New methods to treat strongly correlated systems

Thesis

Reference

New methods to treat strongly correlated systems

LI MANNI, Giovanni

Abstract

This thesis focuses on development of three novel multiconfigurational quantum chemical methods, to study strongly correlated systems: GASSCF, SplitCAS and SplitGAS. By GASSCF it is possible to eliminate ineffective configurations from the CI expansion by choosing an arbitrary number of active spaces and by tuning the interspace excitations. By SplitCAS and SplitGAS, the CI wave function generated by CAS or GAS type of active spaces is partitioned into two parts: a principal part containing few relevant configurations and an extended part containing less relevant configurations. The two methods differ in the criteria for the splitting. Lowdin's partitioning technique is used to reduce the secular problem to the size of the principal space. These methods allow for an enlargement of the active space with respect to the CASSCF method. Theory, algorithmic details and test calculations are presented. Based on this work, the ‘scaling catastrophe' of the CASSCF approach is partially solved.

LI MANNI, Giovanni. New methods to treat strongly correlated systems. Thèse de doctorat : Univ. Genève, 2013, no. Sc. 4535

URN : urn:nbn:ch:unige-286789

DOI : 10.13097/archive-ouverte/unige:28678

Available at:

http://archive-ouverte.unige.ch/unige:28678

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

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Section de chimie et biochimie

Département de chimie physique Professeur L. Gagliardi

Professeur T. Wesolowski

New Methods to Treat Strongly Correlated Systems

THÈSE

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

par

Giovanni LI MANNI

de

San Giuseppe Jato – Palermo (Italie)

Thèse N^o 4535

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"New Methods to Treol Slrongly Correlofed Syslems "

Lo Foculté des sciences, sur le préovis de Modome L. GAGLIARDI, professeure ordinoire et d i r e c t r i c e d e t h è s e ( U n i v e r s i t y o f M i n n e s o t o - DeportmenT o f C h e m i s t r y - Minneopolis, U . S . A . ) , M o n s i e u r T . A . W E S O L O W S K I , p r o f e s s e u r o s s o c i é e t c o d i r e c t e u r d e t h è s e ( D é p o r t e m e n t d e c h i m i e p h y s i q u e ) , e t T . S A U E , p r o f e s s e u r ( U n i v e r s i t é d e T o u l o u s e - L o b o r o t o i r e d e c h i m i e e t p h y s i q u e q u o n t i q u e s - Toulouse, F r o n c e ) , o u t o r i s e I ' i m p r e s s i o n d e l o p r é s e n t e t h è s e , s o n s e x p r i m e r d ' o p i n i o n s u r l e s p r o p o s i t i o n s q u i y sont énoncées.

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N . B . L o i h è s e d o i t p o r t e r d o n s l e s " l n f o r m o t i o n s

l o d é c l o r a l i o n p r é c é d e n t e e t r e m p l i r r e l o l i v e s o u x t h è s e s d e d o c t o r o i ù

l e s c o n d i l i o n s é n u m é r é e s I ' U n i v e r s i t é d e G e n è v e " .

1. G. La Macchia, G. Li Manni, T. Todorova, M. Brynda, F. Aquilante, B. O. Roos and L. Gagliardi, On the Analysis of the Cr-Cr Multiple Bond in Several Classes of Dichromium Compounds, Inorg. Chem., vol. 49, p. 5216,2010

2. G. Li Manni, F. Aquilante and L. Gagliardi,Strong Correlation Treated Via Effective Hamiltonians and Perturbation Theory, J. Chem. Phys., vol. 134, p. 034114,2011

3. D. Ma, G. Li Manni and L. Gagliardi, The Generalized Active Space Concept in Multiconfigurational Self-Consistent Field Methods, J. Chem.

Phys., vol. 135, p. 044128,2011

4. G. Li Manni, J. R. Walensky, S. J. Kraft, W. P. Forrest, L. M. P´erez, M. B. Hall, L. Gagliardi and S. C. Bart,Computational Insights into Uranium Complexes Supported by Redox-Active α-Diimine Ligands, Inorg. Chem., vol. 51, p. 2058,2012

5. G. Li Manni, A. L. Dzubak, A. Mulla, D. W. Brogden, J. F. Berry and L. Gagliardi, Assessing Metal–Metal Multiple Bonds in Cr–Cr, Mo–

Mo, and W–W Compounds and a Hypothetical U–U Compound: A Quantum Chemical Study Comparing DFT and Multireference Meth- ods, Chem. Eur. J., vol. 18, p. 1737, 2012

6. G. Li Manni, M. Pederson, L. Gagliardi,Multiconfigurational analysis of the Cu₃Mg₃F₁₂ single molecule magnet model, (Submitted)

7. G. Li Manni, D. Ma, J. Olsen and L. Gagliardi, The SplitGAS-CI method, (Submitted)

i

I would like to acknowledge my PhD advisors Professor L. Gagliardi and Pro- fessor T. Wesolowski for their constant scientific support. Professor T. Saue for agreeing to be a member of the committee for my thesis dissertation.

Professor J. Olsen who I consider to be my role model in science as well as a good collaborator and friend. All of my knowledge has been anchored in his lectures, continuous e-mails, and his code Lucia. A thanks to Lucia follows.

Also, thanks to Francesco Aquilante for introducing me to L¨owdin’s work that is the basis of one of the methods we developed. I thank him also for his help implementing the first rudimentary version of the code that was already promising despite its limitations. A special thanks to Dongxia Ma,

‘the scientist’, for sharing her knowledge with me, for working together to combine the pieces to solve a huge puzzle, and also for assisting me in under- standing the most complicated steps of the theory and developments as well as in the tedious task of debugging. Special thanks to Dongxia, as friend and wife this time, with whom I have shared the most exciting events of my life. And of course I thank my son, Francesco Yu Chen, who has completed my life with a special touch of joy. Thanks to the MOLCAS community for giving me the opportunity to learn how to code in a large-scale software package. And among them, I thank Valera Veryazov for teaching me special large-scale code analysis methods. Thanks also to Tanya, Cesar, Xiuwen, Piotr, Little Laura, Georgios and Giovanni (my colleagues and friends in Geneva) who made my weekends enjoyable while boosting my research and Bess, Allison, Nora, Toni, Pere, Daniel, David, Will, Laura (LOOR-ruh), Josh, Chad, R`emi, Abbas, Mike, Danylo and Christine (my colleagues and friends in Minneapolis) with whom I have been able to share my enthusiasm for science and ‘survive’ the tough Minnesotan winters. And finally thanks to my parents and brother who have supported me in doing research abroad undeterred by their own wish of having me closer.

