1

Chapter 8

Relativistic Pseudopotential Calculations for ElectronicExcited States

Christian Teichteil, Laurent Maron a andValerie Vallet b

aLaboratoire de Physique QuantiqueI.R.S.A.M.C, Universite Paul Sabatier and CNRS (UMR 5626)118, route de Narbonne, 31062 Toulouse cedex, FranceElectronic-mail: christian.teichteil@irsamc.ups-tlse.fr

bInstitute of Theoretical and Physical ChemistryTechnical University of MunichLichtenbergstrasse 4D-85747 Garching, Germany

AbstractRelativistic and electron correlation effects play a important role in the electronic structure of

molecules containing heavy elements (main group elements, transition metals, lanthanide and

actinide complexes). It is therefore mandatory to account for them in quantum mechanical

methods used in theoretical chemistry, when investigating for instance the properties of heavy

atoms and molecules in their excited states. In this chapter we introduce the present state-of-

the-artab initio spin-orbit configuration interaction methods for relativistic electronic structure

calculations. These include the various types of relativistic effective core potentials in the scalar

relativistic approximation, and several methods to treat electron correlation effects and spin-

orbit coupling. We discuss a selection of recent applications on the spectroscopy of gas-phase

molecules and on embedded molecules in a crystal environment to outline the degree of maturity

of quantum chemistry methods. This also illustrates the necessity for a strong interplay between

theory and experiment.

1. INTRODUCTION

Modelling molecular systems of chemical interest is still a daunting chal-lenge for theoretical chemists. Present methodological developments are aim-

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ing at defining more and more efficient approaches able to handle very largemolecules, as well as improving the accuracy of existing methods. The treat-ment of relativity and electron correlation enters the latter theme. Althoughrelativistic methods have been exhaustively reviewed in the first part of thisbook [1], it remains to go closely into the matter of the spin-orbit configura-tion interaction methods (SOCI) following scalar relativistic SCF calculations,in connection with the pseudopotential techniques. In this field we also refer tothe recent excellent review article by Heß and Marian [2] which mainly focuseson all-electron approximations rather than on pseudopotential techniques.

In the frame ofab initio methods, much attention was paid in the last decadesto the fields of relativity and correlation. Relativistic approximations have beendefined in all-electron schemes [1] able to handle correlation, even using sophis-ticated CI methods. Unfortunately such methods are very computer resourcedemanding, and are therefore severely limited to systems containing at mosttwo heavy atoms. Three ingredients can be employed, separately or altogetherto improve the efficiency in the calculations :

(a) the elimination of the chemically inactive electrons,

(b) the split of the relativistic operator into a scalar and a spin-orbit operator,

(c) the use of effective Hamiltonian techniques to make easier the compu-tationally demanding methods which treat electron correlation and spin-orbit coupling simultaneously.

Among the numerousab initio tools devoted to extend the capability of the-oretical methods, one of the first proposed [3, 4] dealt with the elimination ofthe inactive electrons in the bonding description, or in valence spectroscopyof molecules. One can discriminate two kinds of inactive electrons: the onesbelonging to inactive molecular fragments which can be represented by Effec-tive Group Potentials (or Group Pseudopotentials) [5], and others belonging toatomic cores, which can be represented by Effective Core Potentials (ECPs). Apractical implementation of the Effective Group Potential method has only beenrecently developed, and few chemical groups have been defined up to now [6].Apart from first very promising spectroscopic tests reported in Ref. [6], thereare practically no spectroscopic applications and we do not discuss this themeany further. Effective core potentials are divided into two families:

• the Model Potentials which keep the nodal structure of the orbitals fromall-electron reference calculations intact, whose modern implementation

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is the Ab Initio Model Potential (AIMP) approximation presented in thisbook (cf. chapter 7)

• the atomic pseudopotentials for which nodeless valence pseudoorbitalsare defined (for the general theory of pseudopotentials we refer to chap-ter 14 in part 1 of this book [7]).

