“The measure of greatness in a scientific idea is the extent to which it stimulates thought and opens up new lines of research.” − P. A. M. Dirac

U N I V E R SITÉ LIBRE DE BRUXELLES

Relativistic study of electron correlation effects

on polarizabilities, two-photon decay rates, and

electronic isotope-shift factors in atoms and ions

Ab initio and semi-empirical approaches

Th`ese pr´esent´ee en vue de l’obtention du titre de

Docteur en Sciences de l’Ing´enieur et Technologie

Livio Filippin

Promoteur

Prof. Michel Godefroid

Co-Promoteur

Dr. J¨orgen Ekman (Malm¨o H¨ogskola)

Service

Chimie quantique et Photophysique

Ann´ee acad´emique

Relativistic study of electron correlation effects

on polarizabilities, two-photon decay rates, and

electronic isotope-shift factors in atoms and ions

Ab initio and semi-empirical approaches

Th`ese pr´esent´ee en vue de l’obtention du titre de

Docteur en Sciences de l’Ing´enieur et Technologie

Livio Filippin

Promoteur

Prof. Michel Godefroid

Co-Promoteur

Dr. J¨orgen Ekman (Malm¨o H¨ogskola)

Service

Chimie quantique et Photophysique

Acknowledgments

Aucun travail ne peut être réalisé sans le bon-vouloir, l’énergie et la collaboration de

plusieurs personnes. A l’heure où je rédige ces lignes concluant un chapitre de quatre

années, je voudrais remercier toutes les personnes ayant contribué à ce projet.

J’adresserai tout d’abord mes remerciements à Michel Godefroid, mon promoteur de

thèse mais également de mémoire, pour son enseignement, son engouement et son

enthou-siasme tout au long de ces années de recherche. Dès mon arrivée en tant que mémorant,

il a su m’accueillir parfaitement au sein du service de Chimie quantique et Photophysique

dont il est co-directeur. Je voudrais le remercier pour son soutien et son aide, sa présence

et ses conseils, mais surtout pour la confiance qu’il m’a accordée dans mon travail au

quo-tidien. En effet, en tant que doctorant je me suis senti libre de mener à bien les projets qui

m’intéressaient, et ce de manière autonome mais toujours sous sa bienveillance. En plus

de féliciter l’excellent promoteur et professeur, je voudrais aussi remercier la personne aux

grandes qualités humaines. Toujours optimiste et faisant preuve d’attention, il est

con-sidéré à juste titre comme le “papa gâteau” du service CQP. En outre, je me souviendrai

longtemps de nos soirées de conférence, qu’elle se soient passées autour d’un prosecco

à Trento, d’un whisky à Lund, ou d’une caipirinha à São Paulo. De tout cœur merci

pour ces moments de complicité partagés hors du cadre de la recherche. Je trouve que

nous avons formé une superbe équipe, de loin la plus petite unité du service puisqu’elle

n’a compté que deux membres.

Je lui souhaite d’ailleurs quatre années de recherche

fructueuse avec Sacha Schiffmann, qui sera vraisemblablement le dernier doctorant de

son impressionnante carrière académique.

J’en viens à présent à la deuxième personne qui a largement contribué à ce travail. Je

remercie Daniel Baye, co-promoteur durant mon mémoire mais également collaborateur et

conseiller très précieux durant ma thèse, pour son expertise inestimable, sa disponibilité

ainsi que la grande attention qu’il a portée à mes différents projets.

Il m’a souvent

indiqué la voie à suivre, a corrigé mes défauts, et m’a appris à réfléchir en projetant mon

travail au-delà de mes ambitions premières. Ses nombreuses idées et inspirations lors de

discussions scientifiques avec Michel m’ont grandement éclairé dans la compréhension de

la méthode des réseaux de Lagrange, et ont augmenté mon envie d’aller plus loin. Ce fut

également un plaisir d’étendre à ses côtés cette méthode à la fois simple et élégante au

calcul relativiste de propriétés atomiques. Cette collaboration s’est révélée très fructueuse,

ayant abouti à la rédaction de quatre articles.

I would like to warmly thank Per Jönsson and Jörgen Ekman for welcoming me during

two months in their offices at Malmö Högskola. You both made my stay in Sweden very

enjoyable and memorable thanks to your kind attentions to me. I will cherish our baths

in the cold water of the Öresund sound, but also those during the CompAS meeting

in summer, much more convenient for a rookie like me. I will also remember my last

dinner at Monster restaurant, eating those massive triple-layer burgers, as well as our

relaxing times at the beautiful skybar of Malmhattan. Turning to science, your expertise

in computational atomic physics helped me a lot to get my hand in the grasp

2

k code,

and to understand subtleties in electron correlation models. Our collaboration was very

flourishing, leading to the writing of four papers.

Acknowledgments

Je termine mon doctorat dans le bureau où j’ai commencé mon mémoire. En cinq

ans, j’ai eu l’occasion de connaître trois personnes hautes en couleur, qui sont tour à

tour devenues des amis, à savoir mes collègues de bureau Ludo, Milaim et Guillaume.

Je voudrais les remercier pour les excellents moments que nous avons passés ensemble.

Ludo finissant sa thèse dans les mêmes temps que moi, je souhaite à Milaim ainsi qu’à

Guillaume une excellente dernière année de travail, et un bon courage pour la rédaction.

Je voudrais également remercier tous les membres du CQP pour l’ambiance unique et

excellente qui y règne. Celle-ci n’est certainement pas étrangère au bon déroulement de

ma thèse et au plaisir que j’ai eu à la réaliser. Je souhaite bon courage aux doctorants

qui finiront leur thèse d’ici un an.

Un grand merci à mes amis doctorants extérieurs au CQP, à savoir Michael, Robin

et Piotr, pour les agréables moments de détente sur le campus mais également hors de la

vie universitaire.

Je tiens à présent à remercier mes amis et ma famille, en particulier ma mère, pour

m’avoir toujours soutenu et entouré, pour m’avoir toujours poussé à réaliser mes projets.

Son amour indéfectible m’a sans conteste porté là où je suis aujourd’hui, et je tiens à

exprimer la chance immense de la compter dans ma vie. Elle m’apporte cette sérénité au

quotidien, cette insouciance qui m’a permis de me concentrer sur ce projet de thèse sans

devoir faire face aux tracas et aux imprévus de la vie. Mais c’est promis, je prendrai mon

envol très prochainement.

Abstract

List of publications

This thesis includes the results presented in the following peer-reviewed papers. The

latter are classified into two main groups, labeled A and B.

Group A: Relativistic calculations of polarizabilities and two-photon decay rates in

hydrogenic and alkali-like ions.

A

_I

Accurate solution of the Dirac equation on Lagrange meshes

D. Baye, L. Filippin, and M. Godefroid,

Phys. Rev. E

89, 043305 (2014).

A

_II

Relativistic polarizabilities with the Lagrange-mesh method

L. Filippin, M. Godefroid, and D. Baye,

Phys. Rev. A

90, 052520 (2014).

A

_III

Relativistic two-photon decay rates with the Lagrange-mesh method

L. Filippin, M. Godefroid, and D. Baye,

Phys. Rev. A

93, 012517 (2016).

