O N T H E U S E O F N O N - O RT H O G O N A L PA RT I T I O N C O R R E L AT I O N F U N C T I O N S I N AT O M I C P H Y S I C S : T H E O RY A N D A P P L I C AT I O N S

O N T H E U S E O F N O N - O RT H O G O N A L PA RT I T I O N

C O R R E L AT I O N F U N C T I O N S I N AT O M I C P H Y S I C S : T H E O RY A N D A P P L I C AT I O N S

simon verdebout

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences de l’Ingénieur

Service de Chimie Quantique et Photophysique Ecole Polytechnique de Bruxelles

Université Libre de Bruxelles

Promoteur: Prof. Michel Godefroid Année académique 2012-2013

R É S U M É

Dans notre thèse, nous abordons le problème polyélectronique dans un contexte non-relativiste et relativiste en adoptant une approche orbitalaire. En utilisant les suites de programmes reconnues ATomic- Structure Package ( ^ATSP ) et General-purpose Relativistic Atomic Struc- ture Program ( ^GRASP ), nous sommes aptes à approcher des fonctions d’ondes polyélectroniques au-delà du modèle des particules indépen- dantes en utilisant une superposition de configuration state functions ( ^CSFs ). Le processus d’optimisation, basé sur le principe des variations,

fournit la meilleure estimation possible des coefficients de mélange, fix- ant la combinaison linéaire de ^CSFs et la meilleure base de spin-orbitales sur laquelle on impose la condition d’orthonormalité entre les fonctions appartenant au même sous-espace l ou κ . En adoptant cette approche dans un cadre relativiste, nous évaluons des énergies de transition, des probabilités de transition, des déplacements isotopiques ainsi qu’une transition induite par mélange hyperfin pour l’atome d’antimoine trois fois ionisé (Sb ³⁺ ).

Dans le but de relâcher partiellement les contraintes d’orthogonalité entre les orbitales de corrélation, nous utilisons le principe des variations afin de cibler des effets précis de la corrélation en taillant l’espace des configurations. Les ensembles indépendants d’orbitales de corrélation sont obtenus via la méthode multiconfiguration Hartree-Fock ( ^MCHF ).

Les espaces de ^CSFs , exprimés sur ces fonctions mono-électronique non-

orthogonales, sont couplés en résolvant le problème aux valeurs propres

généralisé associé. Les matrices Hamiltonienne et de recouvrement sont

déterminées au moyen de la technique des transformations biorthonor-

males et de la contre-transformation des vecteurs propres associés. Cette

méthode originale est utilisée avec succès pour décrire des systèmes

atomiques légers comme Li I, Be I, B I, C II et Ne I. Un effet indésir-

able, appelé “effet de contrainte”, est décrit et étudié pour ces derniers

systèmes atomiques. Même si ces contraintes peuvent-être en principe

totalement levées au travers de la méthode Deconstrained Partitioned

Correlation Functions Interaction ( ^DPCFI ), les ressources nécessaires à

l’application de cette dernière approche nous ont conduit à la recherche

de stratégies simples et efficaces autorisant leur levée partielle. Pour

ce faire, dans le cadre de nos calculs réalisés sur l’atome de béryllium,

nous avons envisagé deux stratégies particulières: l’une basée sur les co-

efficients de mélange et l’autre basée sur le type d’excitation. Avant de

conclure, nous proposons quelques développements combinant le pro-

cessus auto-cohérent et la condition de biorthonormalité dans le but

de relâcher les contraintes d’orthogonalité appliquées lors du processus

d’optimisation de la base de spin-orbitales.

A B S T R A C T

Our thesis tackles the many-electronic problem considering a non-rela- tivistic and a relativistic orbital approach. Using the suites of pro- grams ATomic-Structure Package ( ^ATSP ) and General-purpose Rela- tivistic Atomic Structure Program ( ^GRASP ), we are able to approximate many-electron wave functions beyond the independent particle model by considering a superposition of configuration state functions ( ^CSFs ). The optimization process, based on the variational principle, provides the best possible mixing coefficients fixing the linear combination of ^CSFs and spin-orbital basis on which we impose the orthonormality condition between functions of the same l or κ subspace. Using this conventional approach within the relativistic framework, we estimate different prop- erties of the triply ionized antimony atom (Sb ³⁺ ), namely transition energies, transition probabilities, isotope shifts and a hyperfine-induced transition.

In the aim of partially relaxing the orthogonality constraints be- tween correlation orbitals, we use the variational principle for target- ing specific correlation effects by tailoring the configuration space. In- dependent sets of correlation orbitals, embedded in Partition Correla- tion Functions ( ^PCFs ), are produced from multiconfiguration Hartree- Fock ( ^MCHF ) calculations. These non-orthogonal functions span ^CSF spaces that are coupled to each other by solving the associated general- ized eigenvalue problem. The Hamiltonian and overlap matrix elements are evaluated using the biorthonormal orbital transformations and the efficient counter-transformations of the configuration interaction eigen- vectors. This original method is successfully applied for describing dif- ferent light atomic systems such as Li I, Be I, B I, C II and Ne I. An unwanted effect, called the “constraint effect”, is described and stud- ied for these particular atomic systems. Even if this constraint can be completely relaxed through the Deconstrained Partitioned Correlation Functions Interaction ( ^DPCFI ) method, the computational resources re- quired by such an approach lead us to study some simple strategies relaxing partially this constraint. This study takes it place in the con- text of neutral beryllium for which we test two particular strategies:

one based on a weight criterion and one based on the type of excita-

tions. Before concluding, we expose some developments combining the

self-consistent field ( ^SCF ) process and the biorthonormal condition to

relax the orthogonality constraints that are presently applied to the

optimization process of the spin-orbital basis.

P U B L I C AT I O N S

Using the conventional methodology:

• P. Jönsson, S. Verdebout, and G. Gaigalas. Spectral properties of Sb IV from MCDHF calculations . Journal of Physics B: Atomic, Molecular and Optical Physics, 45 (16):165002, 2012.

• J. Grumer, S. Verdebout, P. Jönsson, G Gaigalas and M. Gode- froid. Spectral properties of Sn II from MCDHF calculations . (work in progress)

• C. Nazé, S. Verdebout, P. Rynkun, P. Jönsson, G. Gaigalas and M. Godefroid. Isotope shifts of beryllium-, boron-, carbon- and nitrogen-like ions from relativistic configuration interaction calcu- lations . Atomic Data and Nuclear Data Tables (work in progress)

• S. Verdebout, C. Nazé, P. Rynkun, P. Jönsson, G. Gaigalas and M. Godefroid. Hyperfine structures and Landé g _j factors of be- ryllium-, boron-, carbon- and nitrogen-like ions from relativistic configuration interaction calculations . Atomic Data and Nuclear Data Tables (work in progress)

Using the methodology developed in our thesis:

• S. Verdebout, P. Jönsson, G. Gaigalas, M. Godefroid and C. Froese Fischer. Exploring biorthonormal transformations of pair-correla- tion functions in atomic structure variational calculations . Jour- nal of Physics B: Atomic, Molecular and Optical Physics, 43 :074017, 2010.

• E. Cauët, T. Carette, C. Lauzin, J. Li, J. Loreau, M. Delsaut, C. Nazé, S. Verdebout, S. Vranckx, M. Godefroid, J. Liévin and N. Vaeck. From atoms to biomolecules: a fruitful perspective . Theo- retical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), 131 (8):1-17, 2012-08-01.

• S. Verdebout, P. Rynkun, P. Jönsson, G. Gaigalas, M. Gode- froid and C. Froese Fischer. A partitioned correlation function approach using the Multiconfiguration Hartree-Fock method . Jour- nal of Physics B: Atomic, Molecular and Optical Physics, 2012.

(in preparation)

• P. Rynkun, S. Verdebout, P. Jönsson, G. Gaigalas, M. Godefroid

and C. Froese Fischer. A partitioned correlation function approach

for targeting the 2s ² 2p ² P ^o −2s2p ^{2 4} P transition in neutral boron .

(in preparation)

A C K N O W L E D G M E N T S

Avant toute chose, je voudrais profondément remercier Michel Gode- froid. Ce fut un réel plaisir de pouvoir côtoyer une personne d’une telle humanité durant les quatre années de ma thèse. Je le remercie égale- ment pour sa patience, son écoute, son humour, sa disponibilité et sa volonté de prendre le temps de faire les choses. Cette expérience aura été des plus enrichissantes aussi bien d’un point de vue scientifique que personnel.

It is a real pleasure to acknowledge Per Jönsson who was constantly present all along my PhD thesis. I will never forget the enthusiasm that you shown during each of my wonderful visit in Malmö. I really appreciated all the work and relaxation moments that we shared during these four years. I am specially grateful for all our fruitful discussions on the developments of the computer programs.

I would also like to thank Gediminas Gaigalas and Pavel Rynkun for the useful and interesting exchanges and collaborations that we had all along this PhD thesis. I am really glad to have benefited from all the experience of Gediminas Gaigalas in angular algebra and code develop- ment.

I gratefully thank Charlotte Froese Fischer for the constructive ex- changes that we had during these four years. It was a real pleasure to benefit from all her knowledge and expertise in variational atomic structure calculations.

I specially thank Jiguang Li with whom it was a real pleasure to have daily discussions and collaborations. It was really nice to share some time with you during all our common conferences and collaboration trips.

Je voudrais également remercier Georges Destrée sans qui le développe- ment et l’optimisation de beaucoup de nos programmes informatiques n’auraient pu aboutir. Je le remercie sincèrement pour sa patience et la pédagogie dont il a fait preuve tout au long de ma thèse.

Je remercie Nathalie Vaeck et Michel Godefroid de m’avoir fait con- fiance pour encadrer les séances d’exercices liées à leurs activités d’en- seignement.

Je remercie Brigitte pour sa gentillesse, sa cuisine et nous avoir prêté Michel pendant de longs moments ces dernières semaines.

Cette aventure n’aurait pu être aussi agréable sans le concours de

tous les membres du Service de Chimie Quantique et Photophysique.

Tout d’abord Cédric, mon compagnon de thèse, avec qui ce fut un réel bonheur de partager ces quatre années de doctorat. Je tiens à le remercier sincèrement pour toutes nos discussions ainsi que pour tous les fous rire que nous avons pu avoir sur des sujets aussi vastes que l’inoubliable Scorpion Kick, le teunnis, etc... Merci à Maxence pour toutes ses blagues décalées, sa musique chinoise et ses conseils pour Diablo. A Thomas et Jérôme, les Anciens, je vous remercie pour votre accueil, vos conseils, votre sens de l’humour et pour tous les moments passés devant Civilization, le football américain ou autour de la table de ping-pong.

Je tiens également à remercier mes collègues expérimentateurs “de l’autre bout du couloir”. Merci à Keevin pour toutes les vidéos improb- ables qu’il arrive à dénicher sur internet ainsi que pour ses connaissances encyclopédiques qui nous ont permises de devenir plus d’une fois cham- pion d’or ! Je remercie Clément pour sa motivation quotidienne, sa bonne humeur et ses anecdotes atypiques.

Je tiens également à remercier Stéphane pour toutes les promenades, et les discussions en découlant, partagées en direction de Watermael- Boitsfort. Je le remercie également pour son aide dans la relecture de ma thèse.

Je n’oublie pas Badr, Sophie, Emilie, Lieven, Xavier, Valentin, Tomas, Darius, Jean-Lionel, Yasmine, Julie, Yasmina, Marcella, Martin, Robin, Catherine, Thomas, Ariane et Séverine avec qui j’ai eu la chance de partager d’agréables moments.

Merci aussi pour le soutien apporté par les amis de l’université et d’ailleurs. Merci à Laurent, François, Thomas, Johan, Sophie, Cédric, Julien, Dominique, Max, Franck, Nico, Anne, François, Annick, Manon, Antonello, Pierre, et les autres.

Je tiens profondément à remercier mes parents pour leur soutien per- manent, pour avoir toujours cru en moi et pour avoir éveillé en moi l’envie de comprendre les choses qui m’entourent. Je remercie également mon frère, Thomas, ainsi que sa compagne, Marie, et ma filleule, Rose, pour leur motivation sans limite, leurs conseils et leur aide dans la relec- ture de ma thèse. Je voudrais également remercier mes grand-parents pour leur intérêt constant porté à mon travail ainsi que pour leur sou- tien. Je remercie tout particulièrement Bonne-Maman de m’avoir aidé à préparer l’examen d’admission. Je n’oublie pas Ariane, Jacques et Juliette qui m’ont encouragé durant toutes ces années.

Et enfin, je remercie sincèrement la personne avec qui je partage ma vie, Amélie, pour son soutien quotidien, sa patience ainsi que pour la confiance indéfectible qu’elle me porte. Elle a su trouver les mots pour me remotiver dans mes moments de doute et me redonner confiance.

Je tiens enfin à remercier le Fonds pour la formation à la Recherche

dans l’Industrie et dans l’Agriculture (F.R.I.A.) pour son soutien fi-

nancier à la réalisation de ce travail.

C O N T E N T S

1 i n t ro d u c t i o n 1

i u s e f u l c o n c e p t s 5

2 h a rt r e e - f o c k a n d i t s n at u r a l m u lt i c o n f i g u -

r at i o n a l e x t e n s i o n 7

2.1 Introduction . . . . 7

2.2 The variational principle . . . . 8

2.3 The Hartree-Fock model . . . . 9

2.4 The Multiconfiguration Hartree-Fock model . . . . 13

2.4.1 Some numerical considerations . . . . 14

2.4.2 A useful property of some MCHF wave func- tion: invariance under orbital rotations . . . . . 16

2.5 Computation of atomic properties . . . . 17

2.5.1 Isotope shifts . . . . 18

2.5.2 Fine structure - Breit-Pauli approximation . . . 21

2.5.3 Hyperfine structure . . . . 24

2.5.4 Transition probabilities . . . . 28

3 a f u l l r e l at i v i s t i c a p p roac h - t h e d i r ac - h a rt r e e - f o c k a p p roac h 33 3.1 Introduction . . . . 33

3.2 The Dirac-Hartree-Fock equations . . . . 36

3.3 Computation of atomic properties . . . . 39

3.3.1 Isotope shift . . . . 39

3.3.2 Hyperfine-structure . . . . 40

3.3.3 Landé g _J -factors . . . . 41

3.3.4 Transition parameters . . . . 41

ii u s i n g t h e “ c l a s s i c a l” c o d e s 43 4 a t wo va l e n c e e l e c t ro n s y s t e m - s b i v 45 4.1 Introduction . . . . 45

4.2 Calculations . . . . 46

4.2.1 Spectrum . . . . 46

4.3 Results and discussion . . . . 47

4.3.1 Spectrum properties . . . . 47

4.4 Targeting the five lowest states . . . . 56

4.4.1 Transition probabilities . . . . 56

4.4.2 Isotope shifts . . . . 58

4.4.3 Hyperfine-induced transition . . . . 59

iii b e y o n d t h e o rt h o g o n a l i t y c o n s t r a i n t 61 5 t h e o rt h o g o n a l i t y c o n s t r a i n t 63 5.1 Introduction . . . . 63

5.2 How the orthogonality constraint modifies a many-electron

wave function? . . . . 64

5.3 The Löwdin method . . . . 66

5.4 The biorthonormal transformation . . . . 67

6 t h e pa rt i t i o n e d c o r r e l at i o n f u n c t i o n s i n t e r - ac t i o n a p p roac h 75 6.1 Introduction . . . . 75

6.2 What is the Partitioned Correlation Function Interac- tion method? . . . . 76

6.2.1 A Partition Correlation Function . . . . 76

6.2.2 The interaction problem . . . . 81

6.3 The constraint problem - deconstraining the interaction space . . . . 83

6.4 Implementation of the Partitioned Correlation Function Interaction method . . . . 90

6.4.1 lscud . . . . 90

6.4.2 biopair . . . . 93

6.4.3 bpci_pair . . . . 94

6.4.4 matrix_mult & matrix_construct . . . . 94

6.4.5 diag_tri . . . . 95

7 t h e p c f i m e t h o d : d e v e l o p m e n t s a n d a p p l i c a - t i o n s 97 7.1 Neutral beryllium - a “pair” approach . . . . 97

7.1.1 The three lowest terms of Be I - from PCFI to DPCFI . . . . 97

7.1.2 Partially deconstrained wave functions . . . . . 102

7.2 Neutral lithium - core-polarization effect . . . . 114

7.2.1 A global core description . . . . 114

7.2.2 A PCF dedicated to core-polarization . . . . . 116

7.3 Neutral lithium - spectrum calculation . . . . 122

7.4 Neutral neon - coupling different LS terms . . . . 125

7.5 Neutral boron - Quartet-Doublet transition energy . . . 130

8 n o n - o rt h o g o n a l m c h f : a r e ac h a b l e d r e a m ? 137 8.1 Introduction . . . . 137

8.2 Non-orthogonal MCHF equations . . . . 138

8.3 New possible strategies . . . . 144

8.3.1 Fully variational PCFI approach . . . . 145

8.3.2 The growing onion picture . . . . 145

9 c o n c l u s i o n s a n d p e r s p e c t i v e s 147 iv a p p e n d i x 151 a s i n g l e o r d o u b l e e xc i tat i o n ? 153 a.1 A single excitation 2s → 3d ? . . . . 155

a.2 A double excitation (2s2p) → (2p3d) ? . . . . 155

b g e n e r a l i z e d o r n o t ? 157

c d e ta i l e d t e s t ru n 161

b i b l i o g r a p h y 175

A C R O N Y M S

ASF Atomic State Function . . . . 13

ATSP ATomic-Structure Package . . . . 1

CAS complete active space. . . . 17

CI configuration-interaction. . . .13

CSF configuration state function. . . .10

cud closed under de-excitation . . . . 72

cude closed under de-excitation and excitation . . . . 139

DPCFI Deconstrained Partitioned Correlation Functions Interaction 84 EAL Extended Average Level . . . . 38

EM electromagnetic . . . . 24

EOL Extended Optimal Level. . . .38

FS field shift. . . . 18

GRASP General-purpose Relativistic Atomic Structure Program . . . . . 2

HF Hartree-Fock . . . . 1

KG Klein-Gordon . . . . 33

MCDHF multiconfiguration Dirac-Hartree-Fock . . . . 2

MCDF-gme MultiConfiguration Dirac-Fock-General Matrix Elements22 MCHF multiconfiguration Hartree-Fock . . . . 1

MS mass shift . . . . 18

MR multi-reference . . . . 78

NMS normal mass shift . . . . 19

QED quantum electrodynamic . . . . 39

PCF Partition Correlation Function. . . . 3

PCFI Partitioned Correlation Functions Interaction . . . . 3

RAS restricted active space . . . . 17

RCI relativistic configuration-interaction . . . . 38

RIS relativistic isotope shift. . . . 40

SCF self-consistent field . . . . 4

SMS specific mass shift . . . . 19

SMMP sparse matrix multiplication package . . . . 94

SD Slater determinant. . . .2

1

I N T R O D U C T I O N

The study of the atoms, the building blocks of any material or molecule, was one of the leading fields of physics in the past century. The under- standing of these small systems and their interaction with the light, was one of the main motivations underlying the great revolution of the physical sciences. With the birth of the quantum mechanics and the spe- cial relativity theories, the physicists were ready to tackle the general description of an atomic system.

Unfortunately, even if we keep aside the special relativity to consider the non-relativistic Schrödinger equation, only few particular systems may be described by an analytical solution. Among these systems, we count the hydrogenic atoms with their unique electron. All the other atomic systems require the use of techniques which provide approxima- tions of the analytical solution. The leading difficulty of these many- electron systems arises from the Coulomb interaction between the elec- trons. Unfortunately, the so-called electronic correlation associated with the Coulomb interaction plays an important role to forecast the prop- erties of an atom. As a direct consequence, it cannot, in general, be neglected.

In the early days of quantum mechanics, Hartree developed a tech- nique based on the variational principle to approach the wave functions associated with an atom. After that Vladimir Fock pointed out that the Hartree method was not respecting the antisymmetry principle, the famous Hartree-Fock ( ^HF ) technique provides an approximation of the N -body problem taking into account the antisymmetry principle. The

HF technique, which is used in many different fields of physics such as nuclear and solid state physics, is presented in chapter 2. In 1969, Froese Fischer, a former student of Hartree, published her original computer program adopting the multiconfiguration Hartree-Fock ( ^MCHF ) method in the seminal first volume of the Computer Physics Communications journal. This program is still used today to approach the solution of the Schrödinger equation.

Due to the growing power of the computers and the development of

the methodology, different versions of the ^MCHF program, embedded in

the ATomic-Structure Package ( ^ATSP ), have been published along the

last forty years. The last version of this package, released in 2007 [52],

allows to compute many different spectroscopic properties using the ab

initio many-electron wave functions. A non-exhaustive overview of these properties together with their physical origin is presented in the second part of chapter 2.

Simultaneously to the development of the Hartree numerical method, Dirac proposed his famous equation which combines quantum mechan- ics and special relativity. This equation, which naturally includes the coupling between the spin and the orbital angular momenta, provides an accurate description of the fine-structure splitting of the spectroscopic terms. As for the Schrödinger equation, only the single-electron systems can be solved analytically, as briefly described in chapter 3. Parpia and Grant with the collaboration of Froese Fischer transposed the Hartree- Fock methodology to the more complicated case of the Dirac equa- tion. In 1996, they published together the first version of the General- purpose Relativistic Atomic Structure Program ( ^GRASP ) package in which they provide a computer program treating the multiconfiguration Dirac-Hartree-Fock ( ^MCDHF ) equations, presented in chapter 3.

As the ^ATSP , the ^GRASP package is in permanent evolution and its last version will be available soon [77]. To understand the inherent prob- lems associated with the current methodology and to test the limits of the new ^GRASP package, we, in collaboration with Jönsson and Gaigalas, initiated a series of calculations targeting elements with an intermedi- ate atomic number. Our first results, concerning the three times ion- ized antimony, are presented in chapter 4. For this particular atomic system, we compute both the relative positioning of the sixty lowest energy levels in the aim of evaluating transition energies and probabili- ties, and more specifically target the five lowest states for studying the isotope shifts and one hyperfine-induced transition. The strategy used to perform these calculations, which are described in details in [79], is currently adopted for investigating the isoelectronic sequence of neutral indium.

Even if the reported spectrum calculations on the Sb IV provide

transition energies which are in majority only few hundreds cm ⁻¹ away

from the reference values, it is clear that the use of a common orthog-

onal orbital basis is a limiting factor. To allow the use of specifically

optimized orbital bases and representing different but interacting states,

it is necessary to deal with non-orthonormal spin-orbital bases. Chap-

ter 5 exposes two methodologies allowing the evaluation of matrix ele-

ments involving non-orthonormal spin-orbitals. The first one is the well-

known Löwdin cofactor method which extends the conventional Slater-

Condon rules to the non-orthonormal case. Even if this methodology is

used in different renowned computer codes, it requires the use of Slater

determinants ( ^SDs ) as a basis which is not naturally adapted to the to-

tal spin and spherical symmetries. The second one is the biorthonormal

transformation method which modifies the representation of the two

states involved in a matrix element to allow the use of the conventional

Fano-Wigner-Racah algebra. We choose to base our methodology on

this biorthonormal transformation method since it does not require the

use of ^SDs and avoids the search of the explicit expression of Racah states in ^SDs . Nevertheless, the price to pay is the loss of the left-right exchange symmetry of the radial integrals.

Using the efficient biorthonormal transformation to deal with the non- orthogonalities, we developed the Partitioned Correlation Functions Interaction ( ^PCFI ) method. It consists in expressing a many-electron wave function as a superposition of Partition Correlation Functions ( ^PCFs ), each of them bringing a spin-orbital basis optimized on a given correlation effect. After the definition of the useful notations for the se- quel, we provide our methodology in chapter 6. During the development of our work, we faced an issue that we named the “constraint” problem.

This difficulty led us to adapt our initial strategy by “deconstraining”

the ^PCFs appearing in the expression of the many-electron wave func- tion. The end of chapter 6 consists in a brief description of the different computer programs developed during our thesis to allow the use of the

PCFI method with an arbitrary degree of deconstraint.

To test this ^PCFI approach, we consider neutral beryllium, the sim- plest atom for which three kinds of correlation appear: core, core-valence and valence. Targeting the three lowest states of this particular atomic system, we test the convergence pattern of many different properties such as the total energy, the specific mass shift, the hyperfine parame- ters and the oscillator strength. In the aim of pushing our understand- ing of the constraint effect, we test different possible schemes for decon- straining the ^PCFs . A systematic study of the evolution of the previously mentioned properties with respect to different strategies of deconstraint is proposed. This particular atomic system has been our “laboratory”

during all our thesis.

Another benchmark of the ^PCFI method is then realized by focusing on the lithium atom. To represent the two lowest states of this three electron system, we modify the strategy used for the beryllium atom to determine the ^PCFs . In the aim of controlling the convergence of the hyperfine parameters, we dedicate a ^PCF for capturing the polarization effects associated with the single excitations from the core. Combining this particular function with the other ^PCFs targeting the double and triple excitations, we are able to improve significantly the rate of conver- gence of the spectroscopic properties associated with each state. This ability of the ^PCFI approach to efficiently capture the desired correlation effects opens new perspectives for atomic structure calculations.

To prove the usefulness of the biorthonormal transformation and the

ability of the ^PCFI method to capture the correlation effects more effi-

ciently, we propose two additional test cases. The first one shows the effi-

ciency of the ^PCFI method in a spectrum calculation. By considering the

five lowest levels of a Rydberg series of neutral lithium, we compare the

absolute energies obtained using the ^PCFI and the conventional ^MCHF

methods. In the second one, we consider the interaction between two in-

dependently optimized terms, in neutral neon. These states, which are

expressed on a basis of specifically optimized ^PCFs , are coupled through the Breit-Pauli Hamiltonian, to determine the levels of a given total an- gular momentum. For both of these tests, the ^PCFI method improves significantly the convergence rate of the considered properties.

We finally use the ^PCFI method to study a particular spin-forbidden transition in neutral boron. Since that transition has never been ob- served experimentally, the transition energy has been estimated by using data fitting along the isoelectronic sequence and by theoretical atomic structure calculations. Even if these two methodologies provided results in very good agreement, a more recent work using the data fit- ting technique revised significantly the transition energy. We present some preliminary results associated with that particular transition. In order to estimate the transition energy, we use the ^PCFI method to get an accurate representation of both levels. The relativistic shift is then evaluated by including the non-fine structure operators appearing in the Breit-Pauli Hamiltonian.

All the developments realized during our thesis lead us to imagine a generalization of the ^MCHF method to the case of non-orthonormal orbitals. Even if such an attempt was made in the past, the resulting methodology suffered from several limitations which make it difficult to use in practice. By adopting the biorthonormal transformation during the self-consistent field ( ^SCF ) process, we are convinced that it is possi- ble to relax the orthonormality constraint applied between spin-orbitals belonging to the same symmetry. In chapter 8, we expose the methodol- ogy that we imagine to tackle the problem of a general non-orthogonal

SCF calculation. We finish this chapter by proposing two computational

strategies that the existence of such a methodology would make possi-

ble.

Part I

U S E F U L C O N C E P T S

2

H A RT R E E - F O C K A N D I T S N AT U R A L M U LT I C O N F I G U R AT I O N A L E X T E N S I O N

2.1 i n t ro d u c t i o n

In this chapter we present the theory underlying our non-relativistic calculations. As stressed in the general introduction, the stationary Schrödinger equation ¹





N

X

i=1

p ² _i 2m e

− Ze ² 4π 0 r i

+

N

X

i>j

e ² 4π 0 r ij



 |Ψi = E |Ψi , (2.1) related to a N -electron system cannot be solved exactly. Equation (2.1) introduces the N -electron wave function |Ψi , on which we will come back below, and the Hamiltonian ² obtained under the assumption that the nucleus may be treated as a point charge of infinite mass. The first summation corresponds to the one-electron (one-body) operators in which the first term corresponds to the kinetic energy of each elec- tron ( p i is the momentum of each electron and m e the electron mass) and the second one (where r _i is the distance between the electron i and the nucleus and 0 the vacuum permittivity) is the potential energy of each electron in the Coulomb field of the nucleus of charge ( +Z ). The second summation over i and j introduces the potential energy due to the Coulomb interaction between two different electrons, r ij represent- ing the distance between the electrons i and j . This last part of the Hamiltonian operator represents the main difficulty in the description of many-electron systems since it introduces a correlation between the position of each electron [30, 18, 35].

In this work, we follow the well-known development proposed by Froese Fischer et al. [52] in the ATomic-Structure Package ( ^ATSP ). As mentioned in the introduction, these authors adopt a general method- ology allowing the treatment of any N -electron system. To make it possible they use spin-orbitals φ _α defined as

φ α (q) = 1

r P (nl; r)Y _lm

(Ω)χ

2

m

(σ) , (2.2)

1 We will refer it as the Schrödinger equation since we will always consider a stationary problem

2 In atomic units we have e = m

= ¯ h = 4π

= 1

where one may recognize the spherical harmonics Y lm

(Ω) , the spin χ

2

m

function and the radial function P (nl; r) satisfying the boundary con- ditions

P(nl; 0) = 0 P (nl; r) −−−→

r→∞ 0 .

We introduce the contracted variable q associated with the spatial space r and the spin space σ ( q = (r, σ )) and the contracted label α = nlm _l m _s . This assumption may be linked to the well-known independent particle model. The underlying idea of such a model is that each electron is moving in an effective spherically symmetric potential created by the nucleus and all the other electrons [18, 92]. Adopting it, we may rewrite the N -electron Hamiltonian appearing in (2.1) as

H = H ^cf + ∆ , (2.3)

where the central field Hamiltonian H ^cf (r, p) =

N

X

i=1

− 1

2 ∇ ² _i − Z

r _i + V (r i )

, (2.4)

corresponds to the independent particle model including pure radial potentials V (r i ) [18, 92]. This central field hypothesis leads to a tremen- dous simplification since the eigenfunctions of the Hamiltonian (2.4) can be written as an anti-symmetric product of N spin-orbitals (2.2) that constitutes a ^SD

ψ(q ₁ , . . . , q _N ) = 1

√ N !

φ _α

(q ₁ ) φ _α

(q ₁ ) · · · φ _α

(q ₁ ) φ _α

(q ₂ ) φ _α

(q ₂ ) · · · φ _α

(q ₂ )

... ...

φ α

(q N ) φ α

(q N ) · · · φ α

(q N )

. (2.5)

From this central field Hamiltonian, we can define the perturbation ∆ corresponding to the remaining part of the complete non-central [18]

Hamiltonian (2.1)

∆(r, p) =

N

X

i>j

1 r _ij −

N

X

i=1

V (r i )

. (2.6)

In the following section we will successively present the ^HF model, cor- responding to the best way of constructing the radial functions P (nl; r) , and the ^MCHF model which goes beyond this central field approxima- tion [18]. Before presenting these two models, we should present the fundamental theory that underlies them: the variational principle.

2.2 t h e va r i at i o n a l p r i n c i p l e

Two main methods are used nowadays to approach a many-electron

wave function:

• the perturbation method which uses the exact solution of another similar but solvable problem as a starting point;

• the variational method which finds, within a selected family of solutions, the closest one to the real solution.

The multiconfigurational approach that we will extensively use in this work is based on the variational method.

This methodology requires the expression of an energy functional E[χ]

formally written as E[χ] = hχ|H|χi

hχ|χi , (2.7)

which remains valid for any state χ and any Hamiltonian operator H . Using the Ritz Theorem expressing that the mean value of the Hamilto- nian H is stationary in the neighborhood of its discrete eigenvalues [30], one may show that any state χ is an eigenstate of the Hamiltonian op- erator H if and only if the functional E is, on the first order, stationary for this state. We show the equivalence between solving the Schrödinger equation - for bound states - and finding the state χ that leaves the en- ergy functional E stationary for any variation δχ of state χ satisfying the boundary conditions. Let us take a variation of the energy functional

δE hχ|χi = hδχ|H − E[χ]|χi + hχ|H − E[χ]|δχi (2.8)

= 2hδχ|H − E [χ]|χi . (2.9)

If the energy functional E is stationary then δE = 0 and since this is correct for any variation of the state χ we find

(H − E[χ])|χi = 0 . (2.10)

It demonstrates that the energy functional is stationary if and only if the state χ is an eigenfunction of H and E[χ] is the corresponding eigenvalue [30, 18].

2.3 t h e h a rt r e e - f o c k m o d e l

This model, presented in many textbooks [18, 92, 124, 44], is at the origin of the definition of the correlation energy proposed by Löwdin [96]

E ^corr = E ^exact − E ^HF , (2.11)

where E ^exact does not correspond to the observed energy but to the ex- act solution of the Schrödinger equation (2.1). The Hartree-Fock model directly follows from the variational principle and the independent par- ticle approximation.

The detailed study of the expression of the Hamiltonian in (2.1)

brings useful observations on the many-electron wave function that we

try to evaluate. The latter should indeed be simultaneously an eigen-

function of several operators [18, 92] which are:

• the parity operator Π with the corresponding eigenvalue Π = even ( +1 ) or odd ( −1 );

• L ² and L z associated with the total angular momentum L = P N

i l i with the eigenvalues ³ L(L + 1) and M L ;

• S ² and S _z associated with the total spin momentum S = P N i s _i with the eigenvalues ³ S(S + 1) and M _S .

The ^SD defined previously disregards these two last conditions. For this reason we build a new function named configuration state function ( ^CSF ), written |Φ(γ LM _L SM S Π)i and characterized by the quantum numbers γ , L , M _L , S , M _S and Π . Even if M _L and M _S are good quantum numbers, we will often omit them, keeping the more compact notation

|Φ(γ LSΠ)i , since the corresponding energies are degenerated thanks to the commutation laws [H, L _± ] = [H, S _± ] = 0 with L ± = L x ± iL y and S ± = S _x ± iS _y . Note that γ is associated with a collection of quantum numbers that determines the ^CSF unambiguously, typically the ones specifying the spin-angular coupling tree. A ^CSF may be constructed following two equivalent ways: either expressed as a linear combination of ^SD belonging to the same configuration or using expressions based on the angular momenta algebra [35, 81, 92].

Due to its properties, it is natural to use a ^CSF as the Hartree- Fock trial wave function. Since a ^CSF is expressed by means of a spin- orbitals (2.2) basis, for which the angular and the spin parts are fixed, the set of radial functions {P(nl; r)} remains undetermined. These func- tions are defined by the requirement that the energy functional (2.7) associated with a given ^CSF is left stationary for any variation. The energy functional can be written as a linear combination of one- and two-electron integrals [44, 119].

• One-electron radial integrals I(a, b) = − 1

2 Z ∞

0

P (n a l; r) L P(n b l; r) dr , (2.12) with the following differential operator

L = d ² dr ² + 2Z

r − l(l + 1)

r ² , (2.13)

where one may recognize the centrifugal potential in the last term.

The condition l a = l _b = l comes from the integration over the spin- angular coordinates, and the “natural” orthogonality property of the spherical harmonics.

• Two-electron radial integrals are defined using the development of the 1/r _ij operator over the renormalized spherical harmonics

1 r ij

=

∞

X

k

r ^k _<

r ^k+1 _>

k

X

q=−k

(−1) ^q C _−q ^(k) (Ω _i )C _q ^(k) (Ω _j ) , (2.14)

3 In atomic units.

where r < and r > correspond respectively to the smallest and the largest component between r _i and r _j . The use of this development leads to the definition of the R ^k Slater integrals

R ^k (ab, cd) = Z Z ∞

0

P (n _a l _a ; r)P (n _b l _b ; s) r ^k _<

r ^k+1 _> P (n _c l _c ; r)P (n _d l _d ; s) drds , (2.15) for which we distinguish some particular cases corresponding to the direct F ^k (a, b) = R ^k (ab, ab) and the exchange G ^k (a, b) = R ^k (ab, ba) integrals [44]. Once again the integration over the spin- angular coordinates gives rise to some conditions on the possible values for each l i and k. From the Wigner-Eckart theorem

hl _a m _l

|C _q ^(k) |l _c m _l

i = (−1) ^l

^−m

l a k l c

−m _l

q m l

!

hl _a ||C ^(k) ||l _c i , (2.16) one can find the value of the reduced matrix element

hl _a ||C ^(k) ||l _c i = (−1) ^l

p

(2l a + 1)(2l c + 1) l _a k l _c 0 0 0

!

. (2.17) The last 3-j symbol introduces the constraint that l a , l c and k should satisfy the triangular relation |l _a − l c | ≤ k ≤ l a + l c and that the sum l _a + k + l _c should be even. Similar conditions appear on l _b , l _d and k [18, 44].

The weights of these radial integrals are fixed by the spin-angular inte- gration and the occupation numbers of each subshell (nl) in the consid- ered configuration state function. Some rules have been established to allow a systematic treatment for building the analytical expression of the energy functional. Some, as the one proposed by Racah [110, 81], require that the spin-orbitals form an orthonormal basis set

hφ _a |φ _b i = δ n

,n

δ l

,l

δ m

,m

δ m

,m

, (2.18) where even if the three last Kronecker’s delta are “naturally” satisfied, thanks to the properties of spherical harmonics and spin functions, the first one δ n

,n

is not. Others, as the one proposed by Löwdin [95], do not work on this assumption with the unfortunate consequence that the resulting rules are much harder to apply.

Due to the natural link between the Racah algebra [110] and the

angular momenta coupling techniques used to express a given ^CSF , we

impose the orthogonality of the spin-orbital basis. Thanks to the proper-

ties of the spherical harmonics and the spin functions the orthogonality

should be only imposed on the radial functions {P (nl; r)} that belong

to the same l symmetry. This constraint is included through some La-

grange multipliers ij in the functional on which we apply the variational

method. Expressing the stationary condition of the latter functional for any variation of the radial functions leads to the well-known Hartree- Fock equations

d ² dr ² + 2

r [Z − Y (n _a l _a ; r)] − l(l + 1)

r ² − _n

_l

_,n

_l

P(n _a l _a ; r)

= 2

r X(n a l a ; r) +

m

X

b=1,b6=a

n

,n

P (n _b l a ; r) , (2.19) where Y (n _a l _a ; r) and X(n _a l _a ; r) are respectively the direct and exchange potentials [18, 44, 35]. These radial potentials are constructed using the Y ^k (ab; r) functions defined by

Y ^k (ab; r) = Z r

0

s r

k

P (n a l a ; s)P(n b l b ; s) ds +

Z ∞ r

r s

k+1

P(n _a l _a ; s)P (n _b l _b ; s) ds . (2.20) The Hartree-Fock problem is a system of coupled differential equations that can be solved self-consistently. Starting from an initial guess for the radial functions {P (nl; r)} , we build the associated potentials that fix the system of equations. Solving the latter, we find a new set of radial functions {P ⁰ (nl; r)} that can be used in their turn for updating the potentials. The strategy used for reaching the ^HF solution is summarized in figure 1.

Initial radial functions

Estimate Y (nl; r) , X(nl; r) and Lagrange multipliers

Determine the new radial functions

Test the convergence

Final radial functions no yes

Figure 1: Global structure of the Hartree-Fock algorithm.

The described procedure leads to the determination of the best set of radial functions {P (nl; r)} ,

H ^HF |Ψ ^HF i =

N

X

i=1

− 1

2 ∇ ² _i − Z

r _i + V ^HF (r _i )

|Ψ ^HF i , (2.21)

which fixes the ^HF wave function noted Ψ ^HF . Even if this wave function is built on an independent particle model, the total energy goes beyond this model by considering the complete Hamiltonian (2.1)

E ^HF = hΨ ^HF | H |Ψ ^HF i = hΨ ^HF | H ^HF + ∆ ^HF |Ψ ^HF i

= E ₀ ^HF + E ₁ ^HF , (2.22) where ∆ ^HF is given by equation (2.6). Accordingly to the perturbation theory we can say that the Hartree-Fock energy contains the contribu- tion of the perturbation ∆ ^HF to the first order or, in another way, that the ^HF equations can be derived by applying the variational principle on the total energy correct to first order.

2.4 t h e m u lt i c o n f i g u r at i o n h a rt r e e - f o c k m o d e l To find a wave function that goes beyond the single configuration Hartree- Fock model, Froese Fischer et al. [52] proposed a method combining the

SCF procedure and the configuration-interaction ( ^CI ) method [35, 44].

The resulting multiconfiguration Hartree-Fock ( ^MCHF ) method, rely- ing on the variational principle, consists in the search of a particular many-electronic wave function, named Atomic State Function ( ^ASF ), de- scribing the atom in a specific electronic state ( γLSΠ ) [48]. The latter function is written as a superposition of configuration state functions

|Ψ(γ _k LSΠ)i =

M

X

i

c ^k _i |Φ(γ _i LSΠ)i , (2.23) and is characterized by the good quantum numbers ⁴ L , S and Π and by other additional quantum numbers γ required to univocally specify the state. Note that theoretically, the ^CSF basis contains an infinite number of elements associated with all the possible bounded and un- bounded [120] configurations characterized by the same set of LSΠ quantum numbers [18, 124, 101]. Practically, we have to truncate the

CSF basis by selecting a subspace of it corresponding to the M elements appearing in the expansion (2.23).

The set of mixing coefficients {c ^k _i } constitutes, with the set of radial functions {P (nl; r)} , all the available degrees of freedom in the ^ASF . These are both determined by applying the variational principle on the energy functional (2.7) using the ^ASF as a trial wave function [35, 44].

Expressing the stationary condition with respect to any variation of the radial functions {P (nl; r)} leads to a differential equation having exactly the same structure than (2.19). Due to the use of an ^ASF as the trial wave function (2.23), instead of a single ^CSF , the resulting direct and exchange potentials are counting contributions arising from all the

CSFs defining the ^ASF .

4 We omit the projections quantum numbers M

and M

.

Expressing the stationary condition with respect to any variation of the mixing coefficients leads to the expression of the configuration- interaction method [18]. The latter is reduced to a matrix diagonaliza- tion problem

Hc ^k = E k c ^k , (2.24)

where

H i,j = hΦ(γ _i LSΠ)|H|Φ(γ _j LSΠ)i . (2.25) Having determined the whole spectrum of the Hamiltonian matrix, one should wisely select the eigenvalue E _k and the corresponding eigenvec- tor c ^k = {c ^k _i } that will determine the ^ASF . Note that we introduce the index k only to distinguish the different eigenpairs {c ^k _i } and E k

associated with the Hamiltonian matrix H .

The ^MCHF equations and the ^CI problem are solved iteratively un- til the self-consistency is reached. In practice, we arbitrarily decide a convergence criterion based on the total energy variation between two cycles; typically we stop the iteration process when the energy value is stable within 10 ⁻¹⁵ E _h . Figure 2 illustrates the structure of the ^MCHF algorithm.

Initial radial functions and mixing coefficients

Estimate Y (nl; r) , X(nl; r) and Lagrange multipliers

Determine the new radial functions

Determine the new mixing coefficients

Test the convergence

Final radial functions and mixing coefficients no yes

Figure 2: Global structure of the multiconfiguration Hartree-Fock algorithm.

2.4.1 Some numerical considerations

Having presented the theoretical ideas that underly the Hartree-Fock

and multiconfiguration Hartree-Fock methods, it is important to present

the main numerical procedures that are used. A complete description

of these procedures can be found in Froese Fischer’s seminal book [44].

We should add that an evolution from a “pure” numerical treatment, based on the finite difference technique, to the use of B -spline basis has been recently initiated by Froese Fischer [47].

To adopt a numerical treatment of a continuous function, i.e. the radial functions P (nl; r) , we should sample it at some selected points r _i that form the radial grid. Such a grid should contain enough points for capturing the entire pattern of the continuous function. Even if many numerical procedures are highly simplified if the points of the grid are equally spaced, the requirement to properly represent the func- tion would request too many points. For this reason, the space between two points of the radial grid is variable: starting from points close to each others near the origin, where the radial function changes rapidly, finishing with spaced points where the function should exponentially reach zero. The resulting grid may be written as

ρ i = −4 + (i − 1)h for i = 1, M max

r _i = e ^ρ

Z , (2.26)

traditionally with h = 1/16 and M _max = 220 . By adopting such a grid, we are able to fulfill the following requirements:

• the ρ variable presents equally spaced points that simplify the numerical treatment;

• the r variable, corresponding to a variably spaced points, allows the treatment of any atomic element, i.e. any atomic number Z . Figure 3 illustrates the radial grid that we use to represent any radial function.

We previously mentioned the usefulness of the Y ^k (ab; r) functions (2.20), not only for building the potentials, which are used for defining the system of differential equations, but also for determining the value of the different two-body radial integrals ( R ^k ). Since these functions should be evaluated at each ^SCF iteration, we should be able to update them efficiently. Direct evaluation, using standard finite difference tech- niques, is a priori not really adapted since one integral over the entire space would be required for each r value of a given Y ^k (ab; r) function.

As suggested by Hartree himself, these functions can be determined by solving a pair of differential equations [44]. Defining the Z ^k (ab; r) functions as

Z ^k (ab; r) = Z ∞

0

s r

k

P (n _a l _a ; s)P (n _b l _b ; s)ds , (2.27) the differential equations

d

dr Z ^k (ab; r) = P (n a l a ; r)P (n b l b ; r) − k

r Z ^k (ab; r) , (2.28)

0 0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2 2.5 3

grid density

r (a

)

Figure 3: Crosses give the position where any radial function is sampled. The line gives the “cumulative density” of points of the grid between 0 and the current point r, i.e. f (r) = #(points ∈ [0, r])/#(points). We see that 40% of the points are found between the origin and r = 1 a 0 .

and d

dr Y ^k (ab; r) = 1 r

(k + 1)Y ^k (ab; r) − (2k + 1)Z ^k (ab; r)

, (2.29) with the boundary conditions

Z ^k (ab; 0) = 0

Y ^k (ab; r) → Z ^k (ab; 0) as r → ∞ , (2.30) fix both the Z ^k (ab; r) and the Y ^k (ab; r) functions. These differential equations may be solved by any numerical method, i.e. the Runge-Kutta method, with a very good accuracy.

2.4.2 A useful property of some MCHF wave function: invariance un- der orbital rotations

This useful property has been extensively used for testing almost every code that we developed during these four years. That property has been extensively presented by Carette in his thesis [23], in which he theoretically studies the behavior of the mixing coefficients with respect to the applied rotation

P ^rot (n ₁ l; r) P ^rot (n 2 l; r)

!

= γ ₁₁ γ ₁₂ γ 21 γ 22

!

P (n ₁ l; r) P (n 2 l; r)

!

, (2.31)

where the transformation matrix γ _ij represents the considered rotation.

This property does not hold for any ^CSF basis but some specific ones,

i.e. the complete active space ( ^CAS ) and restricted active space ( ^RAS ) bases. The latter bases contain all the possible excitations from a given configuration. For such bases, the sets of mixing coefficients {c ^k _i } and radial orbitals {P (nl; r)} are not univocally determined. This implies that a unique many-electron wave function may have several different representations, i.e. different mixing coefficients and radial orbitals [48].

To go around this indetermination, additional constraints may be applied on the radial orbital sets. In the ^MCHF algorithm the radial functions are chosen in such a way that they diagonalize the Lagrange matrix associated with each l -space. Another constraint is used in the density program [17]. The latter code determines the radial functions that diagonalize the density matrix, defining the natural orbitals .

The set of mixing coefficients associated with a given radial function set, are determined through a configuration-interaction problem. Both {c ^k _i } and {P (nl; r)} determine the many-electron wave function. The invariance property ensures that a given many-electron wave function can be represented by different couples of mixing coefficients and radial orbital sets

|Ψ(γ LSΠ)i =

M

X

i

c ^k,MCHF _i |Φ ^MCHF (γ i LSΠ)i

=

M

X

i

c ^k,natural _i |Φ ^natural (γ _i LSΠ)i

=

M

X

i

c ^k,rot _i |Φ ^rot (γ i LSΠ)i . (2.32) In the next chapters, we will intensively use the invariance of a many- electron wave function under some particular transformations.

