Critical evaluation of the effect of anharmonicity and dispersion interactions using density functional theory on structural and spectroscopic properties of selected inorganic compounds

Thesis

Reference

Critical evaluation of the effect of anharmonicity and dispersion interactions using density functional theory on structural and

spectroscopic properties of selected inorganic compounds

SETHIO, Daniel

Abstract

Density functional theory (DFT) in its modern Kohn-Sham formulation provides an efficient framework for the accurate characterization of the properties of many-electron systems in solid-state physics and in chemistry. In this thesis, DFT has been applied to the prediction of the structural and spectroscopic properties of selected inorganic compounds.

SETHIO, Daniel. Critical evaluation of the effect of anharmonicity and dispersion interactions using density functional theory on structural and spectroscopic

properties of selected inorganic compounds. Thèse de doctorat : Univ. Genève, 2017, no.

Sc. 5108

DOI : 10.13097/archive-ouverte/unige:96319 URN : urn:nbn:ch:unige-963190

Available at:

http://archive-ouverte.unige.ch/unige:96319

Disclaimer: layout of this document may differ from the published version.

UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES

Section de Chimie et Biochimie Prof. Dr. Hans Hagemann Département de Chimie Physique Dr. Max Latévi Lawson Daku

Critical Evaluation of the Effect of Anharmonicity and Dispersion Interactions using Density Functional Theory

on Structural and Spectroscopic Properties of Selected Inorganic Compounds

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès Sciences, mention Chimie

par

Daniel Sethio d’Indonésie

Thèse N° 5108

To my dear mother

Dwi Susi Hastuti

Acknowledgements

I would like to express my sincere gratitude towards my supervisors Prof.

Dr. Hans Hagemaan and Dr. Latévi Max Lawson Daku for giving me the opportunity to study here and their supervision during my PhD at University of Geneva. I am grateful to Prof. Dr. Andreas Hauser for his constant support as a group leader.

I would like to thank Prof. Dr. Zbigniew Łodziana and Dr. Francesco Aquilante for accepting and evaluating my thesis.

I am thankful to Madame Catherine Ludy and Isabelle Garin for helping me with bureaucratic formalities, and also to Madam Nahid Amstutz and Patrick Barman for providing technical support and to Dominique Lovy for IT support.

I would like to acknowledge all the present and former members of An- dreas Hauser group and my friends whom I met during my PhD: Pradip, Qinchao, Elia, Teresa, Pablo, Andrea, Romain, Jacob, Manish, Angelina, Leo, Jakob, Anne-Laure, Yan, Jiji, Jan Pavlik, Antoine, Alexandra, Enza, Prodipta, Yolanda, Rania, Ani, Roberto, Serena, Martin, Jiri, Alexander Zech, and Marie Humbert-Droz (not in a proper order).

I would like to acknowledge all my Indonesian friends, PMNI and PTRI, in Geneva: Bu Vivi, Hakan, Evelyn, Ivy, Zen, kak Jill, Chef Eli, Luc, Lee, Johan, Dini, Marcell, Kalyani, Patrick, Dita, Gwen, Andy, Indah, Mbak Evi, Pierre, Mbak Lillih, Jan Teribillini, Evelyn, Nora, Monica, Esa, Andrea, Alexa, Pierre Jameson Ashoka Lolong, tante Mieke, Dorothea, Oom Farry, tante Kristina, Mikha, Malvino, mbak Santi, Daniel, tante Maya, Pak San-

Annick, Oma Lien, mbak Nesty, mbak Jima, tante Aaltje, Andre, Mardo, Carolyn, Ayen, Chika, Marcell, Mariska, Awandha, Anna, Alexa, mbak Ing- gried, Bernard, kak Rian, Oom Ludy, Raya, Jendra, Agung, Liling, Jeremy, Kaira, Amy and Danny (not in a proper order).

Finally, I would like to thank my wife, Desis Natalia, and my family for having supported me during these years.

Résumé

Dans sa formulation moderne due à Kohn et Sham la théorie de la fonction- nelle de la densité (DFT) est un excellent outil pour caractériser de façon précise les propriétés de systèmes multi-électroniques en physique du solide et en chimie . Dans ce travail de thèse, la DFT a été utilisée dans le but de prédire les propriétés structurales et spectroscopiques de différents composés inorganiques.

Dans une étude préliminaire, nous avons calculé les structures de dif- férents conformères de l’ion tétrabutylammonium (TBA) et déterminé leurs spectres de vibration dans l’approximation harmonique. Les résultats présen- tent une excellente adéquation avec les données expérimentales obtenues par spectroscopie IR et Raman pour les cristallites de TBABr et TBAI. Ceci démontre les bonnes performances de la méthode DFT pour la prédiction des propriétés de spectroscopie vibrationnelle en utilisant l’approximation harmonique.

Nous avons par la suite étudié plus en détail une série de 21 composés à base de borohydrures avec pour objectif d’obtenir une signature spectro- scopique précise de ces systèmes, (i) qui sont de possibles intermédiaires obtenus lors de réactions de décomposition de borohydrures métalliques pou- vant être utilisés comme matériaux pour le stockage d’hydrogène, ou (ii) qui sont impliqués dans des matériaux pouvant servir d’électrolytes solides dans les batteries rechargeables.

Une première série de calculs DFT a montré que les fréquences vibra- tionnelles tendent à être surestimées lorsque l’approximation harmonique

le spectre Raman de NaB₃H₈ présenté dans la figure ci-dessous, qui compare les spectres harmoniques et anharmoniques calculés au spectre expérimental.

Comparaison entre les spectres harmoniques, anharmoniques calculés pour B3H8 et expérimental pour Na(B₃H₈).

Les fréquences d’élongation B-H, qui sont typiquement surestimées dans l’approximation harmonique, sont systématiquement déplacées de 120 cm^-1 vers des plus petites fréquences pour les calculs anharmoniques. Ceci cor- respond à une amélioration de l’accord entre les spectres calculés et expéri- mentaux, plus particulièrement visible dans la zone de fréquence 1000 – 1500 cm^-1. Pour la totalité des composés de la série étudiée, la comparaison avec les données expérimentales disponibles révèle un excellent accord entre les fréquences vibrationnelles anharmoniques calculées et expérimentales dans la zone typique 500 – 1100 cm^-1.

De plus, le déplacement chimique calculé du¹¹B est en bon accord avec les données expérimentales, même s’il est systématiquement déplacé d’environ 4 ppm, comme illustré sur la figure ci-dessous.

Corrélation entre les déplacements chimiques calculés et expérimentaux du

11B.

Dans la dernière partie de cette thèse, nous avons étudié une série de cristaux MFX (M = Ca, Sr, Ba, Pb ; X = Cl, Br, I ; groupe d’espace P4/mmm) au moyen de calculs DFT périodiques. Ces matériaux, lorsqu’ils sont dopés avec des terres rares, présentent de nombreuses propriétés intéres- santes pouvant trouver application dans des domaines tels que le stockage optique de l’information ou l’imagerie par rayons X. Cette famille de cristaux présente une double couche d’halogénures perpendiculaire à l’axe c. En util- isant les méthodes DFT standards, la description de ce feuillet n’est pas satisfaisante, et le paramètre de maille c tend à être systématiquement sures- timé. Cette observation indique que les interactions de dispersion jouent un rôle clé dans la description de ce double feuillet.

Nous avons par la suite étudié en détail l’influence des corrections de dis- persion en caractérisant les structures de 11 différents cristaux MFX (M = Ca, Sr, Ba, Pb ; X = Cl, Br, I) au moyen de la DFT en utilisant la fonc- tionnelle standard PBE et la fonctionnelle PBE-D incluant la correction de dispersion D2 [Grimme D2]. Les résultats ont été comparés aux nombreuses

à une amélioration par rapport à la PBE, mais au contraire tend à détériorer la description structurale : les forces attractives sont surestimées, ce qui con- duit à des distances de liaisons trop courtes et des paramètres de maille trop petits. En revanche, de bons accords entre la théorie et les mesures expéri- mentales sont obtenus en variant les paramètres entrant dans la définition du schéma de correction D2.

En particulier, un très bon accord entre la théorie et l’expérience est observé lorsque les rayons atomiques de Van der Waals sont augmentés de 30% en utilisant le facteur d’échelle Srvdw= 1.3. En plus des structures, nous avons également étudiés les propriétés vibrationnelles de ces systèmes et nous avons trouvé, en utilisant la fonctionnelle PBE-D* défini avec Srvdw= 1.3, un excellent accord entre la théorie et les résultats expérimentaux. Finalement, nous avons étudié l’évolution de la structure et des spectres Raman de PbFCl en fonction de la pression hydrostatique en utilisant la fonctionnelle PBE- D*, et nous avons trouvé un bon accord avec les données expérimentales.

Nous croyons que ces résultats peuvent également trouver des applications plus générales étant donné l’existence de nombreux matériaux inorganiques possédant une double couche d’halogénures.

Summary

Density functional theory (DFT) in its modern Kohn-Sham formulation pro- vides an efficient framework for the accurate characterization of the prop- erties of many-electron systems in solid-state physics and in chemistry . In this thesis, DFT has been applied to the prediction of the structural and spectroscopic properties of selected inorganic compounds.

In a preliminary study, we have calculated the structures of different conformers of the tetrabutylammonium (TBA) ion and determined their vi- brational spectra in the harmonic approximation. The results show an excel- lent agreement with the experimental IR and Raman spectra of crystalline TBABr and TBAI, illustrating the good performance of DFT methods for the prediction of vibrational spectroscopy properties in the harmonic approx- imation.

We have then studied in detail a series of 21 boron-hydrogen species with the aim to obtain accurate spectroscopic signatures of these systems, (i) which are possible reaction intermediates in the decomposition reaction of metal borohydrides that may be used as hydrogen storage materials, or (ii) which are involved in materials that are potential solid electrolytes for use in rechargeable battery applications. A first series of DFT calculations showed that the vibrational frequencies tend to be overestimated when the calculations are performed in the harmonic approximation. The inclusion of anharmonicity led to a significantly improved agreement between experiment and theory. This is illustrated for the Raman spectrum of NaB₃H₈ by the Figure below, which helps compare the calculated harmonic and anharmonic

Calculated anharmonic IR spectra of B3H8 compared to experimental IR spectra of Na(B₃H₈).

The B-H stretching frequencies, which are typically overestimated in the harmonic approximation, are systematically shifted downwards by 120 cm^-1 in the anharmonic calculations. This corresponds to an improved agreement with experiments which is particularly visible in the spectral region from 1000 to 1500 cm^-1. For the whole series of studied species, the compari- son with available experimental data shows an excellent agreement between experimental and calculated anharmonic vibrational frequencies in the 500- 1100 cm ¹ fingerprint region.

Moreover, the calculated ¹¹B chemical shifts correlate well with experi- mental data, but they are systematically shifted by about 4 ppm, as illus- trated in the Figure below.

Correlation between calculated and experimental¹¹B chemical shifts.

In the last part of this thesis, we have studied a series of MFX crystals (M

= Ca, Sr, Ba, Pb; X = Cl, Br, I; space group: P4/nmm) using periodic DFT calculations. These crystals, when doped with rare earth ions, present many interesting properties which can be made use of in applications as diverse as X-ray imaging and optical data storage. This family of crystals presents a double layer of halogen (Cl, Br, I) ions perpendicular to the c axis. Using standard DFT calculations , the description of this double layer is not sat- isfactory, and the lattice parameter ctends be systematically overestimated.

This observation suggested that dispersion interactions play a key role in the description of this double layer.

We have thus studied in detail the influence of dispersion correction by characterizing the structures of a set of 11 different crystals MFX (M = Ca, Sr, Ba, Pb, X = Cl, Br, I) within DFT using both the standard PBE and the D2-dispersion-corrected [Grimme-D2] PBE-D functional. The results have been compared to many experimental data available for these MFX crystals.

The comparison shows that inclusion of the D2 dispersion correction does not lead to an improvement over the PBE results but rather tends to deteriorate

agreements between experiments and theory could however be achieved by varying the parameters entering the definition of the D2 correction scheme.

In particular, a very good agreement between experiments and theory was observed when the atomic van der Waals radii were increased by 30% using a scaling factor of Srvdw = 1.3. In addition to the structures, we have also studied the vibrational properties of these systems and found that a very good agreement between experiments and theory is also achieved using the PBE-D* functional defined with Srvdw = 1.3. Finally, using PBE-D*, we have studied the evolution of the structure and Raman spectra of PbFCl as a function of the hydrostatic pressure and found a good agreement with recent experimental data. We believe that these results may also find more general applications, as there are many inorganic layered compounds with double halogen layers.

List of publications

1. D. Sethio, L. M. L. Daku, and H. Hagemann. Computational study of the vibrational spectroscopy properties of boron-hydrogen compounds: Mg(B₃H₈)₂, CB₉H₁₀^–, and CB₁₁H₁₂^–. Int. J. Hydrogen Energy, in press

2. D. Sethio, L. M. L. Daku, and H. Hagemann. A theoretical study of the spectroscopic properties of B₂H₆and of a series of B_xH_y^{z –} species (x = 1-12, y = 3-14, and z = 0-2):

from BH₃to B₁₂H₁₂^{2 –}. Int. J. Hydrogen Energy, 41:6814, 2016

3. M. Sharma, D. Sethio, V. D’Anna, and H. Hagemann. Theoretical study of B₁₂H_nF_12-n^{2 –} species. Int. J. Hydrogen Energy, 40:12721, 2015

4. M. Sharma, D. Sethio, V. D’Anna, J. C. Fallas, P. Schouwink, R.Černý, and H. Hagemann.

Isotope exchange reactions in Ca(BH₄)₂. J. Phys. Chem. C., 119:29, 2015

List of conferences

2017 11th Triennial Congress of the World Association of Theoretical and Computational Chemists (WATOC) 2017, Munich, Germany (Poster)

Geneva Chemistry & Biochemistry Days, Geneva, Switzerland (Oral Presentation)

2016 15th International Symposium on Metal-Hydride Systems, Interlaken, Switzerland (Oral Presentation)

MSSC2016 Summer School: Ab Initio Modeling in Solid State Chemistry, Turin, Italy Swiss Association of Computational Chemistry, Bern, Switzerland

2015 Joint Indonesia-UK conference on Computational Chemistry, Bandung, Indonesia (Poster)

2014 14th International Symposium on Metal-Hydride Systems, Manchester, England (Poster)

Fall Meeting, Swiss Chemical Society (SCS), Zürich, Switzerland (Poster)

2013 Gaussian Workshop: ’Introduction to Gaussian: Theory and Practice’. Wroclaw, Poland CODECS Summer School: Theoretical Spectroscopy, Geneva, Switzerland

Contents

General Introduction 1

1 Theoretical Background 4

1.1 Solving the non-relativistic time-independent Schrödinger equa-

tion . . . 4

1.1.1 Basics . . . 4

1.1.2 Wavefunction-based methods . . . 12

1.1.3 Density functional theory . . . 28

1.2 Noncovalent interactions . . . 36

1.2.1 Failure of standard density functionals for noncovalent interactions . . . 36

1.2.2 Grimme’s type semiclassicalC6 based methods . . . 39

1.3 Vibrational analysis . . . 44

1.3.1 Harmonic approximation . . . 46

1.3.2 Anharmonic approach . . . 52

1.4 List of methods . . . 54

I Boron-Hydrogen Compounds 55

2.4 Vibrational properties of TBA conformers . . . 67

2.4.1 The vibrational spectra of the three lowest-energy con- formers (A, D, and E) . . . 67

2.4.2 The vibrational spectra of conformers B, C, and E . . . 70

2.5 Concluding remarks . . . 73

2.6 Outlook . . . 73

3 The influence of anharmonicity on the vibrational spectro- scopic properties of boron-hydrogen compounds 74 3.1 Diborane, B2H6 . . . 75

3.1.1 Structure of diborane . . . 75

3.1.2 NMR chemical shifts of diborane . . . 76

3.1.3 Vibrational spectroscopy of diborane . . . 77

3.2 B3H8 . . . 82

3.2.1 NMR chemical shifts of B3H8 . . . 83

3.2.2 Vibrational spectroscopy of B3H8 . . . 85

3.3 Mg(B₃H₈)₂. . . 87

3.4 B10H14 . . . 92

3.5 CB₁₁H₁₂^–, B₁₂H₁₂^{2 –}, CB₉H₁₀^–, and B₁₀H₁₀^{2 –} . . . 92

3.6 General trends in vibrational and in the NMR spectroscopy data . . . 97

3.6.1 NMR ¹¹B chemical shifts . . . 98

3.6.2 Vibrational frequencies . . . 98

3.7 Concluding remarks . . . 101

3.8 Outlook . . . 103

II Alkaline-earth Fluorohalides 105

4.2 Topological analysis of electron density . . . 113

4.3 Crystal structure . . . 114

4.4 Vibrational frequencies . . . 124

4.5 Mechanical and Spectroscopic properties of PbFCl . . . 129

4.5.1 Vinet equation of state . . . 129

4.5.2 The mode Grüneisen parameters . . . 131

4.6 Concluding remarks . . . 135

General Conclusions 136

General Introduction

Density functional theory (DFT) in its modern Kohn-Sham formulation [1,2]

provides an efficient framework for the accurate characterization of the prop- erties of many-electron systems in solid-state physics and in chemistry. DFT has been used as a powerful method for predicting the properties of medium- to large-sized systems. Moreover, DFT shows many success in predicting molecular and solid-state properties, such as structural parameters, atomiza- tion energies, structural energies differences, energy barrier, binding energies, ionization potential, and electron affinities, and band gap [3].

DFT has been widely employed in modern research to support the ex- perimental results. The “predicting power” of theoretical DFT calculations empowers the aplicability of DFT to a wide range of research interests. Here in this thesis, DFT has been applied to the prediction of the structural and spectroscopic properties of selected inorganic compounds.

In the last 10 years, the research interest of our research group has fo- cussed on two quite diverse projects. This has also led to a separation of this thesis into two parts: the first one adresses structural and spectroscopic properties of boron hydrogen species.

Metal borohydrides have been proposed as potential candidates for hydro- gen storage applications. Our research group has been studying the synthesis and caracterization of many new compounds of this type. In particular, vi- brational spectroscopy has been extensively used as a tool to obtain a better understanding of the local symmetry and structure of the BH₄^– ion in these compounds, as hydrogen atoms are not easily observable using X-ray diffrac-

LiSc(BH₄)₄ which revealed the presence of complex Sc(BH₄)₄^– ions [4]. The theoretical structure agreed with the one obtained from X-ray diffraction and the calculated IR spectrum matched the experimental one, which validates the theoretical approach. Another successful application of DFT was the study of Al₃Li₄(BH₄)₁₃ [5]. This unusual stoichiometry arises from the pres- ence of 3 complex ions Al(BH₄)^{4 –} and a complex ion (BH₄)Li₄³⁺. The mini- mization using periodical DFT calculations of the structural model proposed by an initial X-ray diffraction refinement led to a rotation of the Al(BH4)^{4 –} ions. Using this new structure as input for the X-ray refinement, a crystal structure with significantly improved R values was finally obtained. More recently, the effect of distortions of the tetrahedral BH₄^– on the vibrational frequencies of the deformation modes was quantitatively studied [6]. These combined results provide a good understanding of the relations between spec- tra and structure of compounds with BH₄^– ions.

During the thermal decomposition reactions of metal borohydrides (which liberate hydrogen), many different intermediate boron-hydrogen species can be formed. The identification of these intermediate species is an essential part to unravel the complex reaction mechanisms involved. This is the subject of the first part of this thesis. It is well known that modern DFT calcu- lations can predict with some accuracy vibrational and NMR spectra, and this is demonstrated in an initial study of different conformers of the tetra- butylammonium ions, which cleary confirm the conformation of this ion in the crystalline bromide and iodide. In the case of several boron hydrogen species, a significant difficulty arises from the presence of bridging hydrogen atoms which form two electron-three center bonds. These hydrogen atoms are strongly anharmonic, as has been previously established by high level ab initio calculations for diborane [7]. This type of calculations is currently not possible for species such as B10H14, which makes it necessary to find the best practical theoretical method to obtain as reliable as possible spectro- scopic predictions. Diborane has previously been extensively studied both experimentally and theoretically. For this reason, this molecule was chosen to probe the effect of different functionals, basis sets and other parameters to establish the method to be applied in the following for a series of species

up to B₁₂H₁₂^{2 –}. Finally, this study was extended to CB₁₁H₁₂^–, CB₉H₁₀^– and the molecular compound Mg(B₃H₈)₂.

The second project adresses the family of MFX (Matlockite) crystals, where M = Ca, Sr, Ba, Pb and X = Cl, Br, I. Rare earth-doped crystals of this family find many applications such as X-ray detection (BaFBr :Eu²⁺), g ray dosimetry over many orders of magnitude and optical data storage.

Sm²⁺- doped mixed SrFCl_0.5Br_0.5crystals were the first compounds reported to be subject to spectral hole burning at room temperature (and above) [8].

A recent PhD thesis in our group (P. Pal) adressed in detail the effects of high pressure on the crystal field experienced by Sm²⁺ in this family of compounds. From the theoretical side, the effect of physical and chemical (by substitution of Ba by Sr) pressure on BaFCl has been studied [9].

The MFX host compounds crystallize in the tetragonalP4/nmmmatlock- ite structure [10]. They exhibit a layered ionic structure which corresponds to a simple F^– M²⁺ X^– X^– M²⁺ F^– stacking of the ion layers along the c axis. Due to their layered structure and especially the presence of the anionic double layer, the description of the structures and properties of the MFX compounds within density functional theory (DFT) may suffer from the inaccurate description of the dispersion interactions, which can be observed with local, semilocal, and hybrid density functionals [11,12]. This deficiency of standard approximate functionals could indeed explain the over- estimation of thec parameter reported for PbFI [13]. The motivation of the second project is to find a method which is suitable for the desciption of all MFX compounds, especially the description of the double layer and also to study the impact of the lone pairs of Pb compared to Ca, Sr, and Ba. We have therefore investigated the influence of dispersion correction on the per- formance of selected functionals for the description of the structures of the MFX compounds. The influence of dispersion forces was taken into account by using the Grimme’s D2 type correction scheme [14,15].

In both parts of this thesis, the computed theoretical results are critically compared with available experimental data. These comparisons allow to

Chapter 1

Theoretical Background

1.1 Solving the non-relativistic time-independent Schrödinger equation

1.1.1 Basics

The non-relativistic time-independent Schrödinger equation

One of the main goals in molecular quantum mechanics is to solve the non- relativistic time-independent Schrödinger equation [16].

Hˆ (R,r) =E (R,r) (1.1)

where R and rcorrespond to all the coordinates of the nuclei and the elec- trons, respectively, Hˆ is the Hamiltonian operator, E is the energy of the system, and is the total many-electron wavefunction which contains all the information about the system. In the absence of external electrostatic or magnetic fields, the molecular Hamiltonian can be expressed as

Hˆ = ˆTN + ˆTe+ ˆVN,e+ ˆVee+ ˆVN N (1.2)

where the indexesN anderefer to the nuclei and electrons, respectively;TˆN

is the kinetic energy operator of the nuclei TˆN = 1

2 X

↵

~²

M↵r²↵, (1.3)

with ~ the reduced Planck’s constant and M↵ the mass of the ↵^th nucleus.

Tˆe is the kinetic energy operator of the electrons, Tˆe= 1

2 X

i

~²

mer²i, (1.4)

with me the electron mass. VˆN,e is Coulombic attraction between the nuclei and the electrons,

VˆN,e = X

↵

X

i

1 4⇡"o

Z↵e²

|R↵ ri|, (1.5)

with e the elementary charge and ¹/^4⇡"0 the Coulomb constant. Vˆee is the Coulombic repulsion between the electrons,

Vˆee =X

i

X

j>i

1 4⇡"0

e²

|ri rj|, (1.6)

VˆN,N is the Coulombic repulsion between the nuclei VˆN,N =X

↵

X

>↵

1 4⇡"o

Z↵Z e²

|R↵ R | . (1.7)

The↵, indexes andi, jindexes run over the nuclei and the electrons, respec- tively. Hartree atomic units are introduced because they are more convenient for quantum chemical calculations. In this system of units, the four funda- mental physical constants: e, me,~ and ¹/^4⇡"0 are all unity by definition:

e= 1, me= 1, ~= 1, 4⇡"0 = 1.

equation, whereEis an eigenvalue and an eigenfunction of the Hamiltonian Hˆ, which is a linear second-order differential operator. Solving this equation for a system with K nuclei and N electrons allows to characterize all its stationary states. This is extremely difficult and several approximations have to be introduced.

The Born-Oppenheimer Approximation

The first approximation to solve the time-independent Schrödinger equation is the Born-Oppenheimer approximation [17], which allows to decouple the motion of the electrons from that of the nuclei. The proposed separation of the nuclear and electronic motions is based on the fact that the nuclei are much heavier than the electrons and that they consequently move slower than the electrons. Therefore, one can consider that the electrons move in the field of the fixed nuclei. This is the Born-Oppenheimer approximation, wherein the total wavefunction (R,r) is formulated as the product of an electronic wavefunction e(r;R)and a nuclear wavefunction N(R):

(R,r) = e(r;R) N(R), (1.8) and the molecular Hamiltonian is written as the sum of the kinetic energy operator TˆN of the nuclei and the electronic Hamiltonian Hˆe:

Hˆ = ˆTN + ˆHe (1.9)

with

Hˆe= ˆTe+ ˆVN,e+ ˆVee+ ˆVN N, (1.10) which describes the motion of the electrons in the field of the fixed nuclei, and whose contributionVˆN N is thus a constant. The electronic wavefunction

e(r;R)is solution of the electronic problem:

Hˆe e(r;R) =Ee e(r;R), (1.11)

e depends explicitly on the electronic coordinates r; but it depends para- metrically on the nuclear coordinates R, which is also the case for the total electronic energy:

Ee=Ee(R). (1.12)

Within the Born-Oppenheimer approximation, for the system in theI^thelec- tronic state, the nuclei experience the potentialVI(R) = EeI(R), which, with the formulation adopted for the electronic Hamiltonian (Eq. 1.10), is the sum of the internuclear repulsion VˆN N and of the average field of the fast-moving electrons. Consequently, for the system in theI^th electronic state, the nuclear wavefunctions N(R)are the solutions of the nuclear Schrödinger equation:

HˆN N(R) =E N(R). (1.13) where the nuclear Hamiltonian

HˆN = ˆTN+VI(R). (1.14) describes the motion of the nuclei on the potential energy surface (PES) VI(R) of the I^th electronic state. E is the total energy; it includes the electronic, vibrational, rotational and translational energies.

The Born-Oppenheimer approximation gives accurate results when the potential energy surfaces of the different electronic states are well separated and fails in case of degeneracy or near-degeneracy.

The variational method

The variational principle. For a quantum chemical system described by a Hamiltonian Hˆ and any normalizable wavefunction which satisfies the appropriate boundary condition, one defines the functional E[ ] as the expectation value of the Hamiltonian

E[ ] = h |Hˆ | i

h | i . (1.15)

The variational principle states that this expectation value of the Hamilto- nian is an upper bound of the exact ground-state energy E0. That is,

E[ ] E0. (1.16)

The equality hold only when the trial wavefunction is a ground-state wave- function 0 of the system. This principle is used to find approximations to the ground state.

Given a trial wavefunction which depends on a set of parameters, the expectation valueE[ ]is a function of these parameters, which is potentially so complicated that its minimization will be extremely difficult.

The linear variational problem. The variational problem is made simpler if only linear parameterization of wavefunction is considered, that is, if is expanded in a fixed set of K normalized basis functions { i}1iK:

= XK

i=1

ci i. (1.17)

The minimization ofE[ ]becomes the problem of finding the optimum set of coefficients{ci}1iK. This one can be reduced to the linear algebra problem

H c=ES c, (1.18)

where His the Hermitian matrix of the Hamiltonian in the chosen basis set Hij =h ⁱ|Hˆ | ^ji; (1.19) cis the column vector made of the coefficients{ci}1iK;Sis the Hermitian overlap matrix

Sij =h i| ji ; (1.20)

andE is the sought energy. When searching for theK solutions p that can be built from the K basis functions

p= XK

q=1

Cqp q (1.21)

Eq. 1.18 becomes

H C=S C E, (1.22)

where Eis a diagonal matrix of the energies Ep.

The problem given by Eq. 1.22 is turned into an eigenvalue problem by proceeding to the orthogonalization of the basis set, that is, by finding a transformation matrixX={Xrs}1r,sK such that the functions { ⁰l}1lK

0l = XK k=1

Xkl k (1.23)

are orthonormal: ⌦ ₀

i| ⁰j

↵ = ij. In the new basis set, the problem given by Eq. 1.22 becomes the eigenvalue problem

H⁰C⁰=C⁰E, (1.24)

with

H⁰ =X^†H X, (1.25)

where X^† is the adjoint (i.e., the conjugate transpose) of X; and with

C⁰=X ¹C. (1.26)

To obtain the orthogonalization matrixX, one makes use of the fact that Sbeing Hermitian, it can be diagonalized. That is, there is exists a unitary matrix Usuch that

U^†S U =s; (1.27)

where s is a diagonal matrix of the eigenvalues {si}1iK of S, which are positive, and U^† is the adjoint (i.e., the conjugate transpose) of U. Note

that the i^th column vector of U is the eigenvector of S associated with the eigenvaluesi. One can then take as orthogonalization matrix

X=U s ^1/2, (1.28)

where s^1/2 is the diagonal matrix with the diagonal elements {¹/^ps_i}1iK. That is,

Xij = Uij

psj

. (1.29)

The procedure use to obtain an orthonormal basis set this way is referred to as canonical orthogonalization [18].

The many-electron wavefunction

Antisymmetry or Pauli exclusion principle. In the non-relativistic framework, the electronic Hamiltonian (Eq. 1.10) depends only on the spa- tial coordinates of the electrons. However, the description of an electron requires its spin to be specified. This is achieved by introducing the spin functions ↵(!) and (!), which correspond to spin up (") and spin down (#), respectively [18]. The two functions are orthonormal. They depend on the unspecified spin variable !, which actually defines a spin coordinate which with the spatial coordinates r help describe an electron. These four coordinates are denotedx:

x={r,!}. (1.30)

The electronic wavefunction of a N-electron system is thus a function of x1,x2, . . . , xN: (x1, . . .xN)).

Following the antisymmetry principle which is synonymous with the Pauli ex- clusion principle, the N-electron wavefunctions must be antisymmetric with respect to the interchange of the coordinate x of any two electrons:

(x1, . . . ,xi, . . . ,xj, . . . ,xN)= (x1, . . . ,xj, . . . ,xi, . . . ,xN). (1.31)

of the position vectorr and describes the spatial distribution of an electron:

|'(r)|²dris the probably of finding the electron in the volume elementdrat r. A spinorbital (x)describes both the spatial distribution and the spin of an electron. Given a spatial orbital'(r), one can construct two spinorbitals by multiplying it with the↵ or spin function:

(x) ='(r) (!), with =↵or = . (1.32) Slater determinants. A system of N non-interacting electrons is de- scribed by a Hamiltonian of the form

Hˆ = XN

i=1

h(i)ˆ , (1.33)

where the mono-electronic operatorˆh(i)describes the kinetic energy and the potential energy of the i^th electron. Let the normalized spinorbitals{ ⁱ(x)} represent the eigenfunctions of ˆh(i)

ˆh(i) j(xi) ="j j(xi). (1.34) The product of N spinorbitals

⇧ij...k(x1,x2, . . . ,xN) = i(x1) j(x2)· · · ^k(xN), (1.35) called Hartree product, is an eigenfunction of Hˆ associated with the eigen- valueE ="1+"j+· · ·+"k. To have a proper many-electron wavefunction, this Hartree product is antisymmetrized. The resulting antisymmetric wave- function is called a Slater determinant and reads

(x1,x2, . . . ,xN) = 1 pN!

i(x1) j(x1) · · · k(x1)

i(x2) j(x2) · · · ^k(x2) ... ... ... ...

i(xN) j(xN) · · · k(xN)

. (1.36)

1.1.2 Wavefunction-based methods

Analytic solutions to the electronic problem exist for the hydrogen atom, the hydrogen-like ions (e.g., He⁺, Li²⁺ or Be³⁺) or the dihydrogen cation H⁺2 which are one-electron systems. In the most general case of a system of N 2 interacting electrons, the solutions are sought by making use of the variational method in a framework of well-defined approximations.

The Hartree-Fock approximation

The Hartree-Fock approximation is central to quantum chemistry in that it is at the origin of the molecular orbital picture and serves as a good start- ing point for more accurate approaches to solve the electronic Schrödinger equation.

In the Hartree-Fock approximation, given aN-electron system described by the Hamiltonian

Hˆe = XN

i=1

1

2r²i +X

↵

Z↵e²

|R↵ ri|

| {z }

ˆh(ri)

+X

i

X

j>i

1

|ri rj|, (1.37)

the electronic wavefunction of the system is sought in the form of a Slater determinant

(x1,x2, . . . ,xN) = 1 pN!

1(x1) 2(x1) · · · ^N(x1)

1(x2) 2(x2) · · · ^N(x2) ... ... ... ...

1(xN) 2(xN) · · · ^N(xN)

, (1.38)

which in the Dirac bra-ket notation reads

| i=| 1 2· · · Ni . (1.39)

notation for one- and two-electron integrals [18], by

E[ ] = XN

i=1

[i|ˆh|i] +1 2

XN i=1

XN j=1

[ii|jj] [ij|ji] , (1.40)

where the one-electron integral[i|ˆh|i]reads [i|ˆh|i] =

Z

⇤i(x1)ˆh(r1) ^⇤_i(x1) dx1, (1.41) and the two-electron integrals [ii|jj]and [ij|ji]are given by

[ii|jj] = Z

i(x1) ^⇤_i(x1) 1

|r1 r2| ^j(x2) ^⇤_j(x2) dx1dx2, (1.42) [ij|ji] =

Z

i(x1) ^⇤_j(x1) 1

|r1 r2| ^j(x2) ^⇤_i(x2) dx1dx2. (1.43) Taking each spinorbital i(x)as the product of a spatial wavefunction'i(r) and a spin function i(!)

i(x) = 'i(r) i(!), with i =↵or i = , (1.44) and integrating out spin in the expressions of the one- and two-electron in- tegrals, these ones become

(i|ˆh|i) = Z

'^⇤_i(r1)ˆh(r1)'^⇤_i(r1) dr1, (1.45)

Jij = (ii|jj) = Z

'i(r1)'^⇤_i(r1) 1

|r1 r2|'j(r2)'^⇤_j(r2) dr1dr2, (1.46) and

Kij = (ij|ji) = Z

'i(r1)'^⇤_j(r2) 1

|r1 r2|'j(r2)'^⇤_i(r2) dr1dr2, (1.47) where the notation (· · ·) is used in place of [· · ·] to indicate that spin has been integrated out. Jij is termed the Coulomb integral: it corresponds to the classical Coulomb repulsion between the charge distributions |'i(r1)|²

and |'j(r2)|² associated to the occupations of thei^th and j^thorbitals. Kij is termed the exchange integral and has no classical counterpart. Such integrals account for exchange or Fermi correlation, which refers to the fact that, due to the antisymmetry requirement fulfilled by the use of a Slater determinant, the motions of electrons of parallel spin are correlated. Using the integrals expressed in terms of the spatial orbitals, the determinantal energy E[ ] reads

E[ ] = XN

i=1

(i|ˆh|i) + 1 2

XN i=1

XN j=1

Jij Kij. (1.48) Note that the termsJii(1in)associated to the spurious self-interaction of each electron with itself are canceled by the exchange terms inKii.

Within the Hartree-Fock approximation, the variational method trans- lates into finding the spinorbitals { i} which minimize E[ ]. The spinor- bitals are thus varied with the constraints that they form an orthornomal set of functions,

h ⁱ| ^ji= ij = Z

⇤i(x) j(x)dx. (1.49) This leads to the Hartree-Fock integro-differential equation

ˆh(r1) i(x1) + XN

l=1l6=i

"Z

| l(x2)|²

|r1 r2|dr2

#

i(x1) XN

l=1l6=i

Z _⇤

j(x2) i(x2)

|r1 r2| dr2 j(x1) ="i i(x1) (1.50) where "i is the energy of the spinorbital i. The first two-electron term is the Coulomb term and the second two-electron term is the exchange term.

The Coulomb term represents the interaction of the electron in i with the one-electron Coulomb potential of the form

vi(r1) =X

l6=i

Jˆl(r1), (1.51)

where Jˆl(r1), the so-called Coulomb operator defined by Jˆl(r1) =

Z | ^l(x2)|²

|r1 r2|dr2, (1.52) represents the local potential atx1 due to the charge distribution of an elec- tron in l. The exchange term is present because of the antisymmetric nature of the Slater determinant. To deal with non-classical term, one introduces the so-called exchange operatorsKˆl(r1), which are defined by the result their action on a spinorbital i, as follows:

Kˆl(r1) i(x1) =

Z

⇤l(x2) 1

|r1 r2| ⁱ(x2)dr l(x1). (1.53) The exchange operator Kˆl(r1)is nonlocal; the results of its action on i at x1 depends on the value of i throughout all the space. Note that for an electron in i, the expectation values of the Coulomb Jˆl(r1) and exchange Kˆl(r1)operators are the Coulomb an exchange integrals:

h i|Jˆl| ii = [ii|jj], (1.54) h i|Kˆl| ii = [ij|ji]. (1.55) Using the Coulomb and exchange operators, the Hartree-Fock equations read

2

64ˆh(r1) + XN

l=1l6=i

Jˆl(r1) Kˆl(r1) 3

75 ⁱ(x1) = "i i(x1). (1.56)

Because the summation in the above equation runs over l 6=i, the operator defined by the terms in the square brackets seems to depend upon the spinor- bital it is acting on. However, havingJˆi| ⁱi= ˆKi| ⁱi, the restriction on the summation can be lifted. The Hartree-Fock equations take the general form of an eigenvalue problem

fˆ(r ) (x ) =" (x ), (1.57)

where fˆis the Fock operator fˆ(r1) = ˆh(r1) +

XN l=1

Jˆl(r1) Kˆl(r1). (1.58) The Fock operator is the sum of the one-electron core-Hamiltonian operator ˆh(r1)and the effective one-electron potential

ˆ

vHF(r1) = XN

l=1

Jˆl(r1) Kˆl(r1) (1.59) termed the Hartree-Fock potential. Because the Hatree-Fock potential de- pends on the occupied spinorbitals, starting from an initial guess of spinor- bitals, the Hartree-Fock equations are to be solved iteratively till self-consistency is achieved. The Slater determinant | ⁰i built from the N optimal spinor- bitals of lowest energies ("1 "2 · · ·"N) is the best single-determinantal estimate of the ground-state wavefunction | 0i of the N-electron system, with the variational energy

E[ 0] =EHF = XN

i=1

[i|ˆh|i] +1 2

XN i=1

XN j=1

[ii|jj] [ij|ji]. (1.60) In fact,| 0iis also the exact ground-state wavefunction of the Hartree-Fock Hamiltonian

Hˆ0 = XN

i=1

fˆ(ri), (1.61)

with the energy h 0|Hˆ0| 0i=

XN i=1

"i = XN

i=1

[i|ˆh|i] + XN

i=1

XN j=1

[ii|jj] [ij|ji], (1.62) which differs from the variational energy of the real system (Eq. 1.60) by the

with closed-shell systems to use restricted spinorbitals and to use unrestricted spinorbitals. A spinorbital i(x) can indeed be written as the product of a spatial orbital '_i(r)and a spin function (!)( =↵, ):

i(x) =

( '^↵_i(r)↵(!),

'_i(r) (!). (1.63)

For closed-shell systems with an even number N of electrons, a same set of spatial orbitals is used for the spin-up and the spin-down spinorbitals:

'^↵_i(r) ='_i(r) ='i(r); and the restricted spin-up and the spin-down spinor- bitals thus obtained are noted 'i(r)and'¯i(r), respectively. The closed-shell restricted ground-state wavefunction is given by

| ⁰i= '1'¯1'2'¯2· · ·'N/2'¯N/2

↵ ; (1.64)

the closed-shell ground-state energy by

E[ 0] =ERHF= 2

N/2

X

i=1

(i|ˆh|i) +

N/2

X

i=1

N/2

X

j=1

2Jij Kij; (1.65) and the Fock operator reads in the restricted case

fˆ(r) = ˆh(r) +

N/2

X

l=1

2 ˆJl(r) Kˆl(r). (1.66) For open-shell systems with N^↵spin-up and N spin-down electrons (N = N^↵+N ), the spin-up and the spin-down spinorbitals are made from two distinct sets of spatial orbitals {'^↵_i(r)} and n

'_i(r)o

. The Hartree-Fock equations translate into the set of coupled equations

( fˆ^↵(r1)'^↵_i(r) = "^↵_i'^↵_i(r),

fˆ (r1)'_i(r) = "_i'_i(r), (1.67)

where the Fock operatorfˆ^↵(r1) andfˆ (r1)read fˆ^↵(r1) = ˆh(r1) +

N^↵

X

l=1

hJˆ_l^↵(r1) Kˆ_l^↵(r1)i +

XN l=1

Jˆ_l (r1), (1.68)

fˆ (r1) = ˆh(r1) + XN

l=1

hJˆ_l(r1) Kˆ_l (r1)i +

N^↵

X

l=1

Jˆ_l^↵(r1), (1.69)

whereJˆ_l(r1)andKˆ_l(r1) ( = ↵, )are the Coulomb and exchange operators built from the spinorbital'_i(r) (!). The above expressions offˆ ( =↵, ) indicate that, the in Hartree-Fock formalism, a electron experiences the core-Hamiltonian potential, a Coulomb and an exchange interaction with all electrons of the same spin, and a Coulomb interaction with all electrons of opposite spin. The open-shell ground-state energy is given by

E[ 0] =EUHF = X

=↵,

XN i=1

h_i +1 2

X

=↵,

XN i=1

XN j=1

⇥J_ij K_ij ⇤ +

N^↵

X

i=1

XN j=1

J_ij^↵ , (1.70) where h_i is the expectation value of the core-Hamiltonian operator over the spinorbital '_i(r) (!), J_ij ⁰ the one of the Coulomb operator Jˆ_i (r1) over '_j⁰(r) ⁰(!), andK_ij the expectation value of the exchange operator Kˆ_i(r1) over '_i(r) (!).

In practice, the orbitals are expanded in a basis set of normalized func- tions{ ^k(r)}1kK. In the restricted case, the spatial orbitals are given by

'i(r) = XK

k=1

Cki k(r) (i= 1, . . . , K) ; (1.71) and in the unrestricted case by

'_i(r) = XK

k=1

C_{ki k}(r) ( =↵, andi= 1, . . . , K). (1.72)

ear algebra methods. In the restricted case, one thus solves the Roothaan equations

F C=S C", (1.73)

whereFis theK⇥K matrix of the Fock operator (Eq. 1.66) in the auxiliary basis set, C is the K ⇥ K matrix of the expansion coefficients of the K spatial orbitals {'i(r)}, S the overlap matrix of the basis functions, and "

the diagonal matrix of theK orbital energies {"i}. In the unrestricted case, one solves the Pople-Nesbet equations

( F^↵C^↵=S C^↵"^↵,

F C =S C " , (1.74)

where F ( = ↵, ) is the K ⇥K matrix of the Fock operator (Eqs 1.68 and 1.69) in the auxiliary basis set, C is the K ⇥K matrix of the expan- sion coefficients of the K spatial orbitals '_i(r), S the overlap matrix of the basis functions, and " the diagonal matrix of the K energies {"_i}of the spinorbitals .

Beyond the Hartree-Fock approximation

The variational Hartree-Fock energyEHFprovides an upper bound to the ex- act ground-state energyE0of a many-electron system. EHFis lower the larger and more flexible the basis set used in the calculations, and the Hartree-Fock limit refers to the Hartree-Fock results obtained in the complete basis set limit. The difference between E0 and the Hartree-Fock-limit energy E_HF^lim. is called the correlation energy

Ecorr=E0 E_HF^lim. <0. (1.75) Electron correlation refers to the correlated motion of the electrons and sep- arates into two contributions [19]: the Coulomb correlation, which is the correlation between the spatial motions of all electrons due to their instan- taneous Coulomb repulsion; and the Fermi correlation, which arises from the requirements of antisymmetry and spin symmetry of the exact electronic

wavefunction.

The antisymmetry of the wavefunction ensures that two parallel-spin elec- trons cannot be located at the same position in space. The resulting vanishing probability of finding about a reference electron a second electron with the same spin is referred to as a Fermi hole. This is the unique Fermi correlation present in closed-shell systems and is well described by Slater determinants (see further explanation below). In open-shell systems, Fermi correlation is also present between opposite-spin electrons and arises from the spin sym- metry requirement of the electronic wavefunction. This is illustrated by the following example on the first excited states of the He atom, adapted from Ref. [19]. The electronic ground state of He is a singlet state, noted 1¹S, which originates from the configuration 1s², and which can be approxima- tively described by the closed-shell Slater determinant

1¹S(x1,x2) = |'1s'¯1s|='1s(r1)'1s(r2)↵(!1) (!2) ↵(!2) (!1)

p2 , (1.76)

where '1s is the normalized 1s spatial orbital, as obtained from a Hartree- Fock calculation. 1¹S(x1,x2)is an eigenfunction of the total squared (quan- tum number S) and the projected (quantum number MS 2 { S, S + 1, . . . , S 1, S}) electronic spin operators, with S = 0 and MS = 0. The first excited states of Be originate from the1s¹2s¹excited configuration and correspond to the singlet 2¹S (S = 0) and triplet 2³S (S = 1) states. The space generated by the1s¹2s¹ configuration is spanned by the four Slater de- terminants |'1s'2s|,|'¯1s'¯2s|,|'1s'¯2s|, |'¯1s'2s|, where '2s is the normalized 2s spatial orbital. |'1s'2s| and |'¯1s'¯2s| are eigenfunctions of the total and projected spin operators withS = 1andMS =±1. |'1s'¯2s|and|'¯1s'2s|are only eigenfunctions of the projected spin operator, with MS = 0.

Their sum and their difference give proper spin functions, which are also referred to as configuration state functions (CSFs). The 2¹S state is thus described by the CSF

and the 2³S states by the CSFs

(Ms)

2³S (x1,x2) = '1s(r1)'2s(r2) '1s(r2)'2s(r1) p2

⇥ 8>

>>

><

>>

>>

:

↵(!1)↵(!2), MS = +1,

↵(!1) (!2) +↵(!2) (!1)

p2 , MS = 0, (!1) (!2), MS = 1.

(1.78)

For a many-electron system in a given system, the two-electron density (r1,r2), also known as the two-electron probability distribution, gives the probability of finding one electron at r1 and a second one at r2. For He in the 1¹S grounds state described by Eq. 1.76, it reads

1¹S(r1,r2) ='²_1s(r1)'²_1s(r2), (1.79) and is thus the simple product of two one-electron probability distributions:

the spatial motion of the two electrons are uncorrelated (within this ansatz for the wavefunction). For He in the2³Sand2¹Sstates described by Eqs 1.78 and 1.77, (r1,r2)is given by

2³S(r1,r2) = 1

2'²_1s(r1)'²_2s(r2)+1

2'²_1s(r2)'²_2s(r1) '1s(r1)'2s(r2)'1s(r2)'2s(r1), (1.80) and

2¹S(r1,r2) = 1

2'²_1s(r1)'²_2s(r2)+1

2'²_1s(r2)'²_2s(r1)+'1s(r1)'2s(r2)'1s(r2)'2s(r1). (1.81) Thus, in either excited state, the probability of having one electrons at r1

depends on the position r2 of the second electron. Their motions are said to be Fermi correlated [19]. The difference between 2³S(r1,r2)and 2¹S(r1,r2) is in their last term, which have opposite signs and thus drive the two prob- abilities in opposite directions. In particular, when the two electrons get close, the 2³S(r1,r2)triplet probability decreases to zero, which corresponds to a Fermi hole. In contrast, when r2 ! r1, the 2¹S(r1,r2) singlet proba-

bility doubles, which corresponds to a so-called Fermi heap. Hence, due to their Fermi-correlated motions, the electrons tends to avoid each other in the triplet state 2³S and to stay close in the singlet state2¹S, which contributes to the triplet state2³Sbeing lower in energy than the singlet state2¹S. Thus, while Fermi holes are always present for pairs of same-spin electrons and rep- resent the only Fermi correlation in closed-shell systems, Fermi correlation is also present for pairs of opposite-spin electrons in open-shell systems and manifests itself by the existence of Fermi holes and Fermi heaps (See Ref. [19]

for a detailed discussion).

The Hartree-Fock method provides a mean-field treatment of the Coulomb repulsion and thus misses the Coulomb correlation, which can be described in terms of Coulomb holes [20]. Post-Hartre-Fock methods help to recover the missing Coulomb correlation by including more determinants into the description of the electronic system (see below), the Hartree-Fock wavefunc- tion being referred to as the zero-order or reference description. Actually, Coulomb correlation divides into dynamic and static correlation. Static cor- relation refers to situations where a single Slater determinant fails to give a correct zero-order description of the electronic systems. This occurs when there are degeneracies or near-degeneracies among orbitals, resulting in de- terminants of similar energies but different orbital occupations. All the Slater determinants obtained from the different orbital occupations need to be in- cluded in the zero-order description of the system. The zero-order wavefunc- tion is thus a linear combination of these determinants and the combination coefficients and orbitals are determined using the variational principle: this is the multi-configurational self-consistent field (MCSCF) method. The dy- namic correlation is the Coulomb correlation not taken into account by the zero-order description, be it given by a Hartree-Fock or a MCSCF wave- function, and its recovery is mandatory to achieve an accurate description of the electronic system. It should be kept in mind that static and dynamic correlation both arise from the inter-electronic Coulomb repulsion and the distinction between them is somehow arbitrary [19]. In the following, we