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Je tiens `a remercier mes directeurs de th`ese, Professeur L. Gagliardi et Pro- fesseur T. Wesolowski pour leur constant soutient scientifique. Professeur T. Saue pour avoir accept´e d’ˆetre membre du jury de ma soutenance de th`ese. Professeur J. Olsen, que je consid`ere comme un mod`ele en science ainsi qu’un bon collaborateur et ami. Toute ma connaissance est bas´ee sur ses cours, nos ´echanges d’e-mails continus et son code, Lucia. Un merci `a Lucia va de soi. Aussi, je remercie F. Aquilante pour avoir introduit le travail de L¨owdin qui est la base d’une des m´ethodes que nous avons d´evelopp´ees.

Merci aussi pour m’avoir aid´e `a programmer la premi`ere version rudimen- taire du code qui ´etait d´ej`a prometteuse en d´epit de ses limitations. Un merci sp´ecial `a Dongxia Ma, ‘le chercheur’, pour partager avec moi ses con- naissances, pour le travail fait ensemble en combinant les pi`eces de l’´enorme puzzle que nous avons r´esolu et aussi pour m’avoir aid´e dans les ´etapes les plus compliqu´ees de compr´ehension de la th´eorie, de d´eveloppement et dans les p´eriodes fastidieuses de d´ebogage. Un remerciement sp´ecial `a Dongxia, cette fois en tant qu’amie et femme, pour partager avec moi les moments les plus passionnants dans ma vie. Et bien sˆur, notre fils, Francesco Yu Chen, qui a apport´e une touche sp´eciale de joie dans ma vie. Un merci `a la com- munaut´e de MOLCAS, et parmi eux, Valera Veryazov pour m’avoir enseign´e les m´ethodes sp´eciales d’analyse de code `a grande ´echelle. Merci aussi aux coll`egues et amis `a Gen`eve qui ont rendu mes week-ends divertissants et aussi stimulants pour ma recherche, ainsi que mes coll`egues et amis `a Min- neapolis, avec qui j’ai pu partager mon enthousiasme pour la science et avec qui j’ai ´et´e capable de survivre aux rudes hivers du Minnesota. Je remercie enfin mes parents et mon fr`ere qui m’ont soutenu `a faire de la recherche `a l’´etranger, mˆeme malgr´e leur souhait de m’avoir plus pr`es d’eux.

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This thesis focuses on the development of novel multiconfigurational quantum chem- ical methods. Such methods are indispensable to study systems with a complex electronic configuration featuring strong electron correlation. Three methods have been developed as part of this work: GASSCF, SplitCAS and SplitGAS. In the GASSCF method, it is possible to eliminate ineffective configurations from the CI expansion by choosing an arbitrary number of active spaces and by tuning the inter- space excitations. In the SplitCAS and SplitGAS methods, the CI wave function generated by CAS or GAS type of active spaces is partitioned into two parts: a prin- cipal part containing few relevant configurations and an extended part containing less relevant configurations. The two methods differ in the criteria for the splitting.

L¨owdin’s partitioning technique is used to reduce the secular problem to the size of the principal space. These methods allow for an enlargement of the active space with respect to the CASSCF method. The CASSCF/CASPT2 strategy has been applied to several systems: (a) uranium complexes, (b) Cr–Cr, Mo–Mo, and W–W compounds and (c) the Cu3Mg3F12 single molecule magnet model. Both advan- tages and disadvantages were observed, the latter mostly related to the size of the active space. The thesis is organized as follow: in Chapter 1 a general introduction is given that discusses the concept of electron correlation, methods to recover it and routes to overcome the limits of standard methods. In Chapter 2 an overview of multiconfigurational systems that have been investigated at the CASSCF/CASPT2 level of theory is reported. In Chapter 3 and Chapter 4 the GASSCF, the Split- CAS and the SplitGAS methods will be discussed including the theory, algorithmic details and test calculations. In Chapter 5 general conclusions are presented, high- lighting the elements of novelty of the developed methods. In Appendix A the author has collected the equations for an ‘on the fly’ implementation of the Split- GAS method. In Appendix B the reader can find useful details on how a CI space is generate. Based on this work, the ‘scaling catastrophe’ of the CASSCF approach is partially solved allowing larger multiconfigurational systems to be addressed.

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Cette th`ese s’interesse au d´eveloppement de nouvelles m´ethodes multiconfigura- tionnelles de chimie quantique. De telles m´ethodes sont indispensables pour l’´etude de syst`emes fortement corr´el´es `a configurations ´electroniques complexes. Trois m´ethodes ont ´et´e d´evelopp´ees dans le cadre de ce travail : GASSCF, SplitCAS et SplitGAS. Avec la m´ethode GASSCF, en choisissant un nombre arbitraire d’espaces actifs et en contrˆolant le nombre d’excitations inter-espaces, il est possible d’´eliminer efficacement un grand nombre de configurations de l’interaction de configurations (IC). Dans les m´ethodes SplitCAS et SplitGAS, la fonction d’onde r´esultant de l’IC g´en´er´ee `a l’aide d’espaces actifs de type CAS ou GAS est divis´ee en deux parties : une partie principale contenant les configurations les plus importantes, et une partie auxiliaire contenant des configurations moins importantes. Les deux m´ethodes diff`erent par les crit`eres utilis´es pour la partition de l’espace d’IC. La m´ethode de partitionement de L¨owdin est alors utilis´ee pour r´eduire le probl`eme s´eculaire `a la taille de l’espace principal. Ces m´ethodes permettent d’utiliser un nombre d’orbitales plus important dans l’espace actif qu’avec la m´ethode CASSCF traditionelle. L’approche CASSCF/CASPT2 a ´et´e utilis´ee sur plusieurs syst`emes : (a) des complexes d’uranium, (b) les diatomiques Cr–Cr, Mo–Mo et W–W, et (c) l’aimant mol´eculaire Cu3Mg3F12. Les avantages et inconv´enients de cette approche sont discut´es, en particulier en lien avec la taille de l’espace actif. Cette th`ese est organis´ee de la fa¸con suivante : dans le chapitre 1 une introduction g´en´erale est pr´esent´ee, discutant le concept de corr´elation ´electronique, des m´ethodes pour la traiter et lever les limites des approches traditionnelles. Dans le chapitre 2 une vue d’ensemble de syst`emes multiconfigurationnels ´etudi´es au niveau CASSCF- CASPT2 est pr´esent´ee. Dans les chapitres 3 et 4 les m´ethodes GASSCF, SplitCAS et SplitGAS sont introduites d’un point de vue th´eorique et d’un point de vue algo- rithmique, et des tests sont pr´esent´es. Dans le chapitre 5 les conclusions g´en´erales sont pr´esent´ees en mettant en ´evidence les points forts de ces m´ethodes. Dans les annexes des informations additionnelles sont enfin pr´esent´ees.