In the core region the absence of nodes in valence pseudoorbitals is of consider-able advantage since this region can now be described by a small number of ba-sis functions, and one can mainly focus on the quality of the basis set for the va-lence region. It is sometimes argued that pseudopotentials, which do not retainthe proper nodal structure of valence orbitals, ”are not very efficient for com-puting spin-orbit coupling directly” [8]. Of course, if one uses the true operatorfor a core observable, like the spin-orbit interaction on nodeless pseudoorbitals,unphysical results are obtained. In fact, to high accuracy any observable can berepresented by a specific pseudopotential whose action on a given pseudoorbitalmimicsdirectly the action of the true operator on the corresponding true orbital(see section2.1.1). One of the main approximations used in the pseudopotentialtheory is based on the core-valence separation. This separation is well definedfor so-called ”hard core” atoms such as halogens, which have a weakly polaris-able atomic core. However, it is less obvious for transition metals or for the lefthand side of the periodic table, as the atomic core becomes easily polarisable.In these cases, a Core-Polarisation Potential (CPP) can be applied to accuratelycorrect for interactions between the core and valence space (cf. section2.1.3).

Pseudopotentials have been the subject of considerable attention in the lasttwo decades, and they have been developed by a number of different groups.They are also the most widely used effective core potentials in chemical appli-cations either for the study of chemical reactions or spectroscopy. A large va-riety of pseudopotentials are now available, and all the coupling schemes at theSCF step have been implemented: four-component, two-component, and scalarrelativistic along with spin-orbit pseudopotentials. However, it is well knownthat four-component calculations can (in the worst cases) be 64 times more ex-pensive than in the non relativistic case. In addition, the small component of theDirac wave function has little density in the valence region, and pseudopoten-tial calculations can safely be handled within a two-component approximation(cf. chapter 6).

The second point (b) arises at this stage of approximation. The original two-component pseudopotentials [9] were defined as fine-structure pseudopotentialswhose SCF solutions are nodeless two-component spinors|φl , j〉 ( j = l ±1/2).

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Ideally this would be the best solution, apart from the problem of the computerresources, especially for heavy atoms for which one may expect a large spa-tial separation between the two components of the valence spinors. However itwas shown on a well known and difficult case for spin-orbit correlated meth-ods, namely the thallium atom, that appropriate SOCI treatments based on ascalar relativistic pseudopotential approach are able to accurately account forthe rather large spin-orbit splitting of the2P ground state [10, 11]. Moreover,the quest for better accuracy implies to carry out the best electronic correlationtreatment as possible. This is especially a necessity for molecules containingdand f elements, for which the strong intricacy of the molecular states needs toaccount for both electron correlation and spin-orbit coupling. As the bottleneckremains mainly in the correlation treatment and as the CI calculations are muchmore easier performed in non relativistic symmetries, the solutions which arecommonly adopted are based on the first or both following criteria:

• use of scalar relativistic pseudoorbitals instead of two-component pseu-dospinors to treat the SOCI,

• separate the correlation treatment from the spin-orbit coupling.

Working with scalar relativistic pseudoorbitals implies to split the Dirac op-erator into a scalar and a spin-orbit operator. This was first proposed inan all-electron scheme by Dyall [12], and even if it appears somewhat arbi-trary [13,14], it bases the above approximations. It turns out that one can ap-ply it to any valence-only scheme. Moreover, the core electrons which mostlyexperience the relativistic effects, can be mimicked by simple relativistic pseu-dopotentials extracted from fully relativistic all-electron atomic reference cal-culations. The explicitly treated valence electrons have a low average velocity,making the splitting of the relativistic operator into a scalar and a spin-orbitones, if not less arbitrary, at least easier. Finally, it has been argued that electroncorrelation and spin-orbit coupling are so intimately intertwined, that for theheaviest atoms they cannot be treated separately [15]. It was, on the contrary,demonstrated on sixth-row main group hydrides, where the spin-orbit couplingis among the highest in the periodic table, that correlation and spin-orbit ener-gies are very nearly independent [16]. To summarise, these considerations leadto the replacement of the genuine coupleUREP

l j ( j = l ±1/2) of fine-structurepseudopotentials by another representation where the relativistic pseudopoten-tial is defined as a pair containing a scalar relativistic pseudopotential (or Av-eraged Relativistic Effective Potential, AREP),UAREP, and a spin-orbit pseu-dopotential (or Spin-Orbit Relativistic Effective Potential, SOREP),USO. The

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AREPs allow the use of the most sophisticated CI treatments already developedfor non relativistic symmetries,ie. in a Schrodinger-like equation. Moreover,one can take full advantage of the nodeless core representation of the pseudoor-bitals to enlarge the basis set expansion in the valence region. These couples(AREP, SOREP) of relativistic pseudopotentials are actually the most widelyused in theoretical chemistry applications, and are implemented in most com-mercial or free SCF codes [7]. Nevertheless, there is no unique definition of thespin-orbit pseudopotential, which can be problematic when it comes to calcu-late spin-orbit integrals. We will thus have to take stock on the representationof spin-orbit pseudopotentials in a subsequent section2.1.