A

_IV

Relativistic semiempirical-core-potential calculations in Ca

+

, Sr

+

, and

Ba

+

_{ions on Lagrange meshes}

L. Filippin, S. Schiffmann, J. Dohet-Eraly, D. Baye, and M. Godefroid,

Submitted to Phys. Rev. A; arXiv:1709.07672v2 (2017).

Group B: Relativistic multiconfiguration calculations of electronic isotope-shift factors

in many-electron atoms.

B

_I

Core correlation effects in multiconfiguration calculations of isotope shifts

in Mg i

L. Filippin, M. Godefroid, J. Ekman, and P. Jönsson,

Phys. Rev. A

93, 062512 (2016).

B

_II

Multiconfiguration calculations of electronic isotope shift factors in Al i

L. Filippin, R. Beerwerth, J. Ekman, S. Fritzsche, M. Godefroid, and P. Jönsson,

Phys. Rev. A

94, 062508 (2016).

B

_III

Multiconfiguration calculations of electronic isotope-shift factors in Zn i

L. Filippin, J. Bieroń, G. Gaigalas, M. Godefroid, and P. Jönsson,

Phys. Rev. A

96, 042502 (2017).

This thesis does not include the results presented in the following peer-reviewed papers.

The latter are classified in a third group, labeled C.

C

_I

Evolution of nuclear structure in neutron-rich odd-Zn isotopes and

iso-mers

C. Wraith et al.,

Phys. Lett. B

771, 385 (2017).

C

_II

Ab initio

calculations of hyperfine structures of zinc and evaluation of

the nuclear quadrupole moment Q(

67

_Zn)

List of acronyms

AS

active space . . . .

28

ASF

atomic state function . . . .

26

CAS

complete active space . . . .

28

CC

core-core . . . .

28

CI

configuration interaction . . . .

5

CompAS

the Computational Atomic Structure group . . . .

24

CP

core polarization . . . .

5

CSCO

common set of commuting observables . . . .

12

CSF

configuration state function . . . .

21

CV

core-valence . . . .

28

D

double (substitution) . . . .

28

DC

Dirac-Coulomb . . . .

8

DCB

Dirac-Coulomb-Breit . . . .

25

DHF

Dirac-Hartree-Fock . . . .

8

DHFCP

Dirac-Hartree-Fock plus core polarization . . . .

6

DKB

dual kinetic balance . . . .

37

E1

electric dipole . . . .

3

E2

electric quadrupole . . . .

6

FC

frozen core . . . .

29

FS

field shift . . . .

3

grasp2k

the General-purpose Relativistic Atomic Structure Package . . . .

7

HF

Hartree-Fock . . . .

19

IS

isotope shift . . . .

3

ITO

irreductible tensor operator . . . .

39

KB

kinetic balance . . . .

36

LMM

Lagrange-mesh method . . . .

5

LW

Long wavelength . . . .

51

M 1

magnetic dipole . . . .

6

MBPT

many-body perturbation theory . . . .

5

MCDHF

multiconfiguration Dirac-Hartree-Fock . . . .

8

MR

multireference . . . .

27

MRSD

multireference single and double (process) . . . .

28

MS

mass shift . . . .

3

NMS

normal mass shift . . . .

7

PCF

pair-correlation function . . . .

28

QED

quantum electrodynamics . . . .

16

ratip

the Relativistic calculations of Atomic Transition, Ionization,

and recombination Properties package . . . .

8

ris3

the Relativistic Isotope Shift program . . . .

7

S

single (substitution) . . . .

28

SCF

self-consistent field . . . .

24

SE

self energy . . . .

25

SMS

specific mass shift . . . .

7

TP

transverse photon . . . .

25

VP

vacuum polarization . . . .

25

Contents

I

Theory and methodology

1

1 Background and motivation

3

1.1

Polarizabilities . . . .

3

1.2

Two-photon decay rates

. . . .

6

1.3

Electronic isotope-shift factors . . . .

7

1.4

Thesis outline . . . .

8

2 Dirac equation

9

2.1

Relativistic context . . . .

9

2.2

Hydrogenic ions . . . .

12

2.2.1

Radial Dirac equations . . . .

15

3 Dirac-Hartree-Fock method

19

3.1

Relativistic context . . . .

19

3.2

DHF equations . . . .

20

3.3

_{grasp2k package . . . .}

24

4 Beyond the DHF method

25

4.1

Corrections to the DC Hamiltonian . . . .

25

4.2

Electron correlation . . . .

26

4.2.1

MCDHF method . . . .

26

4.2.2

Semi-empirical approach . . . .

29

5 Lagrange-mesh method

31

5.1

Principle . . . .

31

5.2

Lagrange functions . . . .

31

5.2.1

Gauss-quadrature rule . . . .

31

5.2.2

Lagrange conditions

. . . .

32

5.3

Meshes based on orthogonal polynomials . . . .

33

5.4

Regularized Lagrange functions . . . .

34

5.4.1

Regularized Laguerre mesh . . . .

35

6 Static dipole polarizabilities

39

6.1

Definition . . . .

39

6.2

Hydrogenic ions . . . .

39

Table of contents

6.2.2

Tensor polarizability . . . .

44

6.3

Alkali-like ions . . . .

44

7 Radiative multipole transitions

45

7.1

Interaction Hamiltonian

. . . .

45

7.2

Multipole expansion of the radiation field . . . .

46

7.2.1

Gauge transformation

. . . .

47

7.2.2

One-electron transition amplitudes . . . .

47

7.3

One- and two-photon decay rates . . . .

48

7.3.1

Hydrogenic ions . . . .

48

7.3.2

Alkali-like ions

. . . .

51

8 Electronic isotope-shift factors

53

8.1

Isotope shift theory . . . .

53

8.2

Relativistic recoil Hamiltonian . . . .

55

8.2.1

Normal mass-shift factors

. . . .

55

8.2.2

Specific mass-shift factors

. . . .

55

8.3

Relativistic field-shift factors . . . .

56

8.4

Direct diagonalization of the Hamiltonian matrix

. . . .

58

9 Conclusions and perspectives

59

Bibliography

61

II

Papers

71

Paper A

I

73

Paper A

II

85

Paper A

III

99

Paper A

IV

115

Paper B

I

133

Paper B

II

143

Paper B

III

155

List of Figures

2.2.1

Dirac energy spectrum for (a) a hydrogenic system, (b) a free electron, (c) an

anti-hydrogenic system.

. . . .

17

2.2.2

Radial wave functions

P

1s1/2

(r) and Q

1s1/2

(r) of Au

78+

_{(Z = 79).}

_{. . . .}

₁₈

3.3.1

Schematic SCF algorithm solving the DHF equations.

. . . .

24

5.4.1

Basis of 4 Lagrange-Laguerre functions ˆ

f

_i(1)

(x). The 4 mesh points x

i

verify

List of Tables

Part I

1

B

ACKGROUND AND MOTIVATION

This thesis is set within the framework of relativistic quantum mechanics, born from the

union of Einstein’s theory of special relativity and the formalism of quantum mechanics.

The first aim of this thesis is to perform relativistic calculation of polarizabilities and

two-photon decay rates in hydrogenic and alkali-like systems. Polarizabilities are defined

as the response of a system to an external electric field [1]. When immersed in the latter,

an atom or ion undercomes a distorsion of its charge distribution, and behaves as an

electric multipole [2]. Polarizabilities appear in a large number of fields and applications,

namely in cold atoms physics, metrology and chemical physics [3].

Two-photon transitions are part of a priori highly unlikely processes and are therefore

called forbidden radiative processes. The latter occur when angular and parity selections

rules forbid the spontaneous emission of one photon via an allowed electric dipole (E1)

transition from an excited state to a lower one in the energy spectrum. Long lifetimes

from metastable excited states in highly charged Be-like ions were measured in recent

storage-ring experiments, but their interpretation is problematic [4]. Hence, the study of

forbidden radiative processes is all obviously relevant [5, 6].

Another purpose of this thesis is to study the effect of the nuclear mass and charge

distribution on the electronic structure of an atom. Energies of atomic transitions are

slightly shifted from one isotope to another, due to the different masses and charge

distributions of the atomic nuclei [7]. These differences respectively lead to the so-called

mass shift (MS) and field shift (FS). Summing both contributions, the total shift is

referred to as the isotope shift (IS) of a given atomic transition [8]. The second aim of

this thesis is to perform relativistic ab initio calculations of electronic IS factors for a

set of transitions between low-lying levels of neutral systems. These electronic quantities

together with observed ISs between different pairs of isotopes provide the changes in

mean-square charge radii of the atomic nuclei [9].

1.1

Polarizabilities

There has been a renewal of interest in atomic and ionic polarizabilities in the last

decade [2]. Their relevance to various applications, e.g. in cold atoms physics and in

metrology, is discussed in the review of Mitroy et al. [3]. Among the large choice, let us

cite the following fields of research:

1.1 Polarizabilities

• measurement of refractive indices for the determination of the Boltzmann constant

k

B

with a higher accuracy, which is useful for thermometry and macroscopic

stan-dards: the present definition of temperature is based on the triple point of water at

273.16 K. At present, the best estimate of

k

B

is obtained by measuring the speed

of sound in helium (He) gas [10, 11]. Using an accurate theoretical value of the

4

He

polarizability [12] and an accurate measurement of the refractive index for helium

gas [13] yields a value for the universal gas constant,

R, which is proportional to

k

B

through the identity

R = N

A

k

B

, where

N

A

stands for the Avogadro constant,

• estimation of parity nonconservation (PNC) amplitudes in heavy atoms: search for

new physics beyond the standard model of the electroweak interaction. In the

exper-iment [14] on the cesium (Cs)

7s

→ 6s transition, the PNC amplitude was measured

relative to the Stark-induced scalar and tensor transition polarizabilities [15],

• calculation of atomic clock frequency shifts for the research of frequency standards:

redefinition of the SI second, currently based on the microwave transition between

the two hyperfine levels of the ground state of

133

Cs [16]. The transition towards

optical atomic clocks for achieving better relative frequency uncertainties and

sta-bility requires the accurate determination of the dominant systematic shifts, such as

Zeeman, Stark, quadrupole moment and blackbody radiation (BBR) shifts [3, 17],

• determination of magic wavelengths for optical lattices of cold neutral atoms: the

laser field used to create the lattice can be tuned to a magic wavelength [18, 19],

where lattice potentials of equal depth are produced for the two electronic states of

the clock transition. Hence, the quadratic Stark shift induced by the laser, which

is proportional to the frequency-dependent polarizability, is equal for both states of

the clock transition, which cancels the differential shift between the two of them,

• research of decoherence minimization for quantum logical operations: quantum

computation schemes may involve qubits that are realized as internal states of

neutral atoms trapped in optical lattices or microtraps, such as the Rydberg gate

scheme [20] in which the qubit is based on two ground hyperfine states. A

two-qubit phase gate may be realized by exciting two atoms to Rydberg states. The

vibrational state of the atom may change in the latter state, leading to decoherence

due to motional heating. The aim is to minimize the heating by matching the

frequency-dependent polarizabilities of the atomic ground and Rydberg states [21],

• evaluation of radiative line strengths for the determination of abundancy of elements

in astrophysics: line strengths can be extracted from precisely known

polarizabili-ties that are dominated by a single strong transition, as it is the case for Cs where

the dipole polarizability of the ground state

6s is dominated by the 6s

_{→ 6p}

j

transi-tions [22, 23]. Hence, one can extract the line strengths related to these transitransi-tions.

In the particular case of optical atomic clocks, which are based on transitions involving

long-lived metastable states, experiments in this field have reached such a high accuracy

that relativistic effects are visible and must be precisely accounted for in the

calcula-tions [3, 17]. Today’s most advanced atomic clocks report relative systematic frequency

uncertainties below

10

−17

_{[24, 25]. Reaching higher accuracy is limited by small energy}

shifts resulting from BBR and quadratic Stark effect [26–28], highly dependent on the

accuracy of static and dynamic polarizabilities [3].

1.1 Polarizabilities

• Direct method: the application of perturbation theory to the study of an atom or

ion immersed in a uniform electric field involves a response of the system under the

form of a first-order correction to the wave function, solution of an inhomogeneous

equation, weighted by the amplitude of the electric field. The zero- and first-order

wave functions enable a direct computation of the polarizability,

• Sum-over-states method: the second-order correction to the energy is expressed

as a infinite sum of contributions coupling, via an electric multipole operator, the

state whose polarizability is targeted and the stationary states of the zero-order

Hamiltonian. The polarizability, proportional to this correction, is computed by

explicitely summing these contributions, continuous spectrum (spectra) included.

The direct method consists either in solving the appropriate wave equation for different

values of the electric field strength and in extracting the polarizability from the perturbed

energies (finite-field approach) [29], or in adopting the variation-perturbation method by

using the method developed by Dalgarno and Lewis [30,31] for the research of the solution

to the inhomogeneous equation [32].

The sum-over-states method is essentially used for one- and two-valence-electron

sys-tems, for the reason that the polarizability is then dominated by some transitions towards

reasonably excited states. It is thus possible to construct an effectively complete set of

intermediate states, but this method is less suited for atoms with more valence electrons.

Within these two computational means, approaches are distinguished with respect to

the methodology adopted to describe electron correlation and relativistic effects: methods

in bases of explicitely correlated functions [33, 34], multiconfiguration methods combined

to configuration interaction (CI) [35], semiempirical-core-potential approaches [36],

many-body perturbation theory (MBPT) [37], coupled-cluster (CC) method [38], and hybrid

methods combining CI and MBPT [39] or linearized CC [40].

The studied systems are mostly one- and two-valence-electron systems, and the

com-putation of the polarizability is mostly realized with the sum-over-states method. The use

of the finite-field approach is limited to CC computations in Gaussian bases using

molecu-lar codes [29], while the Dalgarno and Lewis method is used in approaches that are better

adapted to the description of systems with more than two valence electrons [32, 41, 42].

With this respect, the calculation of this property with multiconfiguration methods is

currently limited to the nonrelativistic framework [35], using the civ

3

/civpol code [43].

Within this methodological and computational context, this thesis presents results

from relativistic calculations of static multipole polarizabilities in two types of atomic

and ionic systems. In a first step, calculations are performed in hydrogenic ions with the

sum-over-states method, by means of the Lagrange-mesh method (LMM) [44–46]. The

LMM is an approximate variational method taking the form of mesh equations thanks to

the use of a Gauss-quadrature approximation. The Dirac equation for the particular case

of hydrogenic ions is analytically solvable [47, 48], and exact expressions of the

ground-state static scalar dipole polarizabilities were derived in Refs. [49–52]. Besides, accurate

relativistic calculations were reported in Ref. [53] using

B-spline bases [54]. Hence, the

comparison with results from these benchmark references allows to assess the accuracy

of the present numerical method (see Papers A

_I

and A

_II

).

In a second step, this thesis presents results from relativistic

semiempirical-core-potential calculations of static multipole polarizabilities in the alkali-like singly ionized

calcium (Ca

+

), strontium (Sr

+

), and barium (Ba

+

). The latter have been proposed as

candidates for optical frequency standards due to the long lifetimes of their metastable

2

_D

1.2 Two-photon decay rates

to simulate the core-valence correlations for an atom with few valence electrons. The

latter approach is referred to as the Dirac-Hartree-Fock plus core-polarization (DHFCP)

approach, offering simplified expressions and reduced computational times in comparison

with ab initio methods [3, 55, 56]. The underlying methodology is originally discussed in

Refs. [57–61] within the nonrelativistic framework. Later on, nonrelativistic results of

po-larizabilities were published for alkali atoms [62, 63] and alkali-like ions [55, 56]. It is only

very recently that relativistic semiempirical-core-potential calculations were performed in

such systems [64–69]. It should be mentioned that the DHFCP approach is an

approxi-mate method, where the comparison with relativistic ab initio methods [26–28, 38, 70–75]

and experiments [76–78] should provide an assessment of the errors on polarizabilities

due to physical effects that are not included in the model (see Paper A

_IV

).

1.2

Two-photon decay rates

Two-photon transitions are part of the highly unlikely processes, such that they are

clas-sified in a large family containing all the forbidden radiative processes. In this category,

let us mention e.g. the one-photon magnetic dipole (M 1) and electric quadrupole (E2)

transitions, as well as the two-photon electric dipole (2E1) transition. The radiative

de-cay of the metastable

2s

1/2

state to the

1s

1/2

ground state of hydrogenic ions is one of

the most widely studied forbidden atomic transitions. The

2s

1/2

state can decay via two

competing processes: either an

M 1 or a 2E1 transition. The earliest theoretical works on

two-photon processes in hydrogen were performed by Göppert-Mayer [79], and Breit and

Teller [80]. Both (nonrelativistic) studies concluded that the dominant two-photon

2E1

transition is the principal cause of the radiative decay of interstellar

2s hydrogen atoms.

Long lifetimes are attractive in the context of ultraprecise optical frequency standards,

and thus share with atomic and ionic polarizabilities a common field of applications.

Life-times of the

2s2p

3

_P

o

0

state of highly charged Be-like ions (

33

S

12+

and

47

Ti

18+

) were

mea-sured in recent storage-ring experiments, but problems occur in their interpretation [4].

The study of the competition between forbidden radiative processes (one-photon beyond

E1, or multi-photon) and unexpected (induced by hyperfine coupling or by external fields)

is all obviously relevant [5, 6]. Very recently, two-photon transitions have been

theoreti-cally studied in other highly charged Be-like ions, in order to compute the

E1M 1 decay

rate of the

2s2p

3

_P

o

0

→ 2s

2 1

S

0

transition [81, 82].

The ab initio calculation of two-photon radiative decay rates is very rare, although

their mathematical expressions strongly recall the expressions of the polarizability by

the presence of the infinite sum over the intermediate states from the sum-over-states

method [83]. For hydrogenic systems, the decay rates were computed by using

B-spline

bases [84] and more recently

B-polynomial bases [85]. Besides, an interesting physics

appears in the prediction of resonances and extinctions [86].

For many-electron systems, calculations are quasi inexistant [4, 5]. Apart from the

above-cited works on Be-like ions, some studies were carried out on angular distributions

correlation and polarizations of the two photons emitted in heavy He-like systems [87].

Besides, two-photon decay rates have been calculated in Ca

+

, Sr

+

, and Ba

+

ions using the

ab initio all-order CC method [88,89]. Finally, Ref. [90] studied two-photon transitions in

neutral Ba induced by hyperfine coupling and by external magnetic fields. Hence, it is seen

that the evaluation of two-photon transition probabilities remains embryonic, although

there is a real appeal, mostly for highly ionized systems, for estimating the probability

of these decay modes relative to hyperfine- or magnetic-field-induced channels [4].

1.3 Electronic isotope-shift factors

the same theoretical models and numerical methods. The results of the

2s

1/2

→ 1s

1/2

and

2p

1/2

→ 1s

1/2

two-photon decay rates in hydrogenic ions obtained with the LMM

are compared with several benchmark theoretical values [84–86, 91, 92] (see Paper A

III

),

while those of the

nd

3/2,5/2

→ (n + 1)s

1/2

2E1 decay rates in Ca

+

(n = 3), Sr

+

(n = 4),

and Ba

+

(n = 5) ions obtained with the DHFCP-LMM approach are compared with ab

initio values from Refs. [88, 89] (see Paper A

IV

).

1.3

Electronic isotope-shift factors

Differences in mean-square charge radii of nuclear charge distributions along isotope

chains are important for testing nuclear models and interactions [8, 93]. The nuclear

size can be extracted from laser-spectroscopy IS measurements, assuming that the MS

and the electronic densities inside the nucleus can be computed [7,8]. The recent progress

in atomic structure theory enables to perform large-scale calculations for open-shell atoms

and ions, providing the necessary atomic input to analyze laser-spectroscopy data. These

studies have significantly contributed to the understanding of the evolution of nuclear

sizes as a function of neutron and proton numbers along the nuclide chart [94]. However,

the demand from the nuclear physics community is increasing, and further theoretical

atomic investigations are needed for more complex atoms.

For light elements, where the electron density can be considered as constant inside

the nucleus, the transition FS is proportional to differences in mean-square charge radii

between isotopes [95]. In heavier systems, the electron density is no longer constant

inside the nucleus, and the transition FS becomes sensitive to higher-order nuclear radial

moments, or Seltzer moments [96]. The contribution of the latter can be subtracted

from the FS using a pre-determined model of the nuclear charge distribution, in order

to reach better estimates of the mean-square charge radii differences [97]. The nuclear

deformation effects on the transition energies are usually neglected, although theoretical

investigations have recently been realized in heavy ions [98–101].

To determine nuclear properties from IS measurements, the MS must be extracted

from the total IS. The MS of a given atomic transition depends on the nuclear masses and

on the difference of electronic MS factors of the upper and lower levels of this transition.

These MS factors are expressed as expectation values of the so-called nuclear recoil

oper-ator, and are calculated using approximate solutions of the many-electron wave equation.

Since the derivation of relativistic corrections to the recoil operator by Shabaev [102, 103]

and Palmer [104], several papers [99, 105–107] have shown the importance of relativistic

effects on ISs. The tensorial expression of the relativistic recoil operator [108] has been

implemented in the ris

3

(Relativistic Isotope Shift) module [109, 110] of the grasp

2

k

package [111, 112] (General-purpose Relativistic Atomic Structure Package).

1.4 Thesis outline

complex systems. All-order CC methods for relativistic atomic structure calculations have

been applied to predict properties of heavy atoms, from one- to three-valence-electron

systems, but the SMS factor turns out to be the most difficult to accurately evaluate [117].

The combined CI-MBPT approach is attractive for few-valence-electron systems, and the

SMS factor is evaluated with the use of the finite-field technique [118, 119], although it is

currently not part of the CI-MBPT package [120]. With this respect, variational methods

based on relativistic multiconfiguration/CI expansions are more flexible, and have been

applied to few-electron systems to yield highly accurate results [121,122]. MS and FS have

also been estimated in complex neutral systems such as barium [107], copper [123, 124],

manganese [125], polonium [126], and osmium [127].

Within this context, this thesis presents results from relativistic ab initio calculations

of electronic IS factors by using the multiconfiguration DHF (MCDHF) method

imple-mented in the ris

3

/grasp

2

k and ratip [128] (Relativistic calculations of Atomic

Tran-sition, Ionization, and recombination Properties) program packages. Using the MCDHF

method, two different approaches are adopted for the computation of the electronic IS

factors for a set of transitions between low-lying levels of three neutral systems: Mg i (see

Paper B

_I

_{), Al i (see Paper B}

_II

_{), and Zn i (see Paper B}

_III

). The first approach is based

on the estimate of the expectation values of the relativistic nuclear recoil Hamiltonian

for a given isotope, combined with the calculation of the total electron densities at the

origin. In the second approach, the IS factors are extracted from the calculated transition

shifts for given triads of isotopes. Combining the IS factors with observed ISs between

different pairs of isotopes provides the changes in mean-square charge radii of the atomic

nuclei. Within this computational approach for the estimation of the MS and FS factors,

different models for electron correlation are explored in a systematic way to determine a

reliable computational strategy, and to estimate theoretical error bars of the IS factors.

In Paper B

_I

_{, the computed IS factors of some well-known transitions in Mg i are}

com-bined with available nuclear data. It allows to determine transition ISs for the

26

Mg

₋

24

Mg

pair of isotopes, where experimental [129–133] and theoretical values [118, 119, 134–137]

are available for comparison. By contrast, the lack of accurate theoretical calculations

of IS factors in Al i and Zn i must be pointed out. Hence, Papers B

II

and B

III

provide

benchmark values for several transitions involving low-lying states of these two systems.

1.4

Thesis outline

2

D

IRAC EQUATION

2.1

Relativistic context

Relativistic corrections become important when the nuclear charge of an atom increases,

and therefore must be taken into account [138]. Indeed, it is shown that as the number

Z of protons inside the nucleus increases, the electrons of the innermost orbitals reach

speeds near the speed of light [139]. Hence, the framework of relativistic quantum

me-chanics must be adopted. The simplistic relativistic Bohr model, which can only describe

one-particle systems, suffices to illustrate it quantitatively. One of the statements of

special relativity is that the mass of a particle increases towards infinity as its velocity

v approaches the speed of light c in vacuum. The classical theory of special relativity

mathematically formulates the relativistic mass, denoted as

m

rel

in the following, as

m

rel

= γ(v)m =

p

m

1

− (v/c)

2

,

(2.1.1)

where

γ(v) is the Lorentz factor and m the mass of the particle at rest. Within the

relativistic Bohr model of a hydrogenic system, let us consider one electron of mass

m

e

at rest on a circular orbit around a nucleus of positive point charge

Ze and infinite mass.

Its average speed on the Bohr orbit of principal quantum number

n reads

hvi

n

=

Ze

2

4πε

0

n

}

=

Zαc

n

,

(2.1.2)

where

_{−e is the electron charge, ε}

0

the vacuum permittivity,

} the reduced Planck

con-stant (=h/2π), and α the fine-structure concon-stant. The second equality of Eq. (2.1.2) is

obtained by using the expression of the fine-structure constant [140]

α =

e

2

4πε

0

}c

= 1/137.035 999 139(31).

(2.1.3)

Similarly, the average orbital radius

_hri

n

and total energy

hEi

n

of the atomic electron

are respectively given by

hri

n

=

4πε

0

n

2

}

2

m

rel e

Ze

2

=

n

}

m

rel e

hvi

n

(2.1.4)

and

hEi

n

=

−



Ze

2

4πε

0



2

m

rel_e

2n

2

}

2

.

(2.1.5)

2.1 Relativistic context

Let us also consider that its only electron is on the first Bohr orbit (n = 1), which is

closest to the nucleus and subject to strong electrostatic interaction from all 79 protons.

According to Eq. (2.1.2), its speed is

58% of c (v/c

≈ 79/137). Using Eq. (2.1.1) gives a

substantial mass increase

(m

rel_e

_{− m}

e

)/m

e

of

22% which in turn, according to Eq. (2.1.4),

implies a decrease of its Bohr radius of

18%, thus corresponding to a radial contraction.

The linear dependence on the electron mass in the total energy (2.1.5) implies a decrease

of

22% in the total energy of the n = 1 electron, i.e., an increase of its binding energy.

Note that if speeds around

v/c

∼ 0.1 are already considered as relativistic, the n = 1

electron of an hydrogenic ion becomes relativistic for nuclei with

Z

≥ 14. It should also

be mentioned that the nonrelativistic Bohr model [141,142], i.e., in which

m

rel_e

is replaced

by

m

e

, is validated by the Schrödinger equation for one-electron systems [143]. Hence,

these considerations remain valid in the framework of Schrödinger’s theory, considering

the speed of the electron on the

1s orbital.

Furthermore, in order to describe the fine structure of atomic states from fundamental

principles, it is also necessary to treat the bound electrons relativistically [144]. The Dirac

equation describes the behavior of massive elementary particles of half-integer spin, i.e.,

fermions like electrons. This equation is simultaneously compatible with the principles

of quantum mechanics and those of special relativity [47, 48]. This thesis does not focus

in details on the history of the Dirac equation, the latter being fully described in e.g.

Refs. [138, 139].

Atomic units

At this stage, it is necessary to introduce the units used in this thesis, namely, the atomic

units (a.u.), defined by

a

0

= m

e

= e =

} = 4πε

0

= 1,

(2.1.6)

where

a

0

=

4πε0}

2

mee2

is the Bohr radius and

m

e

the electron mass at rest. The atomic unit

of speed

v

B

= αc is defined from expression (2.1.3) of the fine-structure constant α. This

implies that the speed of light

c equals 1/α

_{≈ 137 a.u. The atomic unit of energy is the}

Hartree, defined as

1E

h

=

}

2

/(m

e

a

20

) = e

2

/(4πε

0

a

0

)

≈ 27.2 eV, i.e., twice the ionization

energy of the hydrogen atom in its

1s ground state.

Free particle

The classical theory of special relativity reports that the total relativistic energy

E of a

free particle of mass

m at rest is given by the energy-momentum relation

E = γ(v)mc

2

=

p

m

2

_c

4

_{+ p}

2

_c

2

_,

_(2.1.7)

where

p

2

_{is the square of the relativistic momentum}

_{p = γ(v)mv of the particle, with v}

denoting its 3-velocity. Applying the correspondence rules of quantum mechanics

E

→ i

∂

∂t

,

p

→ −i∇,

(2.1.8)

and considering the square of identity (2.1.7), the Klein-Gordon equation is obtained as

−

∂

2

_ψ

∂t

2

= m

2

_c

4

_ψ

_{− c}

2

_∇

2

_ψ.

_(2.1.9)

2.1 Relativistic context

derivative, which implies that the solutions of this equation do not obey the superposition

principle. Moreover, the wave function at time

t = 0 does not fully determine the system

for any further time, as it was the case for the Schrödinger equation through the evolution

operator

U (t, t

0

= 0) [138, 145]. Finally, the square root of this equation is not invariant

under Lorentz transformations in the Minkowski space-time Mk

4

, and the time and space

coordinates are not treated equally [138]. Indeed, in Mk

4

the expression of the 4-vector

x

µ

in covariant form is written as

x

µ

= (x

0

≡ ct, x

1

, x

2

, x

3

),

(2.1.10)

where each of the 4 components (µ = 0, 1, 2, 3) has the dimension of a length.

The incompatibilities with special relativity occuring in the Klein-Gordon equation

for massive particles led Dirac [47, 48] to propose another Hamiltonian for a free particle

of mass

m and half-integer spin s. Its expression reads

H

D

= cα

· p + βmc

2

,

(2.1.11)

where

α and β are the 4

_{× 4 Dirac matrices. Their expression is given by}

α =



0 σ

σ 0



,

β =



I

0

0

−I



,

(2.1.12)

where

I is the 2

× 2 identity matrix, and σ is a vector operator containing the Pauli

matrices

σ

x

=



0 1

1 0



,

σ

y

=



0

−i

i

0



,

σ

z

=



1

0

0

₋₁



.

(2.1.13)

The link between the operator

σ and the spin angular momentum S is S =

1₂

σ. Matrices

(2.1.13) form with the identity matrix

I a basis of the 2

× 2 Hermitian matrices vector

space [142, 146]. It can be shown that

α and β are Hermitian operators:

α = α

†

_,

_{β = β}

†

_,

(2.1.14)

which leads to the required Hermiticity of Hamiltonian

H

D

[138]. Moreover, they satisfy

the anticommutation relations

α

2_x

= α

2_y

= α

2_z

= β

2

= 1,

{α

x

, α

y

} = {α

y

, α

z

} = {α

z

, α

x

} = 0,

{α

x

, β

} = {α

y

, β

} = {α

z

, β

} = 0,

(2.1.15)

arising from the fact that the solutions of the Dirac equation must also satisfy the

Klein-Gordon equation [138, 144]. The Dirac equation for a free particle reads

cα

_{· p + βmc}

2

ψ(r, t) = i

∂ψ(r, t)

∂t

.

(2.1.16)

Different equivalent formulations of the Dirac equation exist, but the present one is the

closest to the Schrödinger equation. As matrices

α and β are 4

× 4 matrices, the solution

ψ(r, t) must be a 4-column vector.

2.2 Hydrogenic ions

Charged particle in an external field

Let us consider a charged particle of charge

q, mass m and (half-integer) spin s, immersed

in an external electromagnetic field. Its Dirac Hamiltonian is given by [138]

H

D

= cα

· (p − qA) + βmc

2

+ qΦ,

(2.1.17)

where

A is the magnetic vector potential, and Φ the electric scalar potential. Therefore,

the Dirac equation of the particle under consideration reads



cα

· (p − qA) + βmc

2

_{+ qΦ}



_{ψ(r, t) = i}

∂ψ(r, t)

∂t

.

(2.1.18)

2.2

Hydrogenic ions

The present section mainly follows the books by Grant [138] and by Johnson [139]. The

reader is invited to refer to these sources for further details.

In the particular case of a central field, potentials

A and Φ read A = 0 and qΦ = V (r),

where

r =

_{|r|. It is the case for hydrogenic ions, for which V (r) is the Coulomb potential}

V

C

(r) =

−Z/r. The Dirac equation for a hydrogenic ion describes the motion of an

electron of mass

m

e

= 1 a.u., charge

−e = −1 a.u. and spin s = 1/2 in the field of an

atomic nucleus [47, 48]. Its expression reads



cα

_{· p + βc}

2

₋

Z

r



ψ(r, t) = i

∂ψ(r, t)

∂t

.

(2.2.1)

Only stationary states will be discussed here. Hence, by writing

ψ(r, t) with the form

ψ(r, t) = φ(r) e

−iEt

,

(2.2.2)

one obtains the following eigenvalue problem:



cα

_{· p + βc}

2

₋

Z

r



φ(r) = Eφ(r),

(2.2.3)

where

E denotes the binding energy of the electron in the Coulomb potential of the

atomic nucleus.

One notices that

H

D

does not commute neither with

L

2

, the square of the orbital

angular momentum

L, nor with S

2

_{, the square of the spin angular momentum}

_{S. Indeed,}

[L, H

D

] = iα

× p →



H

D

, L

2



6= 0,

[S, H

D

] =

−iα × p →



H

D

, S

2



6= 0.

(2.2.4)

Hence, the common set of commuting observables (CSCO)

_{L

2

, L

z

, S

2

, S

z

} is not

com-patible with

H

D

. However,

H

D

commutes with the total angular momentum

J = L + S.

Indeed,

H

D

is invariant under rotations, therefore it commutes with the generator of the

SO(3) group, namely

J [147]. Hence, because [H

D

, J

x

] = [H

D

, J

y

] = [H

D

, J

z

] = 0, H

D

also commutes with

J

2

_:



H

D

, J

2



= [H

D

, J

z

] = 0.

(2.2.5)

According to Eq. (2.2.5), there exist common eigenfunctions to

H

D

,

J

2

and

J

z

[145]. The

following change of CSCO is thus required:

2.2 Hydrogenic ions

The basis change towards a coupled representation is built by means of Clebsch-Gordan

coefficients:

|j m

j

i =

l

X

ml=−l 1/2

X

ms=−1/2

|l m

l

1/2 m

s

ihl m

l

1/2 m

s

|j m

j

i

=

1/2

X

ms=−1/2

|l m

j

− m

s

1/2 m

s

ihl m

j

− m

s

1/2 m

s

|j m

j

i,

(2.2.7)

where the second equality arises from the fact that the total projection

m

j

must be

the sum of

m

l

and

m

s

so that the Clebsch-Gordan coefficient

hl m

l

1/2 m

s

|j m

j

i is not

null [146, 148]. Hence, one has

m

l

= m

j

− m

s

. According to the triangular relations,

the quantum number

j must verify

|l − 1/2| ≤ j ≤ l + 1/2, thus j is equal to |l ± 1/2|.

Moreover, one has

m

j

=

−j, −j + 1, ..., j − 1, j.

Kets

_{|j m}

j

i are eigenstates of J

2

and

J

z

with respective eigenvalues

j(j + 1) and

m

j

. They correspond to the spherical spinors

χ

jlmj

, constructed from combinations of

spherical harmonics

Y

l

ml

, eigenstates of

L

2

_and

_L

z

with respective eigenvalues

l(l + 1)

and

m

l

, and of Pauli spinors

χ

ms

, eigenstates of

S

2

_and

_S

z

with respective eigenvalues

s(s + 1) = 3/4 and m

s

=

±1/2 [146]. Let us rewrite (2.2.7):

χ

jlmj

(ˆ

r) =

1/2

X

ms=−1/2

hl m

j

− m

s

1/2 m

s

|j m

j

iY

mlj−ms

(ˆ

r)χ

ms

,

(2.2.8)

or, with a spinor form,

χ

jlmj

(ˆ

r) =

hl m

j

− 1/2; 1/2 + 1/2|j m

j

iY

_ml_j−1/2

(ˆ

r)

hl m

j

+ 1/2; 1/2

− 1/2|j m

j

iY

_ml_j+1/2

(ˆ

r)

!

,

(2.2.9)

where

r

ˆ

_{≡ (θ, ϕ). The explicit expression of the spherical spinors reads [138, 139]}

χ

l+1/2,lmj

(ˆ

r) =





q

l+mj+1/2 2l+1

Y

l mj−1/2

(ˆ

r)

q

l−mj+1/2 2l+1

Y

l mj+1/2

(ˆ

r)



 ,

(2.2.10)

χ

l−1/2,lmj

(ˆ

r) =





−

q

l−mj+1/2 2l+1

Y

l mj−1/2

(ˆ

r)

q

l+mj+1/2 2l+1

Y

l mj+1/2

(ˆ

r)



 .

(2.2.11)

Each of these eigenvectors satisfies the following eigenvalue equations:

L

2

χ

jlmj

= l(l + 1) χ

jlmj

,

S

2

χ

jlmj

=

3

4

χ

jlmj

,

J

2

χ

jlmj

= j(j + 1) χ

jlmj

,

J

z

χ

jlmj

= m

j

χ

jlmj

.

(2.2.12)

The CSCO

_{L

2

, S

2

_{, J}

2

_{, J}

z

} is no longer compatible with H

D

because

[H

D

, L

2

]

6= 0 and

[H

D

, S

2

]

6= 0, even if [H

D

, J

2

] = [H

D

, J

z

] = 0. This is the reason why it is interesting to

introduce the operator

K defined as

K =

−(J

2

_{− L}

2

_{− S}

2

_{+ 1)}

=

_{−(2 S · L + 1)}

2.2 Hydrogenic ions

where the second equality arises from

J

2

_{= L}

2

_{+ S}

2

_{+ 2 S}

_{· L. Spherical spinors χ}

jlmj

(ˆ

r)

are eigenfunctions of

σ

· L, and therefore eigenfunctions of K, following

K χ

jlmj

= κ χ

jlmj

,

(2.2.14)

whose (integer) eigenvalues

κ are

κ =











− (j +

1

₂

) =

_{−l − 1, if j = l +}

1

2

+ (j +

1

2

) = l,

if

j = l

−

1

2

.

(2.2.15)

The value of

κ determines both j and l quantum numbers. This is why spherical spinors

are written with the form

χ

κm

in the following [138]. Table 2.1 displays some values of

l,

j and κ, as well as the corresponding spectroscopic notation of some relativistic subshells.

It is worthwhile to notice that

_{|κ| = j + 1/2.}

Table 2.1:

Spectroscopic notation of some relativistic subshells.

s

1/2

p

1/2

p

3/2

d

3/2

d

5/2

f

5/2

f

7/2

l

0

1

1

2

2

3

3

j

1/2

1/2

3/2

3/2

5/2

5/2

7/2

κ

−1

+1

−2

+2

−3

+3

−4

It can be shown that the operator K

= βK commutes with H

D

[138, 139]. The

compatible CSCO with

H

D

is hence

{J

2

, J

z

, K

}. Hamiltonian H

D

is also invariant under

spatial inversion, therefore it commutes with P

= βΠ, where the parity operator Π is the

generator of the

Z

2

group [147]. In spherical coordinates, the operator

Π transforms the

azimutal angle

ϕ into (ϕ + π) and the colatitude θ into (π

_{−θ). When applied to spherical}

harmonics

Y

l

ml

(ˆ

r), this operator provides the eigenvalue π = (

−1)

l

_{, according to}

Π Y

mll

(ˆ

r) = Y

l ml

(π

− θ, ϕ + π) = (−1)

l

_Y

l ml

(ˆ

r).

(2.2.16)

One concludes that spherical spinors are eigenfunctions of

Π with the associated

eigen-value

(

−1)

l

_{. The two spherical spinors}

_χ

κm

and

χ

−κm

, corresponding to the same

j value,

present

l values that differ by one unit. Hence, they are of opposite parity.

Finally, the eigenstates of the Dirac equation (2.2.3) have the form

φ

nκm

(r) =

1

r



P

nκ

(r) χ

κm

(ˆ

r)

iQ

nκ

(r) χ

−κm

(ˆ

r)



,

(2.2.17)

where

P

nκ

(r) and Q

nκ

(r) respectively correspond to the large and the small radial

com-ponent [138, 139]. Each of these eigenvectors satisfies the following eigenvalue equations:

J

2

φ

nκm

= j(j + 1) φ

nκm

,

J

z

φ

nκm

= m φ

nκm

,

K

φ

nκm

= κ φ

nκm

,

P

φ

nκm

= π φ

nκm

.

(2.2.18)

2.2 Hydrogenic ions

As spherical spinors satisfy the orthogonality relations

Z

π 0

sin θ dθ

Z

2π 0

dϕ χ

†_κ0m0

(θ, ϕ) χ

κm

(θ, ϕ) = δ

κ0_κ

δ

_m0_m

,

(2.2.19)

the normalization condition for the eigenstates

φ

nκm

(r),

Z

R3

φ

†_nκm

(r) φ

nκm

(r) d

3

r = 1,

(2.2.20)

reduces to a normalization condition on the radial wave functions

P

nκ

(r) and Q

nκ

(r):

Z

∞

0



P

_nκ2

(r) + Q

2_nκ

(r)



dr = 1.

(2.2.21)

2.2.1

Radial Dirac equations

The operator

cσ

· p can be written as [138]

cσ

· p = −icσ · e

r



∂

∂r

−

σ

· L

r



,

(2.2.22)

where

e

r

is the unit radial vector. The action of the operator

σ

· e

r

on spherical spinors

χ

κm

(ˆ

r) proceeds as

σ

_{· e}

r

χ

κm

(ˆ

r) =

−χ

−κm

(ˆ

r).

(2.2.23)

Hence, as

K + 1 =

−σ · L according to Eq. (2.2.13), one has

cσ

· p f(r) χ

κm

(ˆ

r) = i



d

dr

f (r) +

κ + 1

r

f (r)



χ

−κm

(ˆ

r).

(2.2.24)

This expression, together with expression (2.2.17) of the wave function

φ

nκm

(r) allow to

derive from Eq. (2.2.3) the system of coupled radial Dirac equations [138]:



V

C

(r) + c

2

− E

nκ

−c

_drd

−

κ_r

c

_drd

+

κ_r

V

C

(r)

− c

2

− E

nκ

 

P

nκ

(r)

Q

nκ

(r)



= 0.

(2.2.25)

This eigenvalue problem is analytically solvable. The detailed calculation is available in

Refs. [138, 139]. Here only its solutions are presented, i.e., the energies

E

nκ

of the bound

states, and the corresponding radial wave functions

P

nκ

(r) and Q

nκ

(r).

2.2.1.1 Dirac energies

Solving these eigenvalue equations, the energies of the bound states are given in a.u.

by [138]

E

nκ

= c

2

λ

γ + n

− |κ|

Z

=

c

2

s

1 +

(αZ)

2

(γ + n

_{− |κ|)}

2

= c

2

p

γ + n

r

(γ + n

r

)

2

+ (αZ)

2

|

{z

}

N

,

(2.2.26)

where

λ =

p

c

2

_{− E}

2 nκ

/c

2

and

γ =

p

κ

2

_{− (αZ)}

2

_{. The link between the principal quantum}

number

n and the quantum number n

r

is given by

2.2 Hydrogenic ions

The number of nodes in the large component

P

nκ

(r) is n

− l − 1, as in the nonrelativistic

case, while the number of nodes in

Q

nκ

(r) is n

− l − 1 for κ < 0 and n − l for κ > 0 [139].

The number N involved in Eq. (2.2.26) is called the effective principal quantum

num-ber. Its expression reads

N

=

p

(γ + n

r

)

2

+ (αZ)

2

=

p

n

2

_{− 2n}

r

(

|κ| − γ).

(2.2.28)

Its role is analogue to that of the quantum number

n within the nonrelativistic

frame-work [144]. One notices that for

n

r

= 0, N takes the same value as n.

The energy levels only depend on

_{|κ| = j + 1/2. Hence, for a given n, the solutions}

for

_{±κ are degenerate (e.g. 2s}

1/2

and

2p

1/2

), which preserves the degeneracy in

l as in

the nonrelativistic case. Besides, the Dirac equation does not account for the Lamb shift,

which is a correction arising from quantum electrodynamics (QED) [149,150]. Indeed, the

Lamb shift lifts the degeneracy of these levels, that can be observed experimentally [145].

By contrast, two levels with the same values of

n and l but with different values of j,

e.g.

2p

1/2

and

2p

3/2

, do not have the same energy, and are separated by a fine-structure

splitting [138, 139].

Expanding

E

nκ

(in a.u.) in powers of

(αZ)

2

leads to

E

nκ

= c

2



1

₋

(αZ)

2

2n

2

−

(αZ)

4

2n

4



n

|κ|

−

3

4



+

_{· · ·}



= c

2

₋

Z

2

2n

2

−

α

2

_Z

4

2n

3



1

|κ|

−

3

4n



+

_{· · ·}

(2.2.29)

The first term corresponds to the electron rest energy

c

2

_{, the second one to the}

nonrel-ativistic binding energy, and the next ones to relnonrel-ativistic corrections. Hence, if the rest

energy

c

2

_{is subtracted from}

_E

nκ

, the energy

ε

nκ

is defined as the sum of the

nonrelativis-tic binding energy,

_−Z

2

/2n

2

_{, and of relativistic corrections. The nonrelativistic binding}

energy is therefore recovered at the

c

_{→ ∞ limit.}

For

ε > 0 (or E > +c

2

_{), the electron is no longer bound to the nucleus, and essentially}

behaves as a free particle. Thus, there exists a continuous energy spectrum for these

values. For

_−c

2

< ε < 0 (or 0 < E < +c

2

_{), there exists a discrete spectrum of bound}

states of energy

E

nκ

. Besides, within Dirac’s theory it is mathematically possible to

obtain a negative square root in the expression of

E

nκ

[47,48]. Thus, there exists a second

discrete spectrum of bound states, corresponding to

_−2c

2

< ε <

_−c

2

_(or

_−c

2

_{< E < 0).}

Finally, a second continuous spectrum exists for

ε <

−2c

2

_(or

_{E <}

_−c

2

_{). The existence}

of this negative energy continuum implies that the Dirac Hamiltonian

H

D

does not have

a global lower bound. Hence, one can wonder whether the variational method, which lies

on the existence of such a bound (according to the Ritz theorem) [146], is still applicable

within the framework of the Dirac equation. However, it is possible to demonstrate the

convergence of the variational method for

H

D

[138]. The discrete spectrum

E

1

< E

2

<

· · ·

over

(

−c

2

_{, +c}

2

_{) has an accumulation point at +c}

2

_{. The existence of the continua for}

E > +c

2

_and

_{E <}

_−c

2

_{implies that there exists a lower bound such that}

_−c

2

_{< E < E}

1

for the discrete spectrum, as well as an upper bound

E

_{≤ −c}

2

_{for the negative energy}

continuum.

2.2 Hydrogenic ions

E (a.u.)

+c

2

−c

2

0

Z > 0

Z = 0

Z < 0

p.e.c.

n.e.c.

(a)

(b)

(c)

Figure 2.2.1:

Dirac energy spectrum for (a) a hydrogenic system, (b) a free electron, (c) an

anti-hydrogenic system.

2.2.1.2 Radial wave functions

The calculation of the radial wave functions

P

nκ

(r) and Q

nκ

(r) is detailed in Refs. [138,

139]. Their expressions read

P

nκ

(r) =

p

1 + E

nκ

/c

2

N

nκ

e

−x/2

x

γ

[(N

− κ)

1

F

1

(

−n + |κ|, 2γ + 1, x)

−(n − |κ|)

1

F

1

(

−n + |κ| + 1, 2γ + 1, x)] ,

(2.2.30)

Q

nκ

(r) =

p

1

_{− E}

nκ

/c

2

N

nκ

e

−x/2

x

γ

[

−(N − κ)

1

F

1

(

−n + |κ|, 2γ + 1, x)

−(n − |κ|)

1

F

1

(

−n + |κ| + 1, 2γ + 1, x)] ,

(2.2.31)

where

x = 2λr = 2Zr/N using Eq. (2.2.26). The normalization factor N

nκ

is given by

N

nκ

=

1

N

Γ(2γ + 1)

s

Z Γ(2γ + 1 + n

− |κ|)

2(n

_{− |κ|)!(N − κ)}

,

(2.2.32)

where

Γ is the Eulerian function of the first kind [151]. The functions

1

F

1

involved in

Eqs. (2.2.30) and (2.2.31) are the confluent hypergeometric functions [151]

1

F

1

(a, b, x) = 1 +

a

b

x +

a(a + 1)

b(b + 1)

x

2

2!

+

a(a + 1)(a + 2)

b(b + 1)(b + 2)

x

3

3!

+

· · ·

(2.2.33)

They read, in terms of generalized Laguerre polynomials

L

(2γ)_n−|κ|

(x) and L

(2γ)_n−|κ|−1

(x),