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List of Publications i

Acknowledgments, Remerciements iii

Abstract vii

R´esum´e ix

1 Introduction 1

2 Multiconfigurational Systems 7

2.1 Metal–metal multiple bonds . . . 8

2.2 Uranium Complexes Supported by Redox-Active Ligands . . 12

2.3 The Cu₃Mg₃F₁₂ single–molecule magnet model . . . 14

3 The Generalized Active Space approach 27 3.1 Theory and algorithm . . . 29

3.2 A comparison between GAS and ORMAS . . . 34

3.3 Test calculations . . . 36

3.3.1 Gd atom . . . 36

3.3.2 Gd₂ molecule . . . 38

3.3.3 GAS applied to OxoMn(salen) compound . . . 43

3.3.4 GASSCF method applied to the Cr₂ system . . . 49

3.4 Conclusion . . . 53

4 The SplitCAS and SplitGAS 55 4.1 Theory . . . 59

4.1.1 The CI step . . . 59

4.1.2 The orbital optimization step . . . 63 xi

4.2 Algorithmic details for the SplitCAS . . . 64

4.3 Algorithmic details for the SplitGAS . . . 64

4.4 Size-extensivity in SplitCAS and SplitGAS . . . 67

4.5 SplitCAS test calculations . . . 69

4.6 SplitGAS test calculations . . . 72

4.7 Conclusions . . . 85

5 General Conclusions 87 A Route for an ‘on the fly’ SplitGAS method 91 A.1 Introduction . . . 91

A.2 First term of equation (A.2) . . . 93

A.3 Second term of equation (A.2) . . . 93

A.4 Third term of equation (A.2) . . . 96

B Algorithmic details in Generating CI expansions 119 B.1 Goal . . . 119

B.2 Active Orbitals . . . 120

B.3 Occupation Classes (IOCCLS) . . . 121

B.4 Group of Strings . . . 122

B.5 Types and SuperGroups . . . 124

B.6 Setting up Strings . . . 126

B.7 Combining Strings and SuperGroups Informations . . . 132

B.8 Number of combinations per Symmetry . . . 134

B.9 Summary . . . 138

B.10 CSFs and SDs connections . . . 139

B.11 Counting prototypes . . . 150

B.12 Configurations Distributed per Number of Open Orbitals . . 152

B.13 Configurations and Prototypes . . . 155

B.14 Generating and ordering Prototypes . . . 156

B.15 Partitioning and Baches of the C vector . . . 157

B.16 Generating configurations . . . 161

C Papers 167

Bibliography 227

Meticulous, yes. Methodical. Educated.

They were these things.

There were days of mistakes and laziness and infighting.

And there were days, good days, when by anyone’s judgment...

they would have to be considered clever.

“No one would say

that what they were doing was complicated.

It wouldn’t even be considered new.

Except maybe in the geological sense”

Introduction 1

It is well known that in order to accurately describe a molecular system a quantum chemical method must be able to correctly describe thecorrelation effects for that system. The term correlation effects has been firstly used by L¨owdin [1] to refer to the detailed description of mutual interaction of particles in motion. According to L¨owdin definition, the correlation energy is the difference between theexact non-relativistic energy within the Born- Oppenheimer approximation (achievable using the Full-CI method) and the Hartree-Fock limit, which represents the upper bound to the exact energy:

Ecorr =Eexact−EHF (1.1)

The Hartree-Fock scheme, or any single determinantal based method, are referred to as uncorrelated and in general are not adequate for quantitative accuracy. In single determinantal methodsFermi correlationis the only type of correlation taken into account as the wave function is described as an anti- symmetric superposition of spin-orbital products. Correlation effects have commonly been classified indynamicalandstaticcorrelation. Static correla- tion arises fromdegeneracyornear-degeneracyeffects. Consider for instance the hydrogen molecule at homolytic dissociation. Sometimes people refer to

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this type of correlation also asnon-dynamicalcorrelation as complementary to the dynamical correlation. This latter instead is strongly related to the interaction among particles of opposite spin to count for their continuous and instantaneous repulsion, which is not taken into account in uncorrelated schemes. Sometimes this type of correlation is also referred to asCoulomb effects as it is related to the Coulomb repulsion among particles. The au- thor would like to highlight on the fact that although there are specific cases where one can isolate one or the other type of correlation, in general it is impossible to clearly distinguish between the two effects and a method able to recover correlation will recover both.

From a wave function theory standpoint, static correlation is recovered by going from a single configurational to a multiconfigurational description of the system. In general, but it is not a rule, near-degeneracies are quite localized and involve a limited number of orbitals. For this reason, Multi- configurational Self Consistent Field (MCSCF) methods like the Complete Active Space Self Consistent Field (CASSCF) method [2–4] representad-hoc methods for treating this kind of correlation. In the CASSCF method the Configuration Interaction (CI) expansion is generated as a superposition of all the spin and space adapted functions that may be formed from the dis- tribution of a given number of active electrons in a given number of active orbitals. In CASSCF the CI parameters are optimized by solving the CI sec- ular equations and the orbital parameters are optimized via all possible ro- tations between inactive–active, active–virtual, and inactive–virtual spaces.

CASSCF has become maybe the most popular MCSCF method mostly be- cause the wave function is completely defined once the active space has been selected.

Dynamical correlation is not localized and would require a larger CI expan- sion in order to recover most of it. Due to the small accessible active space, the CASSCF is able to retrieve correlation only within the chosen active space and cannot recover the more delocalized dynamical correlation. The latter has been historically treated bya posterioriperturbative methods ap- plied on an optimized multiconfigurational wave function, like the CASPT2 scheme [5, 6]. The unified CASSCF/CASPT2 strategy is currently among the most widely used ab-initio methods in recovering electron correlation.