Considering now the point (c), a last ingredient comes into play to facilitatethe correlation treatment. In principle, and regardless of the computer feasibil-ity, the best way using the couple (AREP + SOREP) of relativistic pseudopoten-tials is to treat the correlation and the spin-orbit coupling on an equal footing,developing the SOCI matrix on a basis set of either determinants built on av-eraged relativistic pseudoorbitals or double group adapted functions (see refer-ence [17] for double group symmetries). This avoids the question of the leader-ship between electron correlation and spin-orbit coupling, but has the drawbackthat the CI process is performed within relativistic symmetries leading to cum-bersome CI expansions. Moreover in some cases the convergence of the SOCIcould become very difficult [18,19]. As it is underlined above, the CI treatmentis much easier in non relativistic symmetries (referred here as LS-coupling),and as a consequence more approximate methods split the CI calculation andthe spin-orbit coupling, the latter being treated in a last step where the Hamilto-nian is expressed in the basis of a chosen set of correlated LS states. Howeverthe off-diagonal spin-orbit integrals can safely be computed using a severelytruncated expansion of correlated LS states, and expressing the Hamiltonian insuch a basis of truncated LS states makes the SOCI methods much more effi-cient. The diagonal elements are often replaced by the energies obtained at thebest LS-CI levels; this is sometimes improperly called an ”energy shift tech-nique” which is in fact nothing else than the simplest implementation of theeffective Hamiltonian theory [20]. One should not confuse effective Hamiltoni-ans with effective potentials: the lattersimulates the action of a true operatoron the valence true orbitals, while the former fulfils a function ofreductionof the informationcoming from a large Hamiltonian representation. The ef-fective Hamiltonian methods lead to very powerful techniques used in severalSOCI methods. We classify in section2.2 the different SOCI methods accord-ing whether the CI treatment and SO coupling are separated or not. We further

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distinguish them according to whether an effective Hamiltonian technique isused or not.

Finally, some spectroscopic applications for pseudopotentials within SOCImethods are presented in section3. We focus our attention on applicationsrelated to relativistic averaged and spin-orbit pseudopotentials (other effectivecore potentials applications are presented in chapters 6 and 7 in this book). Dueto the large number of theoretical studies carried out so far, we have chosen toillustrate the different SOCI methods and discuss a few results, rather than topresent an extensive review of the whole set of pseudopotential spectroscopicapplications which would be less informative. Concerning the works not re-ported here, we refer to the exhaustive and up-to-date bibliography on relativis-tic molecular studies by Pyykko [21, 22, 23, 24]. The choice of an applicationis made on the basis of its ability to illustrate the performances on both thepseudopotential and the SOCI methods. One has to keep in mind that it is noteasy to compare objectively different pseudopotentials in use since this wouldrequire the same conditions in calculations (core definition, atomic basis set,SOCI method). The applications are separated into gas phase (section3.1) andembedded (section3.2) molecular applications. Even if the main purpose of thischapter is to deal with applications to molecular spectroscopy, it is of great in-terest to underline the importance of the spin-orbit coupling on the ground statereactivity of open-shell systems. A case study is presented in section3.1.4.

2. METHODS

2.1. Spin-orbit integrals and spin-orbit pseudopotentials2.1.1. Spin-orbit integrals

The use of Breit-Pauli or no-pair spin-orbit operators to compute the spin-orbit integrals implies to deal with the full nodal structure of the orbitals [2].When AREPs are used at the SCF step the pseudoorbitals, as eigenfunctionsof the valence Fock-operators, have lost their nodal structure in the core region,exactly where spin-orbit operators essentially act, making it impossible to applysuch operators in pseudopotential schemes. Three solutions can be employed toevaluate the spin-orbit integrals on nodeless pseudoorbitals:

1. extract a semi-empirical one-electron spin-orbit operator,

2. import pre-computed all-electron atomic spin-orbit integrals,

3. define a specific one-electron spