Isotope effects in atomic spectroscopy of negative ions and neutral atoms: a theoretical contribution

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Isotope effects in atomic spectroscopy of negative ions and neutral atoms:

a theoretical contribution

Faculté des Sciences Département de Physique

Thomas Carette

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences Promoteur: Prof. Michel R. Godefroid

Décembre 2010

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A CKNOWLEDGMENTS

Michel Godefroid has been a great supervisor. His diverse advises and criticisms have been of great value; it is impossible for me to discern all I have learned with him. I bow to him for his patience and respectful involvement in my research. Michel Godefroid has been a great provider of encouragements and enriching trials; I should seek his forgiveness for the few times when I opposed a fierce resistance to his teaching. More practically, I am grateful to Michel for proposing me to pursue this thesis, the intense efforts he made for improving the quality of my work and for his final reviewing of this manuscript.

I send warm thanks to Nathalie Vaeck and Jacky Li´evin who have always been there for listening to any weird idea I could come with during these long years, for listening to me and replying with precious comments.

I want to thank Christophe Blondel and his coworkers, Charlotte Froese Fischer, Per J¨onsson and Meassaoud Nemouchi, for fruitful collaborations. I have to acknowledge the few, but highly prized, words of Charlotte Froese Fischer who helped me with her unique insight into MCHF calculations. I am deeply grateful for the interesting and friendly discussions I had with Messaoud Nemouchi. His openness and kindness made the few weeks he spent in the LCQP enjoyable and productive times.

Well earned thanks go to Oliver Scharf for his help with scripts and computing stuffs. I was pretty unlearned at the beginning of my thesis and his knowledge came very helpful in this regard.

I smile warmly at Georges Destr´ee which provided invaluable help as mediator between human and machine, and as a benevolent King of the computer resources.

Additionally, I would like to thanks the members of the Jury, Pierre Gaspard, Paul-Henri Heenen, Thierry Bastin, Jacky Li´evin, Per J¨onsson and Nathalie Vaeck, for accepting to review this work.

Many people contributed to making the life besides work very enjoyable during these few years.

As a matter of fact, Michel Godefroid, Michel Herman, Nathalie Vaeck, Jacky Li´evin and all permanent researchers of the Laboratoire de Chimie Quantique et Photophysique are great people to be around while working. Supportive and open to laugh, the core of the laboratory is joyful and that is maybe the most important factor that makes the time spent in the LCQP so great.

Of course that is not the only factor. I have a great deal of jumping-around-clapping for all my fellow PhD students of the Wild West Wing office. Jérôme Loreau of the Early Settlement, holding the jukebox in a rather wise way, and caring for the Giant Octopus and Little Abandoned Banana Tree in a rather brave way – thank you. The later followers, Cédric Nazé and Simon Verdebout, respectively the bartender and the town’s appointed stargazer, probably sucked in by the lure of power but still, putting up with my divagations with a graceful indifference for the former and understanding for the latter – nice of you. And the Benjamin, our jolly one, Maxence Delsaut, who finished crowding the place – I’ll clean my mess so you can take my seat.

Keevin, Chief Botanist and Fellow Medium-Tastefully 90’s Lover with Thomas D., Clément “Cari- bou” – I hope you will not miss my daily taunts too much–, Ariane – dashing style, great cookies –, Séverine, Badr, Émilie, Catherine, Stéphane, Jean-Lionel, Tomas,. . . : I am happy to have been around them and all people of the LCQP which, shame on me, I did not mention but who are no less responsible for the good time I had in there.

I am grateful to St´ephane Vranckx for the last reading of this whole thesis.

Love to my parents who, despite their many uncertainties, always gave me all the freedom and support that I needed – you made a great many right choices.

I want also to thank all my family and friends which have their share of responsibility in these

years good developments. There are unfortunately too many to refer to and I can only mentally wrap

them all in an huge hug.

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ii

In particular, I want to thank the few that had the courage to read early versions of one or more chapters of this thesis, having (nearly) no clue of what the heck I was talking about: Veronique and Pierre “Bart” Barthelemy, Nicolas and Mathieu Carette.

I also have to acknowledge all the lubs and the flips and the cha’ahoumis, especially their Queen and King, which not only concretely helped, but kept me from insanity.

I keep my last breath for Julie Lesuisse, my wife, who carried me during my studies and PhD. Of course her daily contributions – either her practical help or her mere existence – have been invaluable and this thesis would probably never have existed without her. Of course. But that is not according her nearly enough credit. There is invaluable and invaluable and what Julie brought and keeps bringing me is no doubt of the other kind.

This work was supported by the Belgian Fonds de la Recherche Scientifique – FNRS through a

F.R.S.–FNRS PhD fellowship and by the Communaut´e Franc¸aise of Belgium (Action de Recherche Con-

cert´ee) under the convention ARC ATMOS 2008-2013.

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A BSTRACT

The spectroscopic study of matter is a source of highly accurate atomic data which still appears inex- haustible after about two hundreds years of lifetime. In particular, negative ions have been broadly studied in the last century for their unique properties, equally as interesting physical systems and as elusive objects. With the progresses of experimental techniques, more and more information are ex- tracted from those highly correlated systems but many aspects of their description still remain obscure.

On the other hand, isotope effects, i.e. effects of the nuclear properties on the electronic struc- ture, are small but highly sensitive to exotic properties of the atomic wave function. The modern understanding of atoms is such that it can be used as a laboratory allowing the study of many aspects of particle interaction; in particular, it permits to study the nuclear structure. However, there are many difficulties to overcome on both experimental and theoretical sides. In this context, the calculation of isotope effects in negative ions brings many valuable information to light, often challenging the experiment or questioning common theoretical methodologies.

Our work is focused on the study of isotope effects in neutral atoms and negative ions of the sec- ond and third periods p − block elements (B-F and Al-Cl), which are the best candidates for highly accu- rate binding energy measurements. By using the Multi-Configuration Hartree-Fock (MCHF) method, we show in the second part of this thesis that for light atoms (A < 40), ab initio calculations are a reliable basis for guiding experimental studies and helping in the analysis of measurements.

One of the initial questions motivating our work concerns the periodicity of isotope shift on the electron affinity throughout the Periodic Table of Elements as a way to explore new aspects of inter- electron correlation. In the final chapter we finally identify the main effects intervening in isotope shift differences between a neutral atom and its negative ion. We also assess the current limitations defining the challenges of tomorrow.

During the course of our researches, as often in such process, we encountered many difficul- ties and made a series of observations concerning common methodologies in MCHF calculations. We mostly searched for an efficient way to account for the delicate balance governing the negative ion stability in so-called valence-correlation models and how to include the corresponding core effects without breaking this equilibrium. Practically, our work has been principally one of building proce- dures.

In many ways, this work is largely an exploratory study; in its raw form, it is the story of an apprenticeship. In this report, we try to restitute our understanding of the subtleties of the method we used and of the systems we studied. For this reason, in the first part of this thesis, rather than presenting a thorough and alienated picture of the fundamental equations and models, we choose to take a more subjective path, focusing on aspects that might seem marginal but, we hope, will guide the interested reader toward the understanding of our thought process.

One haunting aspect of this work that kept nagging at us is the concept of quasi-symmetry of the energy functional, the Brillouin theorem and finite rotations in the space of radial functions defining the one-electron states (orbitals). It is only in the final year of our thesis that we finally could explain many of our observations by identifying an interesting property of MCHF solutions. The context is the so-called CAS (Complete Active Set) model, which corresponding ansatz for the particular case of two occupied l-subshells is written

Ψ = �

τ ν

c

_{τ ν}

�( nl )

^τ+ν

( n

^′

l )

^τ−ν

Γτ νLM

_L

SM

_S

� . (1)

The quantum numbers τ ν characterize an abstract isospin and its projection (Kaniauskas et al., 1987),

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iv

related to the orbital rotations in the many-electron state

�( nl )

^τ⁺^ν

( n

^′

l )

^τ⁻^ν

Γτ νLM

_L

SM

_S

� (2) where LM

L

and SM

S

define the usual orbital and spin total symmetries and Γ contains any other information. It is well known that the problem of the minimization of the energy functional � Ψ � H � Ψ � leads to an indetermination of the one-electron states. Many methods recommend one choice or an- other. One of them, introduced in the context of a two-electron CAS, is the so-called GBT method (for Generalized Brillouin Theorem, see Froese Fischer et al., 1997) which is based on the requirement that a specific c

_{τ ν}

is forced to be zero. In Chapter 2, we generalize this method for a two-subshell CAS with an arbitrary number of electrons, and characterize the sets of solutions arising from each choice of c

_{τ ν}

set to zero. We show that the most faithful generalization of the above method is obtained when the Brillouin function of the main component of the wave function, generally a combination of several states of the type (2), is set to zero. We show that, if the highest excitations are removed from the CAS expansion as is often done in practical MCHF calculations, a series of local minima are expected to appear along the direction of the ( nl, n

^′

l ) orbital rotations, regularly distributed around the recommended GBT solution.

This work provides a series of quantitative results which are summarized below.

Chapter 5 presents calculations of the isotope shifts and hyperfine structures of the lowest multi- plets of S, S

⁻

, Cl, Cl

⁻

, Si and Si

⁻

. For the isotope shift on the electron affinity of sulfur and chlorine, we find a good agreement with the experimental values at both level of closed-core and – to a lesser extent in the case of sulfur – open-core calculations (Carette et al., 2010a; Carette and Godefroid, 2010). We predict the IS on the binding energy of the

⁴

S

^o

,

²

D

^o

and

²

P

^o

levels of Si

⁻

using the same procedure.

In addition, we calculate the hyperfine structure of all studied states, but the

²⁹

Si

⁻

excited terms. We find a good agreement with the available experimental data and predict the HFS for various isotopes of the following states:

³

P

₁

of Si,

⁴

S

^o

of Si

⁻

,

³

P

₂

and

³

P

₁

of S,

²

P

₁^o_�₂

of S

⁻

and

²

P

₁^o_�₂

of Cl. Comparing our calculated electron electric field gradients at the nucleus, we deduce, from the experimental elec- tric quadrupole hyperfine constants, the nuclear electric quadrupole moments values of

³³

S,

³⁵

Cl and

37

Cl that are in excellent agreement with the accepted ones.

Chapter 6 confirms the results of calculations initially performed by Per J¨onsson at Malm¨o on the hyperfine structure of the 1s

²

2s

²

2p

²

(

³

P ) 3s

⁴

P state of nitrogen with calculations of our own. The resulting ab initio hyperfine constants of J¨onsson et al. (2010) are in total disagreement with the ones deduced by Jennerich et al. (2006) from saturation spectroscopy on the 3s

⁴

P

J

→ 3p

⁴

P

_J^o

,

⁴

D

_J^o

transi- tions. By noting that the recorded spectra are basically compatible with the simulated spectra using ab initio calculations if one reassigns the weak observed lines to so-called cross-overs, we deduce a new set of experimental hyperfine constants basing our analysis on this reassignation and experimental data only (Carette et al., 2010b). The resulting “new experimental” hyperfine constants are in agree- ment with the theoretical values. Some inconsistencies of the assignation of Jennerich et al. (2006) disappear but we observe that the weak lines in their spectra are systematically suppressed. The large observed change of specific mass shift on the 3s

⁴

P

_J

→ 3p

⁴

P

_J^o

,

⁴

D

^o_J

transitions from one multiplet to another is confirmed. It is explained by a large J -dependency of the 2s

²

2p

²

(

³

P ) 3s

⁴

P mass po- larization due to a strong J -dependency of the mixing of the 2s

²

2p

²

(

³

P ) 3s

⁴

P and 2s2p

⁴

(

³

P )

⁴

P states.

In Chapter 7, we study the hyperfine structures, isotope shifts, and Land´e factors of carbon

³

P ,

1

D,

¹

S and C

⁻⁴

S

_3�2^o

and

²

D

^o_3�2,5�2

bound states (Carette and Godefroid, in preparation). Furthermore, we calculate the M1 and E2 Einstein coefficients of the corresponding intra-configuration transitions.

The used procedure is aimed at the exact non-relativistic wave functions and we find a good agree-

ment of the absolute state energy with the estimated non-relativistic ones (Chakravorty et al., 1993;

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NIST, 2010). We carefully estimate relativistic corrections to our NR results. We obtain an overall good agreement with experiment for the various detachment thresholds and predict the isotope shift on them. We obtain a very good agreement with experiment for the neutral carbon

³

P fine structure and predict the one of C

⁻²

D

^o

to be 1.75 cm

⁻¹

. Our study of C hyperfine structure leads to accurate hy- perfine parameters, allowing a more robust estimation of the B

₁

(

³

P ) experimental hyperfine constant of

¹¹

C and a determination of the

¹¹

C nuclear quadrupole moment of

Q (

¹¹

C ) = + 0.03336 ( 19 )

exp

( 2 )

th

barns, (3) in good agreement with the estimation of Sundholm and Olsen (1992) of + 0.03327 ( 24 ) barns. Fur- thermore, we predict the hyperfine structure of the A = 11, 13 isotopes of the C

⁻ ⁴

S

^o

and

²

D

^o

states, including the off-diagonal (JJ

^′

) interaction. The obtained M1 and E2 transition probabilities between the J -levels of the carbon lowest configuration are in agreement with previous calculations. We predict the ones for C

⁻

, allowing a determination of the important ratio

r (∞) = 3 2

A (

⁴

S

_3�2^o

→

²

D

^o_5�2

)

A (

⁴

S

_3�2^o

→

²

D

^o_3�2

) = 1.10, (4)

with less uncertainty related to the use of the non-relativistic transition operators than in heavier

iso-electronic systems.

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Theory and methodology

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Chapter 1

The atomic problem

“Do not all fix’d Bodies, when heated beyond a certain degree, emit Light and shine;

and is not the Emission perform’d by the vibrating motions of their parts?” I. Newton

“It seems to me that the deliberate utilization of elementary symmetry properties is bound to correspond more closely to physical intuition than the more computational

treatment” E. P. Wigner.

This chapter introduces the fundamental atomic problem from the point of view of the structure of an atomic state. In Section 1.1 we draw a brief sketch of the context

¹

then in Section 1.2 we lay the basics of the non-relativistic theory of atoms, with emphasis on the symmetry of the wave function.

In Section 1.3 we introduce the basic problems arising from the combination of the electro-magnetic interaction and quantum mechanics and Section 1.4 deals with the Dirac relativistic theory and the Breit-Pauli Hamiltonian. Section 1.5 presents the fundamental principles and equations of isotopic effects. In Section 1.6 finally we can discuss the experimental context of our thesis and some general motivations for our work.

1.1 I NTRODUCTION

Since the birth of philosophy, it has been recognized that some structure arises from the nature of bodies and the laws that govern them. It is more recently that the study of patterns intertwined with calculus. Since then, order has often guided the scientists toward the determination of simplifying laws of physics. It has also always been known that those laws could be amended. In the beginnings of quantum mechanics, the discrete spectra of elements were at the heart of discussions and, a posteriori, it seems rather straightforward that once the basic constituents of matter were identified and the link between energy and light was established, the picture of the atom would necessarily be one of stationary electronic waves

²

– the nature of the spin, however, is quite a twist.

Still, once the bodies – the electronic wave functions – are defined, the question of the laws gov- erning them is still to be answered. The complexity of these laws is such that, in 1926, anyone could have foreseen a seemingly endless path leading from one model to another, from one approximation

1

This section establishes a sort of historical picture of our motivations. For the sake of fluidity, we do not generally back our examples by citations to particular works or reviews but wait for sharper occasions to do it. We should however acknowledge the help that the reading of

Pais

(1986) provided us for contextualizing.

2

Eternal states for discreet energy spectra and waves for the quantization of confined states.

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to another. One of the tasks of the theoretical atomic physicists is to test the models and pursue better approximations.

Since the beginnings of spectroscopy, even before anyone understood the underlying phenom- ena, the spectra were used for elements identification and classification. Once the first successes of quantum mechanics were scored, spectroscopy gained power for the forging of theory itself as a mean to test new aspects of the interaction between particles. When Schr¨odinger published his equation, theoreticians directly faced tremendous difficulties solving it for many-electron atoms. We are still at this stage today: theoreticians trying to have a thorough description of atoms, often challenging the experiment and established theories in one way or another.

The rapidly progressing atomic physics field of the twentieth century soon became very appeal- ing in technology. Nowadays, it is useless to try to distinguish what follows from atomic physics from what does not, nor in frontier physics – like particle physics and fundamental interaction or astro- physics and nuclear physics – nor in the daily life – laser technology is a popular item put forward for praising the practical merits of fundamental atomic physics but there is no more meaning in distin- guishing the modern techniques from the underlying knowledge that there is in awarding the credit of the successive progresses of physics of the early XX

^th

century to separate research branches.

As for the present involvement of theoretical atomic physics in research, it is as varied as can be expected from such an old subject. It is not far from truth to say that the electronic structure is decoded: on the one hand our knowledge of the atom can be used as a basis to forge and study very subtle systems with broad applications (quantum information, fundamental interactions, metrology, new –quantum– phases of matter, etc.), on the other hand the aim of atomic structure calculations is now largely to deduce properties from the existing theory. We only know a fraction of the atomic data relevant for instance in plasma physics and astrophysics with a reasonable accuracy. For the latter point there is the limitation of the abundance of the quantities to investigate – think of the number of possibly interesting states of interesting species times the number of interesting properties – but also the limitations in the accuracy of the methods we use. This thesis registers among the works about this latter problematic.

The success of atom-based sciences in the sphere of research cannot be only ascribed to its old success story or to fashion. The atoms combine many advantages that put them on this pedestal.

First is, as one can consider it, their simplicity. We would rather say that they have the exceptional property to be understandable on the most part with today’s knowledge – it is doubtful that atoms were considered simple back at the beginning of the century. Since their structure is so well understood, any small perturbation of their properties can be identified. Their marked order, the hierarchy of the interactions governing them, is the root of their success. On top of that comes the fact that they are the fundamental building blocks of small molecules, organic compounds, common and exotic materials, etc. The fact that they are neutral and that they interact strongly with each other in so many different ways make them the key of the world as we know it, in particular of life – the beaten competitors would be the light, the nuclear matter, black holes for all we know, etc. Furthermore, atoms have a size that permits to build large quantum systems with long coherence time. There are comparatively few ways to observe manifestations of quantum mechanics in a system without any atom. Today, it seems that atomic physics is the backbone of many fields of sciences: it is everywhere, if sometimes hidden in extremely complex systems and hence not at the center of the attention.

Even if this state of affair is largely dependent on our modern knowledge and technology, it is still amazing that the atoms turn out to be at the same time understandable, abundant and interesting.

It could not have been the case. The nucleus, a structure known for more than a century, is a very

simple example that matters could have been much more troublesome if the fundamental interactions

between particles were but slightly different. It is rightful to say that the nucleus is not decoded, at

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1.1. INTRODUCTION 5 least in the above sense. It is interesting, and not bit frightening, to try to imagine a world where the atom is as complex as the nucleus. If the theoretical methods of nuclear physics could serve the same purpose as the atomic structure calculations, it would mean a new observation window, even richer than the atomic one. It should be a general concern to help the nuclear physicists to attain this goal. From the point of view of the atomic physicist, it is necessary to accumulate the nuclear data, isolate the patterns. Isotopic effects – effects of the nuclear properties on the electronic structure, e.g.

its mass and shape – are a gate leading to important properties of the nucleus and to the very nature of the fundamental interactions. Electron polarization near the nucleus and total impulsion of the electrons, for instance, are to be calculated precisely to be useful in nuclear physics. The theoretical investigation of isotopic effects is very challenging (we will meet the complications soon enough) but the stakes are often similar from one system to another. This thesis is aimed at a better way to calculate those quantities, therefore allowing the consideration of more complex atomic systems and states.

1.1.1 The Schr¨odinger equation

The Schr¨odinger equation is written HΨ

_t

= i� h ∂

∂t Ψ

_t

(1.1)

where H is the Hamiltonian operator and Ψ

t

the time dependent wave-function. Even if Schr¨odinger himself probably didn’t imagine right away the latter extensions of his work, the most advanced equa- tions of motion still have this form, or its corresponding Lagrangian form, i.e. the Dirac-Feynman’s path integral formulation of quantum field theory. We will however hardly have the need of such sophisti- cated models as one does not commit a large error by considering only the effective electromagnetic interaction in atoms – at least as far as it is not the fundamental interactions that are investigated. The light is our principal source of information about the universe and the constituents of stars, interstel- lar gas clouds, etc. As a matter of fact, even in the study of terrestrial atmosphere we rely heavily on spectroscopy – that fact indicates the power and success of atom theory. The feedback is that many atomic properties are determined by measuring how atoms interact with light. Any time-dependent state can be developed on a basis of stationary states Ψ

_η

of energy E

_η

obtained by solving the time independent Schr¨odinger equation

H

0

Ψ

η

= E

η

Ψ

η

→ Ψ

t

= �

_η

c

η

( t ) e

^iE^η^t��^h

Ψ

η

(1.2) where H

₀

is a time-independent part of H (see Cohen-Tannoudji et al., 1977, chap. 13). Quantifying the electromagnetic field leads to a representation of the absorption and emission of one or several photons by an atom as an operator acting in the Hilbert space of H

₀

of which the { Ψ

_η

} is a basis set.

Indeed, applying the perturbation theory to ( H − H

₀

) permits to integrate on the time variable (see Johnson, 2007, Chap. 6). As far as we remain in a perturbative regime, non-dynamical aspects of atomic transitions, photodetachment, photoionization, will be open to studies through the knowledge of the involved atomic stationary states and their properties. In this thesis we therefore only consider the unperturbed states Ψ

η

.

As we explained above, the atomic physics is so powerful only thanks to our deep understand-

ing of the structure of equation (1.2). At least for light atoms, we understand the structure of the

H

0

spectrum, i.e. the set of its eigenvalues E

η

, and the underlying structure of the Ψ

η

. Without this

insight, largely grounded in group theory, it would be impossible to analyze the huge quantity of in-

formation given by spectroscopy. However, this insight is correspondingly conditioned by a knowledge

of the background problem. We still need some theoretical tools to be able to lay firmly the aims and

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Table 1.1: Various constants and expressions which are unity in atomic units (CO- DATA values of Mohr et al., 2008). m and e are respectively the electron’s mass and charge in absolute value. h � is the Planck constant h over 2π. α is called the fine structure constant (dimensionless) and c is the speed of light. ε

₀

is the vacuum permittivity. a

₀

is the Bohr radius. E

_h

units stand for hartrees and are equivalent to the atomic units of energy.

Mass m = 9.109 382 15

(

45

)

10

⁻³¹

kg

Charge e = 1.602 176 487

(

40

)

10

⁻¹⁹

A s

Angular momentum

�

h = 1.054 571 628

(

53

)

10

⁻³⁴

m

²

kg s

⁻¹

Speed

αc=

e

²�(

4πε

0

h

�)

= 2.187 691 254 1

(

15

)

10

⁶

m s

⁻¹

Length a

0=

h

��(

mcα

)

= 5.291 772 085 9

(

36

)

10

⁻¹¹

m Energy mα

²

c

²

[E

h

] = 4.359 743 94

(

22

)

10

⁻¹⁸

m

²

kg s

⁻²

motivations of this work. So let us postpone the discussion about our role in this context still a little and begin by introducing the general atomic problem.

1.2 T HE NON - RELATIVISTIC ATOM

In the standard non-relativistic theory of atoms, there are basically two contributions to the Hamilto- nian. First, the kinetic energy of each particle

T = ppp

²

2M = − � h

²

∇ ∇ ∇

²

2M (1.3)

M being the mass of the particle and h � = h � 2π is the Planck constant. The second contribution is the Coulomb potential arising from the electromagnetic interaction between two point-like particles

V = q

₁

q

₂

4πε

0

� rrr

2

− rrr

1

� , (1.4)

the q’s being the charge of the particles, the rrr’s their position and ε

₀

the vacuum permittivity. The non-relativistic atomic problem is usually solved for an infinite mass, point-like nucleus located at the origin so that we have, by summing on the electrons

H = − h �

²

m �

i

∇

∇ ∇

²i

2 − e

²

4πε

₀

��

i

Z r

_i

− �

i<j

1 � rrr

_j

− rrr

_i

�

��

�� (1.5)

where m and e are the electron mass and charge, respectively. Since the potential and kinetic operators appear preceded by constant, universal factors, it is convenient to introduce units in which those coefficients are equal to one, the atomic units (see Table 1.1).

1.2.1 The hydrogen-like atoms

Due to its internal structure, the electron behaves according to the 1 � 2 irreducible representation of the so ( 3 ) Lie algebra

³

. This property suggests that the electron has an intrinsic angular momentum, a spin. However, SO ( 3 ) , the Lie group of rotations in our tridimensional space, that can be considered

3

In the following, we will assume that the basics of group theory and its link to mechanical problems are familiar to the

reader. An exceptionally rigorous, if quite demanding, treatment of the electron symmetry in the hydrogen atom can be

found in

Cornwell

(1995, Chap XII).

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1.2. THE NON-RELATIVISTIC ATOM 7 as the group leading to the classical angular momentum – the one identified by Bohr (1913) in his model of the hydrogen – does not have even-dimensional irreducible representations. Nor do the spa- tial equations defining the spherical harmonics have solutions for l half integer (Cowan, 1981, chap.

2). The spin can truly be considered as a non-spatial property of the electron, a purely quantic charac- teristic of matter, even if it has some indirect effects on its spatial behavior (see also Section 1.4). The group associated to the spin is SU ( 2 ) which is locally isomorphic to SO ( 3 ) , i.e. su ( 2 ) = so ( 3 ) (Corn- well, 1995). There are myriads of exciting issues concerning the nature of the spin, its origin, its consequences. It is not our aim to discuss them. We will limit ourselves to the observation that an electronic state in a spherical potential is defined as an element of an irreducible representation of SU ( 2 ) × SO ( 3 ) . Four variables have to be used for describing an electron state, equivalent to the three spatial coordinates and the choice of a favored direction, m

_s

. The spatial behavior of quantum matter has profuse analogies with classical mechanics and the resulting continuous functions and differential equations. For its part the spin function describes the intrinsic, quantized states of each particle in two dimensional matrix form. At the non-relativistic level, we have formally

ψ ( r, 1 � 2 ) = � ψ ( r )

0 � , ψ ( r, − 1 � 2 ) = � 0

ψ ( r ) � and H = � H 0

0 H � (1.6) where H is given by equation 1.5. It is rightful to drop the matrix form in the notation since it obviously carries little information. As we will see in Section 1.4, things are not always as such.

A single electron orbiting around a nucleus has a well defined orbital angular momentum char- acterized by the quantum number l in addition to its spin:

L

²

� nlm

_l

m

_s

� = l ( l + 1 ) � h

²

� nlm

_l

m

_s

� , (1.7)

L

_z

� nlm

_l

m

_s

� = m

_l

h � � nlm

_l

m

_s

� , (1.8)

S

²

� nlm

_l

m

_s

� = 3

4 h �

²

� nlm

_l

m

_s

� , (1.9)

S

z

� nlm

_l

m

s

� = m

s

h � � nlm

_l

m

s

� . (1.10)

where n is the principal quantum number that accounts for the remaining radial coordinate and L L L and S S S are the usual orbital and spin momenta operators. Furthermore for a single electron, the Hamiltonian 1.5 is separable in angular and radial coordinates so that a single-electron state can be written in spherical coordinates ( r, θ, ϕ )

� r, σ � nlm

_l

m

_s

� = ψ

_nlm_l_m_s

( rrr, σ ) = R

_nl

( r ) Y

_lm_l

( θ, ϕ ) δ

_σ,m_s

(1.11) where the Y

_lm_l

functions are spherical harmonics. We use the usual notation in which a function f ( x ) is the projection of the state � f � on a coordinate system f ( x ) = � x � f � (Cohen-Tannoudji et al., 1977).

According to the usual terminology the quantum numbers nlm

_l

m

_s

characterize a spin-orbital, nlm

_l

an orbital, nl a subshell and n a shell. In the following, we will often use the notation P

_nl

( r ) = 1 � r R

_nl

( r ) for absorbing the r

²

factor arising from the Jacobian of the spherical coordinates ( r

²

sin θ ) that appears in all radial integrals.

Note that since H commutes with the space inversion I = rrr → − rrr, one could expect that an

additional constraint could be put on ψ since we did not use that property yet. On the electron

coordinates, the space inversion is ( r, θ, ϕ, σ ) → ( r, π − θ, π + ϕ, σ ) . It is easy to show that the parity

of ψ is (−)

^l

(Cowan, 1981, chap. 2) and therefore that a single-electron state is naturally an eigen-

function of I. From now on, by the parity of a state, we will refer to its response to space inversion

unless otherwise stated.

(20)

As a matter of fact, the non-relativistic atom is analytically solvable (see Cowan, 1981, Chap. 3) and the energy levels do not depend on l

E

_n

= − Z

²

2n

²

E

h

. (1.12)

It is due to the fact that the Hamiltonian expressed in atomic units H = ppp

²

2 − Z

r (1.13)

commutes with lll and the Runge-Lenz vector defined in quantum mechanics as B B B = 1

2Z [ lll × ppp − ppp × lll ] + rrr

r . (1.14)

It is possible to show that a set of generators of so ( 4 ) = so ( 3 ) ⊕ so ( 3 ) built with { lll, B B B } can be found (Cornwell, 1995, Chap. 12). The two associated angular momenta, denoted M M M and N N N , re- spect M M M

²

= N N N

²

and

N N N

²

� nlm

_l

� = α ( α + 1 ) � nlm

_l

� (1.15)

with n = 2α + 1, the half dimension of the representation of the state. We also have lll = N N N + M M M so that

l = n − 1, n − 2, . . . , 1,0 . (1.16)

Therefore, { N N N

²

, lll

²

, l

_z

, s

_z

} form a Complete Set of Commuting Observables (CSCO) of the single-electron problem. Since it commutes with H as well, the associated quantum numbers n, l, m

_l

, m

_s

are con- served, i.e. they characterize exactly the non-relativistic hydrogenic stationary states. Traditionally, to the electron orbital momentum l, a letter is assigned, i.e. s, p, d, f, g, h, i, k, l, . . . for l = 0 − 8, . . . .

1.2.2 The two electron wave-function

When a second electron is added to the system and that the inter-electronic interaction is turned on, there is no known possibility of constructing a CSCO (excluding H) commuting with the Hamilto- nian. To our knowledge, there is no evidence that one exists. We will first explore the possibilities of constructing a wave function based on the known SCO (omitting the “Complete”) that commutes with H.

This set contains the operator exchanging the electron coordinates. Therefore, if we note the exchange operator P

₁₂

, for any � η � eigenstate of H we have (Cohen-Tannoudji et al., 1977)

P

₁₂²

� η � = � η � ⇒ P

12

� η � = ± � η � . (1.17) For satisfying the Pauli exclusion principle one must choose antisymmetric states for fermions. This is called the symmetrization postulate. It defines according to which irreducible representation of the permutation group a wave function of indiscernible particles should transform under permutation of coordinates.

If one consider an antisymmetrized wave function of independent (uncoupled) electrons

Ψ = � 1; 2 � = � n

1

l

1

m

_l₁

m

s1

; n

2

l

2

m

_l₂

m

s2

� (1.18)

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1.2. THE NON-RELATIVISTIC ATOM 9 the resulting energies will depend on the m

_l

’s through � 1; 2 �

_r¹₁₂

� 1; 2 � . Indeed, the operator

r

⁻₁₂¹

= � rrr

₁

− rrr

₂

�

⁻¹

= �

^∞

k=0

r

^k_<

r

^k+1_>

C C C

^k

( θ

₁

, φ

₁

) ⋅ C C C

^k

( θ

₂

,φ

₂

) (1.19)

= �

^∞

k=0

r

^k_<

r

^k+1_>

�

k

q=−k

(−)

^q

C

₋^k_q

( θ

₁

, φ

₁

) C

_q^k

( θ

₂

, φ

₂

) (1.20) with C

_q^k

= �

2k+14π

Y

_kq

, r

_<

= min ( r

₁

, r

₂

) and r

_>

= max ( r

₁

, r

₂

) , does not commute with lll

₁

, lll

₂

but with L = l

₁

+ l

₂

. It should also be noticed that the indiscernability of the electrons leads to the possible exchange of the particles through their interaction. Concretely, we have the emergence of integrals of the form

σ

�

1,σ2

�

6

ψ

₁^∗

( rrr

1

, σ

1

) ψ

₂^∗

( rrr

2

, σ

2

) r

₁₂⁻¹

ψ

2

( rrr

1

, σ

1

) ψ

1

( rrr

2

, σ

2

) d

³

rrr

1

d

³

rrr

2

∝ δ

ms1ms2

(1.21) that can be compared to the classical mean EM interaction, also direct coulomb interaction, that is written

�

6

� ψ

₁

( rrr

₁

)�

²

� ψ

₂

( rrr

₂

)�

²

r

⁻¹₁₂

d

³

rrr

₁

d

³

rrr

₂

. (1.22) Therefore, � Ψ � H � Ψ � will depend on the m

_s

through the exchange interaction. It is not a consequence of the Hamiltonian but of the structure of the wave function itself. The m

_s

are degrees of freedom that modify how two fermions will interact, for instance without this additional quantum number the Pauli exclusion principle would forbid two electrons to have the same spatial quantum numbers nlm

_l

. Even if H commutes with the electronic spins sss

₁

and sss

₂

in the total Hilbert space, the single particle operators are not internal operators in any symmetric space

⁴

and therefore their commutators with H acting in a fermionic space are not defined. The easiest symmetrical operator that can be constructed with the spins is S S S = sss

₁

+ sss

₂

which commutators with H is then defined – and vanish. A physical state Ψ

_LM_L_SM_S

is constructed to be an angular eigenfunction satisfying relations of the form (1.7) to (1.10).

Let us show how the two-electron wave function satisfying the symmetry constraints can be build.

The angular structure of a two-electron state

The way a wave function or an operator transforms under spin or spatial rotations can be deduced from the Racah algebra (Judd, 1963; Cowan, 1981; Lindgren and Morrison, 1986). In the special case of the wave function of two-electron of fixed ( l

₁

, l

₂

) , we have

�( l

1

l

2

) LSM

_L

M

_S

� = �

ms1ms2

m_l₁m_l₂

� l

1

m

_l₁

m

s1

; l

2

m

_l₂

m

s2

� � l

1

m

_l₁

m

s1

; l

2

m

_l₂

m

s2

� LSM

_L

M

_S

� . (1.23)

� 1; 2 � LSM

_L

M

_S

� is the product of the spin and orbital Clebsch-Gordan coefficients defined in term of a 3 − j coefficient

� j

₁

m

₁

j

₂

m

₂

� j

₁

j

₂

JM � = (−)

^j¹^−j²^+M

( 2J + 1 )

^1�2

� j

₁

j

₂

J

m

1

m

2

− M � . (1.24)

4

In general, a

symmetric space

is a vector space on which an irreducible representation of S

N

(here N

=

2) is defined.

Here, this vector space has to be defined in the Hilbert space of the Hamiltonian.

(22)

Each state will be characterized by a M

_L

= m

_l₁

+ m

_l₂

and a M

_S

= m

s1

+ m

s2

in addition to its total spin and orbital quantum number. A 3 − j factor of which two j (upper line) are inverted is multiplied by (−)

^j¹⁺^j²⁺^J

. The two-electron state can be developed in an infinite sum of functions of the type (1.23).

As we have seen, a single l electron has a parity (−)

^l

. Even though l is not conserved the total parity of the wave function is, and it therefore should be (−)

^l¹⁺^l²

since it is the parity of (1.23).

The symmetrization of the two-electron wave function: example of the 2p3p complex

The electron exchange operator P

12

can be decomposed in the exchange of the spatial coordinates P

₁₂^spat.

and an exchange of the spins P

₁₂^spin

that both commute with H. Therefore, and since H is independent of (σ

1

, σ

₂

), the two-electron state Ψ is a product of a spin function and a spatial function,

Ψ

_LM_L_SM_S

( rrr

₁

, σ

₁

, rrr

₂

, σ

₂

) = Ψ

_LM_L

( rrr

₁

, rrr

₂

) χ

_SM_S

( σ

₁

, σ

₂

) , (1.25) which are both symmetric or antisymmetric under particle inversion.

With two electrons, it is possible to construct three states of spin 1 (triplet), with M

_S

= 1, 0, − 1, the fourth possibility being a state of spin 0 (singlet). If M

_S

= 1 the two single-electron spin functions will be the same and therefore only a symmetric function can be constructed with them. The ladder operators S

_±

(see Cowan, 1981), which respectively raises and lowers the total spin projection, are also symmetrical so that all M

_S

states for a fixed S transform according to the same S

₂

representation.

Since the state with S = 0, M

_S

= 0 is orthogonal to the state with S = 1, M

_S

= 0, it is antisymmetric. If α is the spin up function and β is the spin down

� σ

i

� S = 1, M

_S

= 1 � = α ( 1 ) α ( 2 ) (1.26)

� σ

i

� S = 1, M

_S

= 0 � = 1

√ 2 [ α ( 1 ) β ( 2 ) + β ( 1 ) α ( 2 )] (1.27)

� σ

_i

� S = 1, M

_S

= − 1 � = β ( 1 ) β ( 2 ) (1.28)

� σ

_i

� S = 0, M

_S

= 0 � = 1

√ 2 [ α ( 1 ) β ( 2 ) − β ( 1 ) α ( 2 )] . (1.29) The spatial part should be such as to give an antisymmetric total wave function once multiplied by the appropriate spin function. At this point let us exemplify this with a wave function composed of one 2p electron and a 3p electron both described by a spin-orbital of the form (1.11). There are 9 spatial 2p3p states, with L = 0, 1, 2 (S, P, D). There is only one with ( M

_L

, M

_S

) = ( 2, 1 ) , corresponding to the triplet D, denoted

³

D. Because we chose a state with two electrons of same l, we also have that, in this particular space, P

₁₂^spat.

can be separated in the angular coordinates exchange and the radial coordinates exchange. Therefore, the spatial wave function is itself separable in an angular part and a radial part. For the L = 2, M

_L

= 2 state, the only possible angular factor is Y

₁₁

( 1 ) Y

₁₁

( 2 ) . Then the L = 2, S = 1 term will be characterized by symmetric orbital and spin functions so that it has to have an antisymmetric radial function. We have

� rrr

_i

�

³

D

₂

� = 1

√ 2 [ R

₂

( 1 ) R

₃

( 2 ) − R

₂

( 2 ) R

₃

( 1 )] Y

₁₁

( 1 ) Y

₁₁

( 2 ) (1.30) where we kept only the essential information in the notation, e.g. R

₂

( 1 ) = R

_2p

( r

₁

) . For building the other states, we can use the ladder operator L

₋

(Cowan, 1981) which lowers the orbital angular momentum projection quantum number. We have by applying L

₋

on �

³

D

2

� and after renormalization

� rrr

_i

�

³

D

₁

� = 1

2 [ R

₂

( 1 ) R

₃

( 2 ) − R

₂

( 2 ) R

₃

( 1 )] [ Y

₁₀

( 1 ) Y

₁₁

( 2 ) + Y

₁₁

( 1 ) Y

₁₀

( 2 )] (1.31)

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1.2. THE NON-RELATIVISTIC ATOM 11 The only other state with ( M

_L

, M

_S

) = ( 1, 1 ) is the

³

P

11

which is orthogonal to the

³

D

11

state and has an antisymmetric angular part. Since the total wave function is antisymmetric, the radial part is symmetric

� rrr

_i

�

³

P

₁

� = 1

2 [ R

₂

( 1 ) R

₃

( 2 ) + R

₂

( 2 ) R

₃

( 1 )] [ Y

₁₀

( 1 ) Y

₁₁

( 2 ) − Y

₁₁

( 1 ) Y

₁₀

( 2 )] (1.32) Similarly, we have

� rrr

i

�

³

D

0

� = 1

√ 12 [ R

2

( 1 ) R

3

( 2 ) − R

2

( 2 ) R

3

( 1 )]

× [ Y

₁₋₁

( 1 ) Y

₁₁

( 2 ) + Y

₁₁

( 1 ) Y

₁₋₁

( 2 ) + 2Y

₁₀

( 1 ) Y

₁₀

( 2 )] ,

(1.33)

� rrr

_i

�

³

P

₀

� = 1

2 [ R

₂

( 1 ) R

₃

( 2 ) + R

₂

( 2 ) R

₃

( 1 )] [ Y

₁₋₁

( 1 ) Y

₁₁

( 2 ) − Y

₁₁

( 1 ) Y

₁₋₁

( 2 )] ,

(1.34)

� rrr

_i

�

³

S

₀

� = 1

√ 12 [ R

₂

( 1 ) R

₃

( 2 ) − R

₂

( 2 ) R

₃

( 1 )]

× [ Y

₁₋₁

( 1 ) Y

₁₁

( 2 ) + Y

₁₁

( 1 ) Y

₁₋₁

( 2 ) − 2Y

₁₀

( 1 ) Y

₁₀

( 2 )] .

(1.35) The singlet states are easily obtained by multiplying the above spatial parts by the antisymmetric M

S

= 0 spin function and changing the sign in the radial component sum. Note that the

³

S,

³

D and

1

P have a similar radial factor but the fact that their angular structure is different lifts the degeneracy.

In the same way, the triplet and singlet states of same L have a similar angular function but their radial parts are different. We can also deduce that if we had chosen to consider p electrons of same n, the states corresponding to antisymmetric radial wave functions would vanish (

³

S,

¹

P and

³

D). In general, due to the Pauli exclusion principle, two equivalent electrons (same nl) will only be able to form states with even L + S.

The inter-electronic correlation

Because of the inter-electronic interaction, the single-electron CSCO does not commute with the two- electron Hamiltonian and the nl quantum numbers are not conserved. This effect is called the elec- tronic correlation. We will define it mathematically in Section 2.5 but let us discuss it generally here.

The degeneracy between l states in the hydrogen-like atoms guaranteed, by the commutation of the single electron Hamiltonian with the Runge-Lenz vector, does not hold for states with electrons in fixed shells. Some remnants of this degeneracy persist however (see level database of the NIST, 2010).

A particular orbital energy can be estimated using its effective principal quantum number (Cowan, 1981), with which one can explain the structure of the Periodic Table of the Elements, i.e. the shell filling rule for neutral atoms.

When the electrons are in well spatially separated states, r

₁₂⁻¹

is relatively small compared to the one-body part of the Hamiltonian, resulting in the electrons moving as if in a spherical potential, the single-electron quantum numbers being then well conserved. The situation is however critical when rrr

₁

= rrr

₂

. According to Kato’s cusp condition (Kato, 1957), the electron probability of presence,

� Ψ �

²

, decreases linearly when approaching this hyper-surface, creating a “pointy” region in the wave

function, the cusp. This singularity is especially difficult to reproduce since all orbitals, should they be

developed in gaussians functions or be numerically generated on a grid, will always exhibit smooth

shapes. The description of the inter-electronic cusp is essentially the limiting factor in most atomic

and molecular models.

(24)

One can improve the radial description of the system by using a superposition of npn

^′

p LS states.

Such a wave function will stay separable with unchanged spin and angular part but the radial part will not be reducible to the same form as equations (1.33) to (1.35)

⁵

. One can also relax the angular constraint by allowing the electrons to be of another l (Schwartz, 1962). For example, the

³

S

01

spatial state is a superposition of (1.35) and of

� rrr

_i

� 2s3s S

₀

� = 1

2 [ R

_2s

( 1 ) R

_3s

( 2 ) − R

_2s

( 2 ) R

_3s

( 1 )] Y

₀₀

( 1 ) Y

₀₀

( 2 ) . (1.36) Finally, it should be mentioned that writing the two-electron state as a function of the electron coordinates only is arbitrary. In fact, a very efficient way to build the system’s wave function is to include the inter-electronic distance as a variable (Bethe and Salpeter, 2008).

1.2.3 The many-electron wave function

The Hamiltonian commutes with L L L = ∑

i

lll

i

, L L L

²

, S S S = ∑

i

sss

i

, S S S

²

, all the coordinates permutation (S

_N

group) and I , the space inversion. In general, spin-orbitals of the form (1.11) are used as building blocks for the total many-electron wave function. Let us introduce the concept of configuration which is a class of states with fixed occupations { w

i

} of a set of subshells { n

i

l

i

} . We will see in Section 2.3 that the single-electron quantum numbers are well conserved so that a physical non-relativistic many- electron state can most often be denoted

n

1

l

^w₁¹

n

2

l

^w₂²

. . . n

t

l

^w_t^t ^2S+1

L

^π_M_L_M_S

(1.37) for t occupied subshells. The parity π is either denoted o (odd) or omitted (even) and is equal to (−)

^∑ⁱ^lⁱ

. This notation is not sufficient for isolating an energy level and other quantum numbers must be introduced. These are generally summarized by a single character γ that is supposed to contain all the missing information. γ and (1.37) define what is called a Configuration State Func- tion (CSF). We see below and in Section 2.3 that γ contains only quantum numbers related to the angular structure of the wave function. To go beyond the concept of CSF, we will denote Γ anything that allows to define a general atomic state function.

In general, a system of N indistinguishable particles behaves according to any irreducible rep- resentation of S

_N

, which can be rather complicated. However the symmetrization principle imposes that the wave function transforms only according to one of the two representations of dimension 1.

This choice ensures that any state is always an eigenfunction of all permutation of particles. How- ever, there is no constraint on the behavior of the wave function with respect to the separate S

_N^spin

and S

_N^spat.

permutation groups and the existence of multi-dimensional irreducible representations of these groups cause the wave function to be, in general, not separable in spin and spatial coordinates.

Equation (1.25) is only valid for the particular case of the two-electron wave function. However, the separate commutation of H with the permutations of spatial and spin coordinates permits to write

Ψ

^tot_πΓLM_L_SM_S

= Ψ

^spat._πΓ

spat.LML

⊗

SN

Ψ

^spin_Γ

spinSMS

(1.38)

each of the spin and spatial part of the wave function transforming according to an irreducible repre- sentation of S

_N

. The semi-direct product ⊗

^SN

allows to construct an antisymmetric total state

⁶

.

5

Note that if a superposition of the type 2pnp is constructed, it can easily be shown that it is equivalent to a pure 2p3p state with an arbitrary R

3p

function. That observation is related to the Brillouin theorem that we will discuss in more details in the next chapter.

6

This product is semi-direct since every permutation of coordinates is a combination of spatial and spin inversions and

that the only common element between S

^spin_N

and S

_N^spat.

is the identity. Note that the product of the representation in

equation (1.38) is not direct since it is related to the inversion of any pair of spatial

and

corresponding spin coordinates,

simultaneously. The direct product is the S

2N

group of the permutations of spin

or