Examples of the success of the CASSCF/CASPT2 strategy are reported in Chapter 2.

The CAS paradigm reaches its limit in terms of memory requirements as soon as one reaches an active space containing 16 orbitals and 16 electrons (for a singlet spin state). This limit is due to the scaling of the CI expansion with respect to the size of the active space.

Most of the configurations in the CI space contribute only marginally to the total wave function. Ivanic and Ruedenberg [7–10] systematically investi- gated the amount of ineffective configurations, as known as ‘deadwoods’, and found that they represents more the 99 % of the CI space generated. This feature is also related to the fact that most of the CI-Hamiltonian matrices are sparse, diagonally dominated matrices.

One of the challenges still to be solved is therefore to findab-initiomethods, able to recover most of the electron correlation, static as well as dynamical, able to reproduce quantitative accuracy, and at the same time not penalized by their scaling. If researchers would have the possibility to exceed the cur- rent limits of CASSCF type of calculations and handle larger active spaces, new and complex systems, such as macro-molecules, transition metal clus- ters, metallorganic frameworks, single–molecule magnets, could be better understood in terms of structure and reactivity and thus finely controlled and optimized.

The main focus of this thesis is the development of such kind of methods.

It is opinion of the author that this is one of the rare cases in which a fist- ful of scientists in a relatively limited period of time managed to develop, test and perform computational scale-up of two successful approaches, suc- ceeding also in combining them in a unique unified tool to overthrow the computational costs of Complete Active Space based calculations.

The first method developed is the Generalized Active Space Self Consistent Field (GASSCF) method [11]. In GAS the user has the possibility to choose an arbitrary number of active spaces and control the inter-spaces excitations by means of constraints on the occupation number for each orbital space. No constraints are given for the intra-space excitations, which therefore are all included. By this method most of the configurations which area-prioricon- sidered ineffective (based on chemical considerations) are not generated and thus not included in the CI expansion. The configuration space is reduced and the active space can be enlarged. A greater description of the method is given in Chapter 3 alongside the results obtained for three test calcula- tions, namely the Gd atom, the Gd dimer and the OxoMn(Salen) system

which have been benchmarked against the conventional CASSCF/CASPT2 strategy. A detailed description of the calculations performed and results can also be found in Ref. [11].

The second method developed is named SplitCAS [12]. The basic idea of the SplitCAS is to partition the CI expansion generated by a CAS choice into two parts, a principal part (P) containing few relevant configurations (10⁴), able to recover the most significant portion of the non-dynamical correlation, and an extended part (Q) containing less relevant (but not neg- ligible) configurations (10¹²), necessary to recover the static correlation not treated by the (P) choice and most of the dynamical correlation needed to accurately describe the system. The CI-Hamiltonian matrix is partitioned into four blocks accordingly. L¨owdin’s partitioning technique [13], in Split- CAS approximated to the second order (diagonal approximation), is then used to reduce the full CI Hamiltonian matrix to the size of the principal block (PP). The principal block is dressed perturbatively through the ex- tended blocks and then diagonalized. The keypoint of the SplitCAS is the criterium chosen for the splitting between the (P) and the (Q) space. In Ref. [12] an energy-based splitting has been investigated. The configuration state functions (CSFs) are firstly generated, then ordered according to their approximate energy and finally according to an user specific threshold di- vided into (P) and (Q) parts. It is important to notice that in the SplitCAS method instead of reducing the size of the CI expansion (GAS strategy), one may retain a CAS ansatz, where the principal part is countedin totowhile the extended part is included perturbatively without affecting the memory requirements of the calculation. The two methods have been then combined together in what has been called SplitGAS. In the SplitGAS method, the authors pushed the SplitCAS algorithm to the limit of the GAS strategy.

In SplitGAS an orbital based partitioning has been chosen to separate the principal space from the extended space. By tuning number of active spaces and occupation number constraints the user can perform almost any kind of truncation of the configuration space (feature of the GASSCF strategy) and any kind of orbital-based partitioning of the resulting CI expansion (feature of the SplitCAS strategy); the whole thing accordingly to the chemistry of the system under investigation. This route greatly solves the scaling prob- lems of the CI expansion in CAS-type of calculations. Only entangled and chemically meaningful configurations will effectively contribute to the CI

expansion in the principal and extended space. From the memory point of view the savings are substantial. In fact, as described in better details in Appendix B, the structure of the GAS code is such that the CI elements do not need to be processed all at once, instead they are divided inblocks.

It follows that never a full CI expansion in (P)+(Q) is loaded in memory.

More details on how the blocks are generated are given in Appendix B. The theoretical foundation of the SplitCAS and SplitGAS strategies is described in greater detail in Chapter 4, alongside the some benchmark calculations on previously studied systems against the conventional CASSCF/CASPT2 approach.

With the SplitGAS method in hands it has been possible to successfully attack the Cr₂ system. The Cr dimer in its singlet ground state repre- sents a challenging system for any method aiming at recovering correlation effects. It represents an example of strongly correlated systems in which both static and dynamic correlation play an important role and only by correctly recovering both of them it is possible to accurately describe its dissociation and reproduce the experimental spectroscopic parameters. The minimal choice of active space for the Cr dimer consists of 12 electron dis- tributed in 12 orbitals, CAS(12,12). The potential energy curve generated by the CASSCF(12,12) does not lead to a bound dimer. The reason for this failure is that the size of the active space is too small to fully recover both static and dynamical correlation. Only after a ‘properly calibrated’

CASPT2 correction it has been possible to mimic the experimental poten- tial energy curve. By ‘properly calibrated’ the author critically refers to the necessary parametrization of the CASPT2 method, which has been de- veloped in the years in order to overcome wrong outcoming of the method and which in author’s opinion make this method not fully ab initio. In this respect, several CASPT2 potential energy curves for the Cr dimer have been proposed over the years, corresponding to different parametrizations of the method. The original CASPT2 [14] was affected by a severe intruder states problem which ‘deteriorated’ the results for the Cr₂ system. Using the level-shift technique in the LS-CASPT2 [15] method the intruder state problem was partially solved but a first parameter was introduced. Also an imaginary level-shift was introduced as alternative solution to the intruder states problem [16]. Another and maybe the most recent parametrization was introduced in 2004 by means of a shifted zeroth-order Hamiltonian [17]

(also known as IPEA shift). This approximation corrects for the systematic error of the original formulation of the method for which the correlation en- ergy was overestimated. It has been proved that the shape of the potential energy curve of the Cr dimer is affected by the choice of this parameter [18]

which leads to different spectroscopic parameters accordingly.

In the SplitGAS method any type ofempiricalorsemi-empiricalparametriza- tion is avoided. The results depend only on the initial choice of the active space in the principal (P) and extended (Q) space. This approach guar- antees to the user, as for the CASSCF method, the possibility to judge a-priori what are the limits of the method based on the size of the cho- sen active space, and improve systematically the results by enlarging it if necessary.

Multiconfigurational Systems 2

There are many examples of systems requiring a multiconfigurational treat- ment to reach quantitative accuracy in the description of their electronic structure, properties and reactivity, such as bond breaking and dissocia- tions [19], potential energy hypersurface (PEH) degeneracies (conical in- tersections) [20], symmetry breaking problems (Cope rearrangement) [21], biradical situations [22, 23], organic molecules photophysics [24–27], transi- tion metal bonding [28–34] and spectroscopy [35–39], magnetic properties of single–molecule magnets (SMMs) [40] and actinide chemistry [41–45].

In this chapter several examples of multiconfigurational systems will be briefly discussed [28, 29, 40, 41]. These systems have been studied before the new approaches, GASSCF, SplitCAS and SplitGAS have been devel- oped and tested. Therefore, only the CASSCF/CASPT2 strategy has been applied on most of them. Although the CASSCF/CASPT2 approach has proven accurate in describing these systems, its limits must be mentioned as well. This chapter is not intended to be a comprehensive description of the projects. Instead by this chapter the author wants to highlight on (a) the types of systems requiring multiconfigurational treatment, (b) the limi- tations one can encounter by treating these systems at a CASSCF/CASPT2

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level of theory and (c) the necessity of methods that can give the users the possibility to freely enlarge the active space when required. In describing the metal–metal bonding in Ref. [28, 29] it was not possible to include all the necessary orbitals of the ligands in the active space, as the orbitals of the metal centers were already able to saturate the active space. Beside the lack of accuracy that might occur in describing the metal–ligand interac- tion, a detailed description of the metal to ligand charge transfer (MLCT) excitations resulted unfeasible. In Ref. [28] it was not possible to address the trinuclear heterometallic molecules whose optical properties are of great interest. Only in Section 2.3 of this chapter the reader will find a brief com- parison of CASSCF and GASSCF calculations. More and better discussed examples of multiconfigurational systems treated at GASSCF level of the- ory will be discussed in Chapter 3 and systems analyzed by SplitCAS and SplitGAS methods introduced in Chapter 4.

2.1 Metal–metal multiple bonds

Ever since the discovery of the multiple metal–metal bond in [Re₂Cl₈]²⁻[46], there has been a considerable amount of research dedicated to metal–metal multiple bonding in transition metal compounds. Electron-rich metal–metal units are of general interest because of their unique electronic and optical properties. After Cotton’s discovery, no real breakthrough was achieved in the field of the multiple bonding until the synthesis of Ar’CrCrAr’ [where Ar’ indicates C₆H₃-2,6(C₆H₃-2,6-Prⁱ₂)₂ and Prⁱ indicates isopropyl] (Cr–

Cr=1.835˚A) by Power and co-workers in 2005 (Figure 2.1(a)) [47], bringing to light the first compound featuring a formal quintuple bond. Such accom- plishment was possible by making the right choice of transition metal, as well as providing a ligand assuring sufficient stabilization of the metal–metal bond. In this perspective, the Group 6 elements, are the best candidates for the quintuple metal–metal interaction. Five out of their six valence electrons can be used to form the quintuple bond, leaving one electron free to share a bond with the ligand. The choice of the ligand is the crucial parameter that allows keeping the transition metal in the lowest possible oxidation state, at the same time maximizing the number of valence electrons available for the formation of multiple bonds. The synthesis of Ar’CrCrAr’ gave new momentum to the chemistry of multiple-bonded metal–metal species for the

isolation of increasingly shorter metal–metal interactions. In 2007 Kreisel et al. [48] isolated and characterized the dichromium diazadiene (Figure 2.1(b)) with a Cr-Cr bond length of 1.803 ˚A. This study opened the possibility to isolate species featuring short metal–metal bonds with various transition metal elements by controlling the bond length by ad hocdesigned ligands.

In 2008 Tsai and co-workers [49] showed that amidinate ligands can be used to achieve very short Cr–Cr bond length within a paddlewheel-type archi- tecture (Figure 2.1(c)). The most interesting feature of these ligands is their

(a) Aryl (b) diazadiene (c) amidinate

Figure 2.1: Structures of several compunds studied in this project.

possible modification by attaching different functional groups to the ligand.

Such fine tuning of the ligand allows the ligand to behave as a modifiable pincer, forcing the two transition metals to approach each other to different extents.

These compounds have been computationally studied by the multiconfigu- rational CASSCF/CASPT2 strategy to obtain a theoretical insight into the nature of their metal–metal bonding [29]. The limits on the size of the ac- tive space did not allow the authors for a deeper analysis of possibleπ–type interactions between the ligands and the metal centers. These kind of inter- actions were found for the diazadiene compound synthesized by Kreisel.

The bond multiplicity was evaluated by means of the effective bond order (EBO) concept [50, 51], which reflects the multiconfigurational character of the system. In fact, for a certain bond the EBO is given by eq. 2.1,

EBO= (η_b−η_ab)/2 (2.1)

in whichη_b andη_ab are the sums of the occupation numbers of the bonding and anti-bonding natural molecular orbitals pairs derived from the CASSCF wavefunction. Selected experimental and theoretical (CASPT2) Cr–Cr equi- librium bond distances are summarized in Table 2.1 alongside the computed EBO values.

Compound Exp. [˚A] CASPT2 [˚A] EBO

terphenyl (Ar) 1.835 1.836 3.07

diazadiene 1.803 1.799 3.43

amidinate 1.740 1.738 3.91

Table 2.1: Experimental and CASPT2 Cr–Cr bond lengths of different Cr2 compounds.

Also the EBO value is reported for each compound.

On the basis of the nature of the ligands two groups of compounds can be defined. The first one includes Ar’CrCrAr’ in which the element coordinat- ing the metal center is a carbon atom. In this case, the involved ligand is not able to impose any constraints on the diatomic unit, which results in a larger Cr–Cr bond length. The second group encompasses the species in which nitrogen is the element coordinating the metal center. In contrast to the first group, these are bridging ligands, and the two nitrogens of a single ligand are able to impose constraints on the Cr₂ unit. The pincer effect in these species is evident. A direct correlation between the Cr–Cr bond distance and the EBO value can also be drawn. The reader is invited to read the original paper (Ref. [29]) for further details on this project.

For these species the CASSCF/CASPT2 approach reaches accuracy in the Cr–Cr equilibrium bond distance of the order of the pm or higher. However, it was not possible to assess with higher accuracy the interaction between the bimetallic unit and the ligands; the possibility of aπ-type of interaction between the p orbitals of the ligands and the δ orbitals of the metal core could not be ascertained.

Burdzinski et al. [52] prepared interesting oligomers of empirical formula [Mo₂(TiPB)₂(O₂C(Th)-C₄(n-hexyl)₂S-(Th)CO₂)] (TiPB=2,4,6-triisopropyl benzoate; Th=thiophene) and compounds of formulaetrans–[Mo₂(TiPB)₂L₂] in which the ligand L=Th, BTh (BTh = 2,2’-bi-thiophene-5-carboxylate) and TTh (the corresponding thienylcarboxylate), which are considered as models for the oligomers. The X-ray analysis of trans–[Mo₂(TiPB)₂BTh₂] (Figure 2.2(a)) revealed the presence of Lπ*–M₂δ–Lπ* conjugation, and

density functional theory (DFT) calculations indicated that the HOMO is mainly a M₂ δ orbital and the LUMO is mainly based on the thienylcar- boxylate π* orbitals. They also studied the photophysical properties of these oligomers, which showed metal-to-ligand charge-transfer (MLCT) ex- citations.

Alberding et al. [53] prepared the [MM’(TiPB)₄] compounds, in which M

= Mo or W and M’ = W and characterized them with various techniques.

They also found out that these compounds show a strong MLCT excitations.

Nippe and co-workers [54] reported the synthesis of [W₂(dpa)₄] (dpa = 2,2’- dipyridylamide) (Figure 2.2(b)) and its characterization by X-ray crystal- lography and cyclic-voltammetry. They also synthesized one-electron oxida- tion products of [W₂(dpa)₄] and [Mo₂(dpa)₄], namely [W₂(dpa)₄][BPh₄] and [Mo₂(dpa)₄][BPh₄] (BPh₄ = tetraphenylborate). The molecules [W₂(dpa)₄]

(a)trans–[Mo2(TiPB)2BTh2] (b) [W2(dpa)4]

Figure 2.2

and [Mo₂(dpa)₄] have been utilized along with the [Cr₂(dpa)₄] analogue to prepare linear, trinuclear heterometallic molecules with an M–M–M’ chain where M = Cr, Mo or W, and M’ = Cr, Mn, Fe, Co, Ni, and Zn [55–58].

The heterometallic molecules show rich optical and redox properties, and a better understanding of these properties can be greatly facilitated by a quantum chemical analysis. However, a CASSCF/CASPT2 investigation of these trinuclear species is unfeasible as they would certainly exceed the limits on the size of the active space, if all nd (n= 3, 4 or 5) orbitals were included in the active space, necessary condition to correctly describe the electronic properties of these systems. Only a quantum chemical analy- sis of the Cr₂, Mo₂, and W₂ precursor molecules has been performed at

CASSCF/CASPT2 level of theory [28]. For Burdzinski’s and Nippe’s com- pounds experimental absorption spectra are available. A combined used of TD-DFT and CASSCF/CASPT2 techniques allowed the assignment of the absorption bands. The CASPT2 accurately predicted the metal–metal ex- citations, while the TD-DFT estimated the MLCT and LMCT excitations.

The CASSCF/CASPT2 approach could not be used to predict the MLCT and LMCT excitations as no orbitals of the ligands could be included in the active space. The reader is invited to read the original paper (Ref. [28]) for further details on this project.

2.2 Uranium Complexes Supported by Redox-Active Ligands

Redox-active ligands are becoming popular for use in organometallic chem- istry because of their ability to stabilize reactive metal centers [59, 60].

However, these ligands create ambiguous metal oxidation states; thus, mul- tiple characterization techniques must be used to establish the electronic structure of these complexes. Ligand oxidation states can be determined by examining geometric distortions obtained from the molecular structure, which in turn helps to assign the metal oxidation state. Further compu- tations based on multiconfigurational methods are useful to accurately de- termine the ground state and low-lying excited states often present in sys- tems with these types of complicated redox non-innocent frameworks. Two uranium compounds supported by redox–active α–diimine ligands, namely (^{M es}DAB^{M e})₂U(THF) and Cp₂U(^{M es}DAB^{M e}) with ligand ^{M es}DAB^{M e} = [ArN=C(Me)C(Me)=NAr] and Ar = 2,4,6–trimethylphenyl (Mes) (depicted in Figure 2.3) have been investigated.

Although formally ‘U(0)’ and ‘U(II)’, respectively, characterization by X-ray absorption spectroscopy (XAS) indicates that these species contain tetrava- lent uranium centers, a formulation made possible by invoking two–electron reduction of eachα–diimine ligand to form ene–diamide frameworks [61–63].

The redox–active nature of the ligand is supported by both electronic ab- sorption spectroscopy, which shows characteristic f–f transitions expected for uranium(IV), and X–ray crystallography, which shows the expected bond distortions associated with occupation of the ligand π* orbitals [61–63].

(a) (^{M es}DAB^{M e})2U(THF) (b) Cp2U(^{M es}DAB^{M e})

Figure 2.3: Molecules used for computational study. Aryl groups have been truncated for clarity.

Through a detailed CASSCF/CASPT2 analysis it was possible to ascertain the redox–active nature of the ligand for both compounds of Figure 2.3;

each ^{M es}DAB^{M e} ligand can host two electrons in its anti-bonding π3 or- bital (Figure 2.4). For these two systems the active space included both

(a)π1 (b) π2 (c)π3 (d) π4

Figure 2.4: Schematic representation of the four (HOMO-1), HOMO, LUMO and (LUMO+1) molecular orbitals on the ligand.

metal- and ligand-based orbitals. It was possible to accurately determine at CASSCF/CASPT2 level of theory the electronic structure of the the ground state for the two compounds. Also it was possible to determine the ener- getically low-lying singlet, triplet and quintet spin states and assign the experimentally determined absorption bands.

2.3 The Cu

Mg

F

single–molecule magnet model

Single–molecule magnets (SMMs) have been widely studied both for their fundamental physical properties and for possible applications in magnetic storage and quantum information [64, 65]. By tuning external parameters, such as magnetic field and temperature, these molecules can behave either like traditional bulk magnetic materials or as quantum systems; in fact, they can be magnetized in a magnetic field without any interaction between the individual molecules. This magnetization is a property of the molecules themselves. Due to their dual quantum/classical nature, single–molecule magnets are important to understand the interplay between classical and quantum magnetism. It has been found [66] that in some molecular anti- ferromagnets lacking inversion symmetry, such as the triangular antiferro- magnetic{Cu₃}(Figure 2.5) and other odd–spin rings, an electric field can efficiently couple spin states through the dipole operator, opening the possi- bility to use this objects as tunable quantum bits. Arrays of 3-fold triangular

Figure 2.5: Simplified model of the{Cu3}single-molecule magnet.

spin 1/2 have been made experimentally. Detailed analysis on this type of compounds have been performed by Dalalet al. [67]. In particular they re- ported on the electric-field induced splitting of states for the{Cu₃}SMM of chemical composition Na[Cu₃(AsW₉O₃₃)·3H₂O]·32H₂O. On the theoretical side these compounds have been widely investigated by density functional theory (DFT) based methods (especially the larger models) [68]. Model Hamiltonian can be extracted from single-determinental broken-symmetry analysis. A detailed multiconfigurational analysis of these systems would greatly help the physics community to understand the magnetic proper-

ties of this compounds, as near-degenerate in situ and CT excited states, which represent the key point for these compounds, are properly addressed by MCSCF methods, as opposed to a single-determinantal approach, which would result in a non-accurate description. A multiconfigurational (MC) approach would ascertain certain assumptions of the classic models, like the Hubbard approximation, in which only the CT excitations are considered to be important. In particular an MC approach could confirm or disprove if indeed only CT configurations are sufficient to describe the system or if instead also on-site excitations have to be considered for quantitative accu- racy. A simplified model of the {Cu₃} molecule, namely the Cu₃Mg₃F₁₂, still able to preserve the essential physics of the system has been constructed by the authors of Ref. [68]. It has a D_3h symmetry. Each Cu(II)-d⁹ atom has a local square-planar coordination of F (or O) atoms. The Mg (or Na) counterions are placed in the belt region of the molecule. From a crystal field theory standpoint, the 3d atomic orbitals on each Cu atom, due to the square-planar coordination, are split in (a) a group of three energetically low-lying doubly occupied orbitals (3d_xz, 3d_yzand 3d_z2), (b) the doubly oc- cupied 3d_xy orbital and (c) the singly occupied 3d_x2−y² orbital. The empty 4s orbital is located above the 3d orbitals. The three unpaired spins may couple to form two degenerate doublets (total spin S = 1/2) or a quartet spin state (S = 3/2). In situexcitations (3d–3d and 3d–4s) as well as charge transfer (CT) excitations (from one metal center to another), in principle, may occur. To notice, however, that past efforts have assumed that on-site excitations are not important from the standpoint of the spin-electric effect.

For instance, in the Hubbard approximation, which is the most widely used to describe the spin-electric effect in these compounds, only 3d–3d CT ex- citations are assumed to be of interest.

This fascinating system has been addressed by a CASSCF/CASPT2 anal- ysis in order to give quantitative answers about the electronic structure of ground and excited states and give new insights on the magnetic properties of this kind of systems.

The CASSCF/CASPT2 calculations were performed using the MOLCAS-7.7 package [69]. Basis set of the atomic natural orbital type with double-zeta plus polarization quality (ANO-RCC-VDZP) was used for all atoms. Scalar relativistic effects were included using the Douglas-Kroll-Hess Hamiltonian.

The computational costs arising from the two-electron integrals were dras-

tically reduced by employing the Cholesky decomposition technique [70, 71].

The decomposition threshold was chosen to be 10⁻⁴, as this should corre- spond to an accuracy in total energies of the order of mHartree or higher.

At CASPT2 level of theory orbitals 1s for the F atom, orbitals up to and including 2s for Mg atoms and orbitals up to and including 3s for Cu atoms were kept frozen. Since the symmetry of the system is D_3h, while MOLCAS can only work with abelian groups, it was preferred to work within the C₁ point group in order to equally treat the three centers. An active space of 9 electrons distributed in 12 orbitals was chosen. The active orbitals are mainly linear combinations of the three 3d_xy doubly occupied orbitals and three 3d_x2−y² singly occupied orbitals of the Cu atoms; the remaining six or- bitals are empty correlating orbitals mainly with 4d_xy and 4d_x2−y² character.

The optimized CASSCF(9,12) natural orbitals were subsequently localized as follows: (I) all the inactive and virtual orbitals were kept frozen, (II) the twelve active orbitals were split in 4 groups of three, each group containing same atomic orbital contributions from the three metallic centers (delocal- ized) (III) Boys localization was performed on each group, while the nine left active orbitals were kept frozen. The localized orbitals are depicted in Figure 2.6.

(a) [3d^A_xy] (b) [3d^B_xy] (c) [3d^C_xy]

(d) [3d^A_x2−y²] (e) [3d^B_x2−y²] (f) [3d^C_x2−y²]

(g) [4d^Axy] (h) [4d^Bxy] (i) [4d^Cxy]

(j) [4d^A_x2−y²] (k) [4d^B_x2−y²] (l) [4d^C_x2−y²]

Figure 2.6: Localized active orbitals for the CAS(9,12) of the Cu3Mg3F12 system.

These orbitals were used as input orbitals for: (a) a CAS-CI calculation, (b) a CASSCF/CASPT2 calculation and (c) a GASSCF calculation. In the CAS-CI method only the CI parameters are optimized, while the orbitals are not altered. In CAS-SCF, both CI and orbital parameters are optimized.

All kind of excitations (in situ as well as charge-transfer) are intrinsically included in the CI expansion for the CAS-CI and CAS-SCF calculations.

Also for the GAS-SCF calculations both CI and orbital parameters are op- timized. In the GASSCF(9,12) the orbitals were split in three orbital-spaces, one per Cu center and each containing only the orbitals for that center. No charge-transfer excitations were allowed by GAS choice (they are referred to as disconnected spaces in the original GASSCF paper [11]). The pos- sibility in GASSCF to selectively exclude the CT excitations from the CI expansion gave the possibility, after comparison with the CASSCF results, to learn about the influence of the CT configurations on the total energy for the ground and excited states. For the three kinds of calculation a State- Average optimization was performed by including four quartet and eight doublet spin states. The results are summarized in Table 2.2.

State CAS-CI(9,12) CAS-SCF(9,12) GAS-SCF(9,12) CAS-PT2(9,12) Quartet

I 0.00 0.00 0.00 0.00

II 1.07 1.07 1.07 1.33

III 1.07 1.07 1.07 1.24

IV 1.07 1.07 1.07 1.24

Doublets

I 0.00 0.00 0.00 0.00

II 0.00 0.00 0.00 0.00

III 1.07 1.07 1.07 1.32

IV 1.07 1.07 1.07 1.33

V 1.07 1.07 1.07 1.19

VI 1.07 1.07 1.07 1.19

VII 1.07 1.07 1.07 1.20

VIII 1.07 1.07 1.07 1.20

Table 2.2: CAS-CI, CAS-SCF, CAS-PT2 and GAS-SCF results for the CAS(9,12). Rel- ative energies given in eV.

The CAS-CI, CAS-SCF and GAS-SCF absolute energies are identical. This result might be explained by considering that (a) the further CAS-SCF opti- mization on the localized orbitals does not converge to a different minimum, (b) the result is not altered by excluding the charge transfer configurations from the CI expansion in the GAS calculation. Therefore, the CT configura- tions may be considered ‘dead-woods’ for this model system. The first two

doublet and the first quartet states are degenerate. They are all dominated for 98% by the electronic configuration:

(3d^A_xy)²(3d^B_xy)²(3d^C_xy)²(3d^A_x2−y²)¹(3d^B_x2−y²)¹(3d^C_x2−y²)¹.

The unpaired spins are [u u u] for the quartet and [u u d] and [u d u] for the doublet states. The three excited quartet and six excited doublet states at 1.07 eV above the ground states are also all degenerate. They corresponds to in situ single excitations from 3d_xy to 3d_x²_−y² orbitals as described in Table 2.3.

State 3d^A_xy 3d^B_xy 3d^C_xy 3d^A_x2−y2 3d^B_x2−y2 3d^C_x2−y2 Quartet

I 2 2 2 u u u

II 2 2 u u u 2

III 2 u 2 u 2 u

IV u 2 2 2 u u

Doublets

I 2 2 2 u u d

II 2 2 2 u d u

III 2 2 u u d 2

IV 2 2 u d u 2

V 2 u 2 d 2 u

VI 2 u 2 u 2 d

VII u 2 2 2 d u

VIII u 2 2 2 u d

Table 2.3: Dominant configurations for the eight doublet and four quartet spin states.

Other excitations were identified by increasing the number of states for the SA-CASSCF calculations to eight quartet and sixteen doublet states. The results are summarized in Table 2.4. The new computed excitations are double and triplein situ excitations.

Higher excited states were systematically investigated. In situ excitations from 3d orbitals to 4s orbitals appeared in the energy range between 8.6 eV and 9.8 eV. Increasing the number of orthogonal states to be simultaneously optimized at the state-average CASSCF(9,12) level of theory, resulted in a systematic worsening of the optimized orbitals. This is due to (a) an unavoidable lost of accuracy of the SA-CASSCF calculations over the lower states as new excited states are involved and (b) ‘orbital flippings’ in the states higher in energy with respect to the lower in energy. In fact, for the former states 4s virtual orbitals displaced the 4d correlating active orbitals and the 3d inactive orbitals displaced the 3d active orbitals. In order to

State 3d^A_xy 3d^B_xy 3d^C_xy 3d^A

x²−y² 3d^B

x²−y² 3d^C

x²−y² CASPT2 excitation

energy type

Quartet

1 2 2 2 u u u 0.00 Ground State

2 2 2 u u u 2 1.31 Single

3 2 u 2 u 2 u 1.18 Single

4 u 2 2 2 u u 1.18 Single

5 2 u u u 2 2 2.49 Double

6 u 2 u 2 u 2 2.49 Double

7 u u 2 2 2 u 2.61 Double

8 u u u 2 2 2 3.90 Triple

Doublets

1 2 2 2 u u d 0.00 Ground State

2 2 2 2 u d u 0.00 Ground State

3 2 2 u u d 2 1.31 Single

4 2 2 u d u 2 1.32 Single

5 u 2 2 2 u d 1.18 Single

6 2 u 2 u 2 d 1.18 Single

7 u 2 2 2 d u 1.18 Single

8 2 u 2 d 2 u 1.18 Single

9 2 u u d 2 2 2.48 Double

10 2 u d u 2 2 2.48 Double

11 u 2 u 2 d 2 2.48 Double

12 u 2 d 2 u 2 2.49 Double

13 u u 2 2 2 d 2.62 Double

14 u d 2 2 2 u 2.62 Double

15 u u d 2 2 2 3.91 Triple

16 u d u 2 2 2 3.90 Triple

Table 2.4: Dominant configurations for the eight quartet and sixteen doublet spin states.

The CASPT2 relative energy is given in eV.

describe consistently the ground state as well as the excited states, state- average calculations were performed without orbital optimization, under the assumption that the localized orbitals depicted in Figure 2.6 are suitable to correctly describe CT states. The drawback of this approach is that the 3d–4s excitations are lost as the 4s orbitals are not included into the active space. Moreover, by constraining the 4s orbitals to be virtual in the CI calculation resulted in an ‘intruder-state’ problem at CASPT2 level (vide infra).

SA-CAS-CI(9,12) calculations were performed by including 14 states for the quartet and 28 states for the doublet spin states. At this level of theory the first eight states for the quartet are the above mentionedin situexcitations, while from state 9 to 14 CT excitations appear (Table 2.5). For the doublet, in situexcitations occur up to the sixteenth state, while from state 17 to 28 CT excitations are observed (Table 2.6).