2.5 c o m p u tat i o n o f at o m i c p ro p e rt i e s

So far we described ways to approximate some eigenfunctions of a given

atomic system using a given model, i.e. non-relativistic Hamiltonian

treating the nucleus as a point charge of infinite mass. This prelimi-

nary step is crucial, as it provides information on the energy value of

each eigenfunction, and therefore information on the transition ener-

gies, within the chosen model. Even if these informations are valuable,

they may suffer from an important deviation from the reality. To go

further, we use the perturbation theory to evaluate corrections to the

first order in energy. The three first subsections display some analytical

expressions allowing us to evaluate these corrections. These subsections

successively consider the isotopic shift, due to the volume and the finite

mass of the nucleus, the fine structure splitting, mainly due to the spin-

orbit coupling, and finally the hyperfine splitting, appearing from the

coupling between the total electronic momentum and the nuclear spin.

To be able to supply other informations of spectroscopic interest, the

ATSP includes a program computing the transition probability condi- tioning the intensity of a given spectral line. The last subsection of this chapter presents some formal expressions relevant for that property.

Since the aim of our thesis is not specifically linked to one of these properties, we limit ourselves to the presentation of the basic theoretical concepts.

2.5.1 Isotope shifts

In previous sections we describe the nucleus as a point charge of infinite mass. These assumptions are, of course, far from the truth. We should consider some corrections to this model to take the finite volume and mass of the nucleus into account. These corrections named respectively the field shift ( ^FS ) correction for the finite volume of the nucleus, and the mass shift ( ^MS ) correction for the finite mass of the nucleus, lead to the notion of isotope shift of an energy level [74]. In other words, the energy value of each level depends on the isotope that we investigate.

First consider, the non-relativistic N -electron Hamiltonian of the fi- nite mass nucleus expressed in the “global” coordinates ⁵ ,

H _M

(ρ, π) = π ₀ ² 2 M _A +

N

X

i

π _i ² 2 m _e −

N

X

i

Z e ² 4π ₀ |ρ _i − ρ ₀ | +

N

X

i<j

e ²

4π 0 |ρ _i − ρ j | , (2.33) where ρ ₀ corresponds to the position of the nucleus and ρ _i for i ≥ 1 is the position of the electron i . The π ₀ is the momentum of the nucleus of mass M A and π i is the momentum of the electron i .

We may then express the same Hamiltonian (2.33) using the mass center coordinates [74]. For the position we find

r i = ρ i − ρ 0 , (2.34)

R = M _A ρ ₀ + m _e P

i ρ _i M A + N m e

, (2.35)

and for the momentum

p _i = −i¯ h∇ _r

, (2.36)

P = −i¯ h∇ _R . (2.37)

5 We do not use the atomic unit (m

= 1) explicitly in that case to avoid any confusion.

Using these expressions, one may write the Hamiltonian in the mass center coordinates

H _M

(r, p) = P ²

2 (M _A + N m _e ) +

N

X

i

p ² _i 2 µ + 1

M _A X

i<j

p i · p j

−

N

X

i

Z e ² 4π 0 |r _i | +

N

X

i<j

e ²

4π 0 |r _i − r j | , (2.38) where µ is the reduced mass of the electron given by

µ = M _A m e

M _A + m _e . (2.39)

In equation (2.38) the first term corresponds to the kinetic energy of the mass center ⁶ . The second one is the kinetic energy of each electron with respect to the nucleus. The two last are the Coulombic potential terms.

Due to the separation between the mass center and the nucleus, a new two-body term, named mass polarization, appears in equation (2.38) [9].

Comparing equations (2.1) and (2.38), the corrections are identified:

1. a scaling factor for the kinetic energy of each electron ₂ ^p _m

→ ₂ ^p

_µ which leads to the normal mass shift ( ^NMS ) correction;

2. a new term, the mass polarisation, in the Hamiltonian

1 M

P

i<j p i ·p _j which defines the specific mass shift ( ^SMS ) correc- tion.

With these corrections, we can express the energy correction with respect to the infinite nuclear mass assumption:

∆E = E _M

− E ∞ = hΨ _M

|H _M

|Ψ _M

i − hΨ|H|Ψi . (2.40) This equation stresses a difficulty: the exact evaluation of the energy shift implies that we know the wave functions for both Hamiltonians. To bypass this difficulty, one may use the perturbation theory to evaluate the energy correction to the first order which is given by the mean value of the perturbation using the un-perturbed wave function (2.22) [59].

Strictly following the perturbation theory , we define the perturbation as

H _M

= H + 1

2 µ − 1 2 m e

p ² _i + 1 M A

X

i<j

p i · p j , (2.41) which leads to the ^MS correction at first order in energy

E ₁ ^MS = hΨ|

1 2 µ − 1

2 m e

p ² _i + 1 M A

X

i<j

p _i · p _j |Ψi . (2.42)

6 This part of the Hamiltonian is not interesting, its wave function corresponds to a

plane wave describing the mass center motion.

Even if this approach has the advantage of treating the normal mass shift ( ^NMS ) and the specific mass shift ( ^SMS ) on the same foot, a solution found by Mårtensson and Salomonson in 1982 [97] goes beyond that.

Indeed, one may find the link existing between the exact solution of the Bohr Hamiltonian

H ^B _M

A

= H _M

− 1 M A

X

i<j

p _i · p _j , (2.43)

and the solution of the infinite nuclear mass Hamiltonian that we have considered so far

E _M ^B

= µ

m _e E ∞ ; Ψ ^B _M

(r) = Ψ( µ

m _e r) . (2.44)

In other words, the wave function is more diffuse for a lighter nucleus and the binding energy is reduced ( _m ^µ

< 1 ). Using the adequate scaling factor

µ m e

= M _A M A + m e

, (2.45)

this methodology allows us to treat the ^NMS to an infinite order. The corresponding energy correction is given by

E _NMS = E _M ^B

− E ∞ =

M _A M A + m e

− 1

E ∞

= − µ M _A E ∞

= − m e

M _A E _M ^B

. (2.46) The specific mass shift correction is then treated, to the first order, using the perturbation theory . Using the mean value of the mass-polarisation term, we define the energy correction as

E _SMS = 1 M _A hΨ ^B _M

A

| X

i<j

p i · p j |Ψ ^B _M

A

i (2.47)

= 1

M _A µ

m _e 2

hΨ| X

i<j

p _i · p _j |Ψi (2.48)

= M _A ¯ h ²

(M A + m e ) ² S _SMS , (2.49) where we have introduced the specific mass shift parameter S SMS [5, 59],

S SMS = −hΨ| X

i<j

∇ _i · ∇ _j |Ψi . (2.50)

The normal mass shift and specific mass shift are then combined to express the total energy mass shift of a particular isotope A using the infinite nuclear mass wave function

E _MS ^A = E _M

−E _∞ ≈ E _NMS +E _SMS = − µ M A

E ∞ + M _A ¯ h ²

(M A + m e ) ² S _SMS .