that the diagonalisation for that Lmax was not performed from start to finish (i.e. the diagonalisation was stopped and later restarted due to the length of time of the calculations, with the presented Niter being only that of the final run).

Lmax NCI Fullness(%) Niter 0 1440 29.20 30053 1 5010 16.47 31994 2 12014 11.71 29758 3 19457 9.89 19626 † 4 26833 8.76 26868 5 34161 7.93 19875 6 41569 7.24 17945 7 49121 6.69 10305 † 8 56673 6.24 634 † 9 64225 5.87 8364 † 10 71777 5.56 3815 †

C.3 Kohn scattering linear equations

The form of the scattering trial wavefunction for both the potential scattering and electron-positron scattering problems are given by the general form of Equation (6.2),

NSR Ψ ; LS = α Φ ; LS + α Φ ; LS + c Φ ; LS . (C.7) | t i 0| s i 1| c i n| n i n=1 X The ﬁrst two terms Φ ; LS and Φ ; LS contain the non-square integrable (long- | s i | c i range sinusoidal) continuum functions. The set of N Φ ; LS functions describe the SR | i i shorter-ranged correlations of the system. The general Kohn normalisation factors α0 and α1 were given by Equations (6.15). The construction of the Kohn variational scattering linear equations assumes that the Hamiltonian matrix has already been constructed. This is not required, and means

201 that two large dimensioned matrices are stored in memory, but this made for easier conversion of the CI program to handle a Kohn variational calculation.

For the scattering problem, we want the Kohn functional (αv of Equation (6.16)) to be stationary with respect to variations of the linear parameters in the trial wavefunction, i.e. ∂α ∂ v =0=1 2 Ψ ; LS H E Ψ ; LS ∂α − ∂α h t | − | t i t t (C.8) ∂αv ∂ =0=2 Ψt; LS H E Ψt; LS . ∂cn ∂cn h | − | i Using the trial correction term, I = Ψ ; LS H E Ψ ; LS , we have tt h t | − | t i ∂Itt 1 ∂Itt = and = 0 n = 1,...,NSR . (C.9) ∂αt 2 ∂cn ∀ Rewriting the trial wavefunction of Equation (C.7), to collect the linear parameters

NSR Ψ ; LS = cos τ α sin τ Ψ + sin τ + α cos τ Ψ + c Ψ | t i − t | si t | ci n| ni n=1 X NSR = cos τ Ψ + sin τ Ψ + α sin τ Ψ + cos τ Ψ + c Ψ | si | ci t − | si | ci n| ni n=1 X NSR = Ψ + α Ψ + c Ψ . | 0i t| αi n| ni n=1 X (C.10) and thus the trial correction term can be expanded as

I = Ψ H E Ψ = Ψ H E Ψ + α2 Ψ H E Ψ tt h t| − | ti h 0| − | 0i t h α| − | αi + α Ψ H E Ψ + Ψ H E Ψ t h α| − | 0i h 0| − | αi NSR + α c Ψ H E Ψ + Ψ H E Ψ (C.11) t n h α| − | ni h n| − | αi n=1 X NSR + c c Ψ H E Ψ . i jh i| − | ji i,jX=1 Using the variational conditions (Equation (C.8)) we obtain the following (N = NSR)+1 simultaneous linear equations to solve:

∂Itt 2I11 I21 + I12 ... I1α + Iα1 c1 (I10 + I01) ∂c1 → − ∂Itt  I12 + I21 2I22 ... I2α + Iα2   c2   (I20 + I02)  ∂c2 → − .  . .. .   .   .  .  . . .   .  =  .        ∂Itt       I1N + IN1 I2N + IN2 ... INα + IαN  cN   (IN0 + I0N )  ∂cN →      −  ∂Itt      1   I1α + Iα1 I2α + Iα2 ... 2Iαα  αt   (Iα0 + I0α) ∂αt →      2 −       (C.12)

202 where the short-hand notation I = Ψ H E Ψ has been used. For example, ab h a| − | bi 2 Iα0 = Ψα H E Ψ0 = sin τ cos τ Ψs H E Ψs sin τ Ψs H E Ψc h | − | i − h | − | i− h | − | i (C.13) + cos2 τ Ψ H E Ψ + cos τ sin τ Ψ H E Ψ . h c| − | si h c| − | ci

Linear equations solution

One of the bigger (avoidable) mistakes that was made during this research was to use the Numerical Recipes GAUSSJ subroutine [326] to solve the linear equations. Whilst it works, it was very slow, with the solution of a set of linear equations of dimension 3000 taking well over a day to solve. When solving any linear equation of dimension greater than 2 use LAPACK; calculations will take a couple of hours, not days!

203 Appendix D

Expectation values

To elucidate structural information from the wavefunctions various expectation values were computed.

D.1 Electron and positron density matrices

Firstly, density matrices are used to simplify and speed up evaluation of some of the expectation values [327]. The single particle density matrix is deﬁned with annihilation and creation operators as

ρ = Ψ; LS a† a Ψ; LS . (D.1) ij h | i j | i where i,j are the single particle orbital indices. This can then be used in association with one-body operators to speed up their evaluation when many conﬁgurations are involved, i.e.

Norb Ψ; LS T Ψ; LS = ρ φ T φ . (D.2) h | | i ijh i| | ji Xi,j For the one-body wavefunction, Equation (D.1) contracts the wavefunction coeﬃ- cients as

ρij = cicj , (D.3) i.e. for the single electron (or single positron) cases, there is a one-to-one mapping between these since the number of orbitals is equal to the number of conﬁgurations. For the electron-positron system, Equation D.1 for the electron density matrix

204 expands over the NCI electron-positron conﬁgurations as

NCI ρe = c c φ φ ; LS a† a φ φ ; LS ij I J h aI pI | i j| aJ pJ i I,J=1 X (D.4) NCI

= cI cJ δpI ,pJ δaI ,i δaJ ,j . I,JX=1 where the indexing runs over the electron orbitals i,j N , and I,J denote the ∈ e conﬁgurations indexing. Similarly, the positron density matrix elements are given by

NCI ρp = c c δ δ δ i,j N . (D.5) ij I J aI ,aJ pI ,i pJ ,j ∀ ∈ p I,JX=1

The electron density matrix of an electron-electron system is a bit more complicated due to anti-symmetry of the wavefunction,

NCI e ρij = cI cJ 1+ δaI ,bI 1+ δaJ ,bJ (δbI ,bJ δaI ,i δaJ ,j + δaI ,aJ δbI ,i δbJ ,j) I,J=1 X p p (D.6) ℓ +ℓ +L+S +( 1) aI bI (δ δ δ + δ δ δ ) i,j N . − aI ,bJ bI ,i aJ ,j bI ,aJ aI ,i bJ ,j ∀ ∈ e for the two-electron-positron system with conﬁguration state functions of the form, Φ ; LT ST = φ φ [L S ]φ ; LT ST , the density matrices are | I i | aI bI I I pI i

NCI e ρij = cI cJ δpI ,pJ 1+ δaI ,bI 1+ δaJ ,bJ (δbI ,bJ δaI ,i δaJ ,j + δaI ,aJ δbI ,i δbJ ,j) I,J=1 X p p ℓ +ℓ +L +S +( 1) aJ bJ J J (δ δ δ + δ δ δ ) i,j N − aI ,bJ bI ,i aJ ,j bI ,aJ aI ,i bJ ,j ∀ ∈ e NCI ρp = c c δ δ δ δ (δ δ ) i,j N . ij I J aI ,aJ bI ,bJ LI ,LJ SI ,SJ pI ,i pJ ,j ∀ ∈ p I,JX=1 (D.7)

D.2 r and r h ei h pi

Given the density matrices for the electron and/or positron, the radial expectation values for all of the diﬀerent systems can be evaluated simply as a summation over the

205 N = Ne electron orbitals, or N = Np positron orbitals:

N Ψ(r) 2 = ρ rφ (r) rφ (r) (D.8) | | i,j i j i,jX=1 N Ψ; LS Ψ; LS = ρ rφ (r) rφ (r) dr (D.9) h | i i,j i j i,jX=1 Z N Ψ; LS r Ψ; LS = ρ rφ (r) rφ (r) r dr (D.10) h | | i i,j i j i,jX=1 Z N Ψ; LS r2 Ψ; LS = ρ rφ (r) rφ (r) r2 dr . (D.11) h | | i i,j i j i,jX=1 Z This only requires a slight modiﬁcation for the two-electron and two-electron-positron cases, in that, excepting Ψ(r) 2, the electron expectation values must be divided by | | Tr(ρe) = 2 since there are two electrons involved.

D.3 r2 and cos(θ ) h epi h ep i The evaluation of Ψ; LS r2 Ψ; LS relies on the expansion h | ep| i

r2 = r r 2 = r 2 + r 2 2r r ep | e − p| | e| | p| − e · p = r 2 + r 2 2 r r cos(θ ) (D.12) | e| | p| − | e|| p| ep = r2 + r2 2r r C1(ˆr ) C1(ˆr ) . e p − e p e · p Since only C1(ˆr) terms are involved, it is relatively eﬃcient to evaluate these expectation values explicitly by expanding over one-electron-positron conﬁgurations as

NCI Ψ; LS r2 Ψ; LS = c c Ψ r2 Ψ + Ψ r2 Ψ h | ep | i I J h I | e | J i h I | p | J i XI,J (D.13) 2 Ψ r r C1(ˆr ) C1(ˆr ) Ψ , − h I | 1 0 1 · 0 | J i and which for the two-electron-positron case is

NCI Ψ; LT ST r2 Ψ; LT ST = c c Ψ r2 Ψ + Ψ r2 Ψ h | ep | i I J h I | e | J i h I | p | J i XI,J (D.14) 2 Ψ r r C1(ˆr ) C1(ˆr ) Ψ − h I | 1 0 1 · 0 | J i 2 Ψ r r C1(ˆr ) C1(ˆr ) Ψ . − h I | 2 0 2 · 0 | J i

206 D.4 2-γ annihilation rate and Zeﬀ

The calculation of the spin-averaged 2-γ annihilation rate and Zeﬀ (Equations (3.3) and (6.21)) requires the calculation of the expectation value of the Dirac δep function (also called the ’collision probability’ [115]). This is evaluated here using a multipole expansion over spherical harmonics to separate the radial and angular components [324], i.e.

δ = Ψ; LS δ(r r ) Ψ; LS ep h | e − p | i 1 (D.15) = Ψ; LS δ( r r ) (2k + 1)Ck(ˆr ) Ck(ˆr ) Ψ; LS . 4πr2 | e|−| p| e · p e k X

Note that in order to convert this volume integral from atomic to SI units, one also 3 has to divide by a0. Since spin-averaged 2-γ annihilation rates are required, the spin projector of Equa- tion (3.3) can be removed given that the equation is divided by 4. For the one-electron- positron system, the spin-averaged 2-γ valence annihilation rate (in SI units of s−1) is thus

N 1 CI Γ = πcr2 c c rφ (r) rφ (r) φ (r) φ (r) dr v 0 4π I J aI aJ pI pJ I,JX=1 Z kmax (2k + 1) φ φ ; LS Ck(ˆr ) Ck(ˆr ) φ φ ; LS . × h aI pI k 1 · 0 k aJ pJ i k=Xkmin (D.16)

The summation over k, where k runs over every second integer from k = max( ℓ min | aI − ℓ , ℓ ℓ ) to k = min( ℓ +ℓ , ℓ +ℓ ) is evaluated using Equation (B.10). aJ | | pI − pJ | max | aI aJ | | pI pJ | Although Equation (D.16) contains a multipole expansion, Γv can still be evaluated quicker than the similar two-body Coulomb matrix elements (Equation (B.7)) since the radial integrals are independent of k and also there is no running integral to evaluate. Note that the constant in front of Equation (D.16) is also sometimes given 2 with r0 = α a0 (in atomic units).

For a single positron annihilating with a closed shell of core electrons, δep is eﬀec- tively a one-body operator (see Appendix B.2.3), and the core annihilation rate can be given by

N Np core 2(2ℓ + 1) Γ = πcr2 c ρp rφ (r) rφ (r) φ (r) φ (r) dr , (D.17) c 0 4π ij c c i j c=1 X Xi,j Z 207 where the one-body positron density matrix has been utilised for each positron orbital i,j 1,...,N . ∈ p For the two-electron-positron system, the valence annihilation rate is complicated by a passive electron in the direct and exchange δ10 and δ20 contributions to the overall

Γv. These details are omitted as they are organised in a similar way to the V10 and

V20 two-body matrix elements (see Equations (C.5) and (C.6)).

For evaluating the core and valence electron contributions to Zeﬀ (i.e. Equation 2 (6.21)), Equations (D.17) and (D.16) are recycled, although without the (πcr0) factor.

Ncore Np c p Zeﬀ = Nk>0 2(2ℓc + 1) ρij rφc(r) rφc(r) φi(r) φj(r) dr , (D.18) c=1 X Xi,j Z and

NCI v Zeﬀ = Nk>0 cI cJ rφaI (r) rφaJ (r) φpI (r) φpJ (r) dr I,J=1 Z X (D.19) kmax (2k + 1) φ φ ; LS Ck(ˆr ) Ck(ˆr ) φ φ ; LS . × h aI pI k 1 · 0 k aJ pJ i k=Xkmin

Note that the sum over conﬁgurations includes the Φs, Φc and Φi conﬁgurations of

Equation (6.2). The co-eﬃcients α0 and α1 of these conﬁgurations are evaluated from

αt (using Equation (6.15)) resultant from each τ Kohn variational procedure. In general, two normalisation conditions are applied. The ﬁrst is that the wavefunction is normalised by an extra factor of 4π [260], which cancels the 1/4π found in Equations (D.17) and (D.16). Secondly, for k > 0, the scattering trial wavefunction requires Zeﬀ to have an additional factor [251] of

1 Nk>0 = 2 2 , (D.20) k(α0 + α1)

208 Appendix E

Validations

There were many calculations performed to validate that the algorithms and numerics were accurate for both the CI and CI-Kohn calculations. The results of some of these are included here primarily for reference purposes.

E.1 Bound state validations

Large scale diagonalisations to illustrate the systematic convergence of the single particle orbitals using a pure Laguerre basis were performed (and which also demonstrate the accuracy and stability of the single particle program routines). These are similar to those performed by Bray [150]. Beginning with one LTO per partial-wave (for ℓ 3) λ = λ = λ = λ = 1.0, the ≤ s p d f number of LTOs per partial-wave is increased from 1 to 30. The energy of the lowest states, the oscillator strength of the primary transition, and the ground state dipole polarisability are all reported in Table E.1. Table E.3 shows the convergence of r h ei during the same sequence of calculations. Then for the second series of hydrogen calculations the exponents were reduced to λs = λp = λd = λf = 0.5, the results of which are shown in Tables E.1 and E.4. The convergence of outlying states has been enhanced with close to machine precision reached with the inclusion of 30 LTOs per partial-wave. It should be emphasised that in these results, there are no signs of any numerical instabilities, even with 30 LTOs included per partial wave.

209 Table E.1: Energies of ground and excited states of the hydrogen atom resulting from diagonalising pure LTO λs = λp = λd = λf = 1.0, as the number of LTOs per partial-wave is increased from 1 to 30. The oscillator strength, f corresponds to the 1s 2p transition. The res → 3 dipole polarisability (αd) of the hydrogen ground state is given in units of a0.

1 2 3 4 5 10 20 30 1s -0.50000000 -0.50000000 -0.50000000 -0.50000000 -0.50000000 -0.50000000 -0.50000000 -0.50000000 2s - 0.16666667 -0.077350269 -0.11358979 -0.12216704 -0.12499912 -0.12500000 -0.12500000 2p 1.19262−16 -0.10000000 -0.11890889 -0.12353660 -0.12467436 -0.12499994 -0.12500000 -0.12500000

210 3s - - 1.07735030 0.21401445 0.049701246 -0.052824355 -0.055555254 -0.055555556 3p - 0.50000000 0.11657616 0.017105086 -0.021475118 -0.054669358 -0.055555494 -0.055555556 3d 0.16666667 0.019868451 -0.023068564 -0.040327982 -0.048210643 -0.055410289 -0.055555550 -0.055555556 4s - - - 2.39957530 0.60925927 0.006352175 -0.030849764 -0.031249257 4p - - 1.20233270 0.40146294 0.18343111 -0.011223893 -0.031072126 -0.031249747 4d - 0.59917917 0.22296984 0.10191039 0.046544277 -0.022881154 -0.031197163 -0.031249946 4f 0.25000000 0.087531728 0.031116937 0.004702155 -0.009481280 -0.029100777 -0.031242194 -0.031249994

fres 1.00000000 0.66666667 0.50667181 0.44560258 0.42457436 0.41619993 0.41619672 0.41619672

αd 4.00000000 4.50000000 4.50000000 4.50000000 4.50000000 4.50000000 4.50000000 4.50000000 Table E.2: Same as Table E.1 except the LTOs have λs = λp = λd = λf = 0.5.

1 2 3 4 5 10 20 30 1s -0.375 -0.45833333 -0.48935678 -0.49784989 -0.4996271 -0.49999998 -0.50000000 -0.50000000 2s - -0.12500000 -0.12500000 -0.12500000 -0.12500000 -0.12500000 -0.12500000 -0.12500000 2p -0.12500000 -0.12500000 -0.12500000 -0.12500000 -0.12500000 -0.12500000 -0.12500000 -0.12500000 3s - - -0.010643223 -0.049947802 -0.054663089 -0.055555545 -0.055555556 -0.055555556

211 3p - -0.025000000 -0.051376262 -0.054906075 -0.055468235 -0.055555555 -0.055555556 -0.055555556 3d -0.041666667 -0.053571429 -0.055271583 -0.024199799 -0.055552056 -0.055555556 -0.055555556 -0.055555556 4s - - - 0.17279769 0.009909918 -0.031068302 -0.031250000 -0.031250000 4p - - 0.10137626 0.001215621 -0.020319691 -0.031210784 -0.031250000 -0.031250000 4d - 0.041666667 -0.010829181 -0.055521221 -0.028740447 -0.031244884 -0.031250000 -0.031250000 4f 4.56720−17 -0.022299323 -0.028240730 -0.030233907 -0.030926080 -0.031249699 -0.031250000 -0.031250000

fres 2.00000000 0.22222222 0.42778315 0.41662386 0.41592098 0.41619658 0.41619672 0.41619672

αd 32.00000000 9.69230770 5.76306230 4.79933520 4.56864930 4.50000620 4.50000000 4.50000000 Table E.3: Average electron radius of the ground and excited states of the hydrogen atom resulting from diagonalising pure LTO λs = λp =

λd = λf = 1.0, as the number of LTOs per partial-wave is increased from 1 to 30.

1 2 3 4 5 10 20 30 1s 1.5000000 1.5000000 1.5000000 1.5000000 1.5000000 1.5000000 1.5000000 1.5000000 2s - 2.5000000 3.8660254 4.8615860 5.4771841 5.9990239 6.0000000 6.0000000 2p 2.5000000 3.5000000 4.2339742 4.6720286 4.8834345 4.9999098 5.0000000 5.0000000 212 3s - - 2.1339746 3.7645131 5.5171826 11.853287 13.498667 13.500000 3p - 2.5000000 3.9883624 5.5592023 7.0685066 11.690219 12.499678 12.500000 3d 3.5000000 4.9041944 6.1346622 7.1989842 8.0991382 10.288620 10.499964 10.500000 4s - - - 1.8739009 3.2980167 11.649537 22.765921 23.992015 4p - - 2.2776634 3.6753631 5.2282447 13.054360 22.304439 22.996908 4d - 3.0958056 4.5755579 6.0441297 7.4865370 14.336743 20.726251 20.999226 4f 4.5000000 6.1676481 7.6560265 9.0195649 10.278023 15.113745 17.945322 17.999906 Table E.4: Same as Table E.3 except the LTOs have λs = λp = λd = λf = 0.5.

1 2 3 4 5 10 20 30 1s 3.0000000 2.0000000 1.6277187 1.5200751 1.5011103 1.4999980 1.5000000 1.5000000 2s - 6.0000000 6.0000000 6.0000000 6.0000000 6.0000000 6.0000000 6.0000000 2p 5.0000000 5.0000000 5.0000000 5.0000000 5.0000000 5.0000000 5.0000000 5.0000000

213 3s - - 7.3722813 10.509410 12.413684 13.499899 13.500000 13.500000 3p - 7.0000000 9.9639610 11.629025 12.285792 12.499994 12.500000 12.500000 3d 7.0000000 9.0000000 10.044487 14.124645 10.484582 10.500000 10.500000 10.500000 4s - - - 5.9705147 10.125724 23.178990 23.999999 24.000000 4p - - 6.0360390 10.163948 14.198907 22.741281 23.000000 23.000000 4d - 7.0000000 10.684380 10.402805 16.857470 20.951401 21.000000 21.000000 4f 9.0000000 11.941451 14.222002 15.878309 16.954358 17.995996 18.000000 18.000000 E.1.1 Two-electron αd and f validations

Two two-electron systems were used to validate the oscillator strength and dipole polarisability sum-rule numerics. Firstly, the helium system of Nesbet [328, 329], and secondly, the two-electron calcium calculations of Mitroy [127]. The He calculations by Nesbet [328, 329] used two diﬀerent STO basis sets; a 7s7p3d 1 e 1 o basis [328] which has NCI ( S ) = 62 and NCI ( P )=70, and a compact 5s5p3d2f basis 1 e 1 o [329] which has NCI ( S ) = 39, NCI ( P )=46. These calculations are shown in Table E.5, and also demonstrate that even small two-electron calculations can give reliable values of αd and fres for two-electron systems.

Table E.5: Energy, oscillator strengths and static dipole polarisability of the He atom for two STO basis sets as used by Nesbet [328, 329]. Energies are in units of Hartree. E and f is the energy diﬀerence and oscillator strength of the 1s2 1Se 1s2p diff res → 1 o P transition. (fres in dimensionless units). The ground state dipole polarisability, 3 αd, is in units of a0.

2 1 e 1 o Model E(1s S ) E(1s2p P ) Ediff fres αd 7s7p3d basis [328] Present CI -2.90201945 -2.12343145 0.77858799 0.27453012 1.3849942 CI [328] 0.778588 0.274530 Exp. [180] 0.2762 5s5p3d2f [329] Present CI -2.90291958 -2.10365674 0.79926283 0.46103818 1.3839269 CI [329] -2.90292 1.383926 Calc. [330] 1.3831921793

Table E.6 shows neutral calcium model atom calculations that used exactly the same HF core orbitals, core-polarisation potentials, and valence STO orbitals as that 1 e of the CI calculations of Mitroy [127]. These calculations had Lmax = 4, NCI ( S ) = 1 o 135 and NCI ( P ) = 186. Since the CI program of [127] used an analytical evaluation of all STO matrix elements, this provided a useful check of the present numerical approach for core-direct, core-exchange and one- and two-body core-polarisation matrix elements. Repeating these calculations also revealed an approximately 1% error in the di-electronic polarisation (Vp2 ) treatment of [127]. Upon correction, the results of both programs were equal to all signiﬁcant ﬁgures [241]. Note that the fres calculations

214 reported by Mitroy [127] used experimental energy diﬀerences, hence the discrepancy of fres in Table E.6.

Table E.6: Energy, oscillator strengths and static dipole polarisability of a model calcium atom of Mitroy [127], which used a purely STO basis. The CIpol calculations included semi-empirical core-polarisation potentials. All energies are in units of Hartree 2+ relative to the energy of the Ca core. Ediff and fres is the energy diﬀerence and oscillator strength of the 4s2 1Se 4s4p 1P o transition (f in dimensionless units). → res 3 αd is in units of a0. Note that the CIpol αd includes both core and valence contributions. (Refer to Table 2.9 for extensive Ca results).

2 1 e 1 o Model E(4s S ) E(4s4p P ) Ediff fres αd Present CI -0.63320652 -0.52871556 0.10449096 1.89431 178.42 CI [127] -0.633206 -0.528716 0.10449

Present CIpol -0.65953816 -0.55318899 0.10634917 1.73260 158.69

CIpol [127] -0.659701 -0.553569 0.106132 1.820

E.1.2 Gaussian type orbital calculations of PsH

In 1995 Strasburger and Chojnacki [115] performed a series of CI calculations of PsH. The GTO exponents for their largest 10s4p2d GTO basis are given in Table E.7, and which were kindly provided by Professor Chojnacki [220]. The results of the two GTO calculations can be seen in Table E.8. The 10s4p2d basis has Lmax=2, N = 16, NCI = 938. The 8s4p basis (with Lmax=1, N = 12, NCI = 496) is a cutdown version of the 10s4p2d shown in Table E.7, precluding the two

ℓ = 0,n0 = 3 GTOs as well as the two ℓ = 2,n2 = 3 GTOs. Also given in Table E.8 is the collision probability of Equation (D.15), as well as the annihilation rate (for comparison with the LTO CI calculations of Table 4.1). Two different diagonalisations have been used to perform the calculations; the EISPACK full-matrix routine, and the Davidson sparse-matrix routine. Since the Davidson iterative algorithm was chosen to terminate once the energy has converged to better than 10−10 Hartree, this implies that the wavefunction will generally be converged to 1 part in 10−5, which is reflected in the expectation values. For the purposes of this thesis, this is insignificant when compared to the errors in both the extrapolation process and the specification of the core-polarisation potentials.

215 Table E.7: Gaussian exponents [220] of the 10s4p2d PsH basis as used by Strasburger and Chojnacki [115]. The exponents of the electron and positron orbitals were identical.

n0 λ0 n1 λ1 n2 λ2 1 159.9089 2 1.381601 3 0.3235496 1 23.99469 2 0.4249751 3 0.1080245 1 5.462829 2 0.1669883 1 1.543182 2 0.06399340 1 0.4930697 1 0.1669883 1 0.06399340 1 0.02532292 3 0.3235496 3 0.1080245

Table E.8: Comparison of PsH calculations using two GTO basis sets (deﬁned in Table E.7 and the text). The present CI calculations reproduced the CI calculations of Strasburger and Chojnacki [115, 220], validating a majority of the CI program used in this thesis. The 3-body energy (in Hartree) of the PsH system is denoted by E(PsH). The mean electron-nucleus distance r , the mean positron-nucleus distance r , and h ei h pi the mean electron-positron distance r2 are given in units of a , a and a2. The δ h epi 0 0 0 ep ’collision probability’ is given in the δ(r r ) column (in atomic units). The spin- h e − p i averaged 2γ annihilation rate (Γ) is given in 109 sec−1. Two diﬀerent diagonalisations have been used to perform the calculations; the EISPACK full-matrix routine, and the Davidson sparse-matrix routine.

Calc E(PsH) r r r2 δ(r r ) Γ h ei h pi h epi h e − p i 8s4p basis Full -0.7465397281 2.078892 3.489987 17.16302 0.01527938 0.7711465 Sparse -0.7465397281 2.078896 3.490008 17.16323 0.01527915 0.7711348 [115, 220] -0.74653972 0.0153 10s4p2d basis Full -0.7636938604 2.104622 3.425097 15.64820 0.01968115 0.9933024 Sparse -0.7636938604 2.104624 3.425105 15.64827 0.01968106 0.9932977 [115, 220] -0.76369386 0.0197

216 Being able to repeat the PsH calculations of Strasburger and Chojnacki was in- valuable. This independently veriﬁed that most aspects of the present CI program were operating correctly. Whenever modiﬁcations were made that broke the program, this was the fallback calculation. Note that the recently published GTO-based PsH (LT = 0) calculations of Tachikawa [331] were also successfully reproduced.

E.2 Kohn variational scattering validations

Validations and investigations were performed on a number of systems:

k = 0 and 0.5 a−1 elastic e−-static-H potential • 0

0 k 1 a−1 elastic s-wave electron scattering on He at three levels of approxi- • ≤ ≤ 0 mation; static-direct, static-direct-exchange, and static-direct-exchange with polarisation potential

k = 0 and k = 0.5 a−1 elastic s-wave positron scattering on Ne with static-direct • 0 and also static-direct-polarisation

k = 0.5 a−1 elastic s-wave positron scattering from various hydrogen target states • 0

E.2.1 Potential scattering

Initially a program was written to speciﬁcally calculate the scattering length of s- wave electron scattering from a static hydrogen potential using the Kohn variational method as per an example in Burke and Joachain [283]. The static hydrogen potential for electron scattering is deﬁned as:

1 V (r)= 1 1+ exp( 2r) . (E.1) − r −

At zero-energy the trial radial wavefunction with a trial scattering length, At, was

NSR rχ (r)= r + A 1 exp( βr) + c rχ (r) . (E.2) t − t − − i i i=1 X where the basis functions used are of the Slater type (Equation (A.7)). The results for two calculations, NSR = 2 and NSR = 6, with varying cutoﬀs β are shown in Table E.9. All integrations were performed numerically using the CI program (with no Gram-Schmidt orthogonalising required).

217 The Kohn program was extended to solve the non-zero energy s-wave system, the results of which were validated by electron scattering oﬀ the static-hydrogen potential, with NSR = 6 and β = 2.0. The results of which are shown in Table E.10.

− Table E.9: Scattering lengths (in units of a0) of s-wave e scattering from a static- hydrogen potential as a function of the cutoﬀ parameter β. Two sets of calculations are reported, corresponding to 2 and 6 STO short range basis functions.

β A (NSR = 2) A (NSR = 6) 0.5 -6.499417124 -9.339477522 0.6 -7.524429598 -9.420403587 0.7 -8.276823734 -9.441141111 0.8 -8.776184457 -9.446003648 0.9 -9.082159965 -9.446990827 1.0 -9.258120941 -9.447137855 1.1 -9.354053425 -9.447143332 1.2 -9.403813661 -9.447140871 1.4 -9.439652316 -9.447150877 1.6 -9.446370838 -9.447160641 1.8 -9.447088391 -9.447164861 2.0 -9.447034806 -9.447166219 2.2 -9.446965987 -9.447166584 2.4 -9.446819379 -9.447166668 2.6 -9.446361268 -9.447166684 2.8 -9.445128412 -9.447166685 3.0 -9.442333734 -9.447166670 3.5 -9.420674101 -9.447165626 exact [283] -9.44716

This was further implemented with a ﬂexible 0 τ π/2 where τ = 0 denotes the ≤ ≤ vanilla Kohn method, τ = π/2 is the inverse Kohn Normalisation and the square well scattering calculations of Brownstein and McKinley [298] repeated. This was done to examine the eﬀect of the Schwartz [290, 291] anomalous singularities, the results of which are discussed in Section 6.3.1.

218 Table E.10: s-wave phase shifts of electron scattering oﬀ the static hydrogen potential as a function of projectile momentum k.

k δ(NSR = 6) δexact [332] 0.05 0.434402 0.43440 0.10 0.722220 0.72222 0.15 0.883818 0.88382 0.20 0.972521 0.97252 0.30 1.045552 1.04555 0.40 1.057497 1.05750 0.50 1.044660 1.04466 0.60 1.021032 1.02103 0.70 0.992902 0.99290

E.2.2 Elastic scattering from static atoms

Calculations of electron-helium and positron-neon scattering were performed to gain some indications of the LTO basis set requirements to achieve s-wave convergence. Three levels of static approximation were included in the electron-helium calculations, ﬁrstly only static-direct terms were included, secondly when exchange was 3 added, and thirdly when a semi-empirical polarisation potential (αd = 1.383 a0 and

ρ = 2.4 a0) was added. The basis set included 28 s-wave LTOs in addition to the two continuum orbitals. The results as a function of energy are shown in Table E.11 for two Kohn normalisations τ = 0 and π/4. These results are also compared against previous calculations of Sinfailam and Nesbet [333, 329], as well as calculations by Mitroy [241]. −1 Also apparent was the convergence of the very-low energy k = 0.001 a0 calculations towards the zero-energy results. Calculations of positron-neon scattering were also undertaken to examine the La- guerre convergence in the s-wave for positron scattering oﬀ a polarised system. Tables −1 E.12 and E.13 show the zero-energy and k = 0.5 a0 calculations as the number of s-wave LTOs with λ = 1.0 was increased. The polarisation potential used αd = 2.67 3 a0 and ρ = 1.51 a0. These calculations can be compared against the equivalent model potential scattering calculations of Mitroy and Ivanov [124]. The smallest Np = 7 calculation highlights the impact of a small basis set on the Kohn and inverse Kohn results. These eﬀects become less perceptible as the basis was systematically increased.

219 Table E.11: s-wave phase shifts (and scattering lengths) of elastic electron scattering −1 oﬀ a static-helium atom as a function of momentum k (in units of a0 ). The calculations were performed for two Kohn normalisations: τ = 0 and π/2. Included are three levels of approximation; static-direct, static-direct-exchange and static-direct-exchange with polarisation potentials.

S-D S-D-X S-D-Xpol

k Model δ0 (or A) δ0 (or A) δ0 (or A)

0 τ0 5.2506767 1.4824567 1.1767927 other 1.183511[329] other 1.1784 [334]

0.001 τ0 -0.00525064 -0.00148246 -0.00117696

τπ/2 -0.00525064 -0.00148246 -0.00117696

0.01 τ0 -0.05246909 -0.01482426 -0.01186900

τπ/2 -0.05246909 -0.01482426 -0.01186900

0.1 τ0 -0.49187265 -0.14793961 -0.12681268

τπ/2 -0.49187265 -0.14793961 -0.12681268 other -0.1480 [333] -0.128210 [329]

0.3 τ0 -1.09036113 -0.43669279 -0.40064856

τπ/2 -1.09036113 -0.43669279 -0.40064856 other -1.0904 [241] -0.4379 [333] -0.402098 [329]

0.5 τ0 -1.37943991 -0.70584284 -0.66714286

τπ/2 -1.37943991 -0.70584284 -0.66714286 other -1.3795 [241] -0.70589 [241] -0.668355 [329] other -0.7091 [333]

1 τ0 1.40584724 -1.25147354 -1.21978579

τπ/2 1.40584723 -1.25147354 -1.21978581 other 1.4058 [241] -1.25139 [241] -1.205643 [329] other -1.2516 [333]

220 Table E.12: Zero-energy s-wave scattering length (in a0) and threshold Zeﬀ of positron scattering from a static neon atom with polarisation potential. Np denotes the number of s-wave orbitals, including two continuum orbitals and the remainder a purely LTO basis with λ = 1.0. These are compared with the identical model potential calculations of Mitroy and Ivanov [124] (with the G enhancement factor removed from Zeﬀ). Also shown for comparison are the polarised orbital calculations of McEachran et.al. [239].

c Np A Zeﬀ Itt 7 -0.63662827 2.706519 -0.21272 12 -0.63847323 2.706840 -0.12170 17 -0.63880895 2.707307 -0.08312 22 -0.63890086 2.707296 -0.06261 27 -0.63893401 2.707307 -0.05005 32 -0.63894863 2.707309 -0.04160 37 -0.63895597 2.707310 -0.03556 [124] -0.640 2.707 [239] -0.61412 7.1400

Table E.13: The s-wave phase shifts and Zeﬀ of positron scattering from a static −1 neon atom with a polarisation potential at a projectile momentum of k = 0.5 a0 .

Calculations are shown for the two Kohn normalisations τ = 0 and π/2. Np denotes the number of s-wave positron orbitals as in Table E.12. These are compared with the identical model potential calculations of Mitroy and Ivanov [124] (the s-wave contribution to Zeﬀ [241] has the G enhancement factor removed), as well as the polarised orbital calculations of McEachran et.al. [239].

c c Np δ0(τ0) δ0(τπ/2) Zeﬀ(τ0) Zeﬀ(τπ/2) Itt(τ0) Itt(τπ/2) 7 -0.12087750 -0.12057254 1.5832593 1.4733688 -8.105−3 0.3151 12 -0.12078615 -0.12078570 1.5858767 1.5843045 -7.522−4 4.938−3 17 -0.12077830 -0.12078183 1.5804814 1.5870568 -1.434−2 -2.060−3 22 -0.12078053 -0.12078054 1.5862506 1.5863801 -4.231−5 -2.986−4 27 -0.12078028 -0.12078030 1.5862460 1.5873313 -5.871−5 -2.800−3 32 -0.12078022 -0.12078021 1.5862443 1.5859220 -6.353−5 9.133−4 37 -0.12078021 -0.12078020 1.5862909 1.5863378 5.911−5 -1.826−4 [124, 241] -0.120776 1.5863 [239] -0.12107 5.9249

221 E.2.3 s-wave e+ elastic scattering from h the positron scattering oﬀ an active electron were validated using three standard close- coupling target states, ﬁrstly the static only H(1s), secondly the H(1s2s2p) and thirdly the H(1s2s2p) approximations of the forms given in

rχ = r exp( r) (E.3) 1s − rχ = r exp( 0.5r) √3r2 exp( 0.5r) (E.4) 2s − − − rχ = r2 exp( 0.5r) (E.5) 2p − 8 2 rχ = (2r2 + r3)exp( r) (E.6) 2p 129 − rχ 0.43133105 r2 exp( r)+ 0.59062447 r3 exp( r) . 2p ≡ − − The H(1s) target state was a particularly useful test since it could be run in both static mode (as a static core orbital), or explicitly as an active electron. The H(1s2s2p) target states only includes 66% of the hydrogen dipole polarisability, whilst the H(1s2s2p) basis includes a pseudostate which describes the exact polarisability of the hydrogen ground state (previously used in [335, 310, 269]). Example calculations of these diﬀerent target states can be seen in Table E.14 at −1 an incident positron momentum of k = 0.5 a0 , with the results of two Kohn normalisations compared to other calculations [310, 311, 241] for validation purposes.

The Zeﬀ for each target atom was consistent with those previously determined by a momentum-space T -matrix calculation [121]. These calculations further demonstrate the convergence of the Kohn and inverse Kohn normalisations if enough positron scattering particle states are included, especially when target polarisation is included. Note that the equivalent zero-energy model target state calculations are given in Table 6.3, −1 whilst the full LTO basis calculations at k = 0.5 a0 are given in Table 6.2.

222 −1 + Table E.14: Results of CI-Kohn calculations for k = 0.5 a0 s-wave elastic e -H scattering for diﬀerent hydrogen target state models. The total number of electron and positron orbitals are denoted by Ne and Np. The NCI column is the number of con- ﬁgurations included in each calculation. Note that the 1s2s2p and 1s2s2p calculations included 33s and 30p positron LTOs.

Model Ne Np NCI δ0 Zeﬀ Itt 1s CC(1,0) model −9 present τ0 1 35 35 -0.26353476 0.36779118 1.471 −8 present τπ/2 1 35 35 -0.26353476 0.36779117 4.861 CC [241] -0.2634 T -matrix [121] 0.3678 1s2s2p CC(3,0) model −9 present τ0 3 65 98 -0.19936219 0.44651176 -2.154 −8 present τπ/2 3 65 98 -0.19936219 0.44651178 -9.508 CC [241] -0.1993 CC [311] -0.1986 CC [310] -0.1997 T -matrix [121] 0.4464 1s2s2p model −8 present τ0 3 65 98 -0.117202912 0.59002210 3.395 −6 present τπ/2 3 65 98 -0.117202912 0.59002174 2.571 CC [311] -0.1175

223 Appendix F

Laguerre exponents of positronic systems

As Laguerre type orbitals can in principle form a complete representation of the continuum of target states they have proved successful in scattering calculations (see for example the works of Bray and Stelbovics [145, 265, 266]). Previous scattering calculations on hydrogen-like systems have restricted the use of LTOs to representing the target states. For example, Bray [150] used a decreasing λ series for electron scattering on sodium with the lowest-energy scattering calculations using λs,p,d,f = 1.3, 0.9, 0.75, 0.75. (since highly excited electron states are located further from the nucleus). For positron scattering on atomic-hydrogen λ = 1.5 for ℓ 15 has been used [113]. Mitroy [336] also used a Laguerre basis to represent H ≤ target states with λ = 1.0 for ℓ 3 during positron-hydrogen scattering. Note that all ≤ of these calculations used hydrogenic Laguerres, which have an exponent of ( 0.5λ r). − α The λs given above have been divided by two for comparison against the present LTO exponents. Since LTOs have not previously been used to describe the structures found in positronic atoms, there was no initial guide for choosing the LTO exponents λℓ for each partial-wave. The LTO exponents shown in Table F.1 are the results of the extensive energy optimisations for the various positronic systems, and which are also plotted in Figure F.1. The exponents for some systems have not been completely optimised due to the size and/or time of the calculations, most notably for e+Cd as well as the higher ℓ of e+Li.

224 Table F.1: Optimised Laguerre exponents (λp) of the single particle positron orbitals for the various positronic systems (in order of λp at L = 10) The electron exponents + (λe) are also included where they diﬀered from the positron exponents. Note for e Li the increment of both λ and λ between L = 19 30 was simply chosen to increment e p → up from L = 18 at a constant 0.08 per L.

L PsH e+Be e+Zn e+Cd e+Cu e+Mg CuPs e+Ca e+Li e+Sr

λe (where diﬀerent to λp) 0 - 1.22 1.30 1.60 1.30 1.20 1.00 1.40 1.35 1.50 1 - 1.45 1.48 1.40 1.40 1.50 1.56 1.36 - 1.65 2-1.60------

λp 0 1.52 0.45 0.48 0.47 0.65 0.58 1.02 0.92 1.15 1.10 1 1.75 0.90 1.10 0.85 1.25 1.20 1.60 1.23 1.40 1.45 2 2.02 1.52 1.63 1.50 1.70 1.56 1.56 1.36 1.50 1.60 3 2.12 1.75 1.80 1.75 1.85 1.69 1.61 1.53 1.55 1.60 4 2.25 2.00 1.96 1.98 1.95 1.85 1.73 1.69 1.67 1.63 5 2.40 2.20 2.16 2.14 2.10 2.02 1.88 1.81 1.77 1.74 6 2.60 2.40 2.36 2.32 2.27 2.17 2.03 1.93 1.89 1.85 7 2.85 2.65 2.53 2.48 2.47 2.32 2.18 2.03 1.99 1.99 8 3.05 2.80 2.68 2.62 2.63 2.47 2.33 2.15 2.11 2.08 9 3.25 2.95 2.85 2.78 2.78 2.61 2.49 2.28 2.23 2.21 10 - 3.10 3.02 2.94 2.93 2.76 2.67 2.42 2.35 2.34 11 - - - - 3.05 - - - 2.45 - 12 - - - - 3.15 - - - 2.55 - 13 - - - - 3.25 - - - 2.63 - 14 - - - - 3.33 - - - 2.72 - 15 - - - - 3.41 - - - 2.82 - 16 - - - - 3.48 - - - 2.91 - 17 - - - - 3.55 - - - 3.01 - 18 - - - - 3.62 - - - 3.10 -

225 Figure F.1: Optimised Laguerre exponents (λp) of the single particle positron orbitals as found in Table F.1. The Legend is arranged in order of λp at Lmax = 10.

PsH e+Be + λ 2 e Zn e+Cd e+Cu e+Mg 1 CuPs e+Ca e+Li e+Sr 0 0 2 4 6 8 10 12 14 16 18 L

Some trends emerged that are worth mentioning:

Generally λ increased linearly over the range of ℓ investigated here, which is • p related to the location of where positron-electron clustering occurs. Each LTO has an average radius of r n/λ, which implies that to concentrate the radial h i≈ basis at the same r , that λ increases linearly with ℓ. h i

The values of λ were generally the largest for the systems with the Ps clustering • closest to the nucleus (as occurs in PsH, and e+Be), so that the single particle radial basis is concentrated closer to the nucleus. The converse is found in the systems which have their Ps clusters at large distances from the nucleus (e.g. e+Li and e+Sr).

For all of the systems, (except PsH), further optimisations such that λ > λ • e p resulted in improvements of the system energies. This was performed for the ℓ = 0 and 1 partial-waves (and for e+Be, the ℓ = 2 exponents). This is a result of the electron density being closer to the nucleus than the positron, and also

that the λe trends are aﬀected by the use of the mixed Slater/Laguerre radial basis.

The higher n and ℓ LTOs describe electron-positron correlations. This could be •

226 seen when an extra positron (or electron) LTO basis function with large ℓ or n was added to the calculation, would only have a small eﬀect on the energy. However, once the equivalent electron (positron) basis function was added there was a signiﬁcantly larger combined contribution.

As L was increased, the trend was for the λs of each partial-wave to reoptimise • max as a decrease, i.e. the single particle basis is able to describe the positronium cluster at further distances from the nucleus.

227 Appendix G

Extrapolation details

The L limit for all of the positronic atoms was estimated using a simple ex- max → ∞ trapolation technique outlined in Section 3.3. Of particular interest was the behaviour of the exponents p for the expectation values of each positronic system as Lmax was increased.

The exponents for the energy of the system pE, and for the valence annihilation rate pΓ for each Lmax, are collated in Tables G.1 and G.2. These can be contrasted with the asymptotic values pE = 4 and pΓ = 2 established by Gribakin and Ludlow [214] using second order perturbation theory. Both of these tables highlight the slow convergence of the CI calculations towards this asymptotic region. The present CI calculations also suggests that by the time that the calculations have reached this asymptotic region, for all intents and purposes the expectation values of the positronic system would be converged.

228 Table G.1: Energy extrapolation exponent pE for various positronic systems calculated using the L 2,L 1,L series of CI calculations. max − max − max + + + + + + + + Lmax PsH e Be e Zn e Cd e Cu e Mg CuPs e Ca e Li e Sr 2 1.54 5.22 3.27 2.55 -0.27 2.89 0.87 1.12 -0.33 0.72 3 1.88 0.69 1.28 0.66 -0.86 0.10 1.19 0.54 0.34 0.71 4 2.17 0.52 1.58 1.49 -0.33 0.78 1.76 1.00 0.97 1.01 5 2.41 1.12 1.00 0.96 0.39 1.39 2.00 1.49 1.34 1.46 6 2.61 1.69 1.57 1.54 1.01 1.82 2.19 1.82 1.57 1.78 7 2.79 2.12 2.01 1.98 1.47 2.15 2.38 2.08 1.75 2.03 8 2.95 2.44 2.35 2.31 1.82 2.41 2.55 2.31 1.89 2.26 9 3.09 2.70 2.62 2.57 2.09 2.63 2.71 2.50 2.00 2.45 10 2.91 2.83 2.79 2.31 2.82 2.86 2.67 2.10 2.62 11 2.50 2.19 12 2.66 2.28 13 2.80 2.36 14 2.93 2.43 15 3.04 2.51 16 3.14 2.59 17 3.23 2.66 18 3.31 2.74 19 2.82 20 2.89 21 2.96 22 3.03 23 3.10 24 3.17 25 3.23 26 3.29 27 3.34 28 3.40 29 3.45 30 3.50

229 Table G.2: Valence annihilation rate extrapolation exponent pΓ for various positronic systems calculated using the L 2,L 1,L series of CI calculations. max − max − max + + + + + + + + Lmax PsH e Be e Zn e Cd e Cu e Mg CuPs e Ca e Li e Sr 2 0.46 -2.86 -2.36 -2.39 -2.16 -3.15 -0.27 -2.39 -1.65 -2.03 3 0.88 -2.09 -2.11 -2.33 -2.39 -1.55 -0.12 -0.98 -0.40 -0.96 4 1.16 -0.82 -0.87 -0.83 -1.35 -0.03 0.51 0.00 0.23 -0.10 5 1.32 0.27 0.19 0.23 -0.35 0.62 0.78 0.49 0.53 0.39 6 1.44 0.94 0.80 0.75 0.30 0.95 0.99 0.77 0.72 0.68 7 1.53 1.32 1.20 1.15 0.70 1.19 1.15 1.01 0.86 0.90 8 1.66 1.60 1.51 1.36 0.98 1.39 1.29 1.14 0.97 1.09 9 1.76 1.79 1.63 1.56 1.20 1.55 1.39 1.29 1.06 1.22 10 2.00 1.93 1.77 1.37 1.68 1.48 1.45 1.13 1.35 11 1.52 1.20 12 1.65 1.26 13 1.76 1.32 14 1.86 1.36 15 1.95 1.40 16 2.03 1.44 17 2.11 1.48 18 2.18 1.52 19 1.57 20 1.62 21 1.66 22 1.70 23 1.75 24 1.79 25 1.84 26 1.88 27 1.92 28 1.96 29 2.00 30 2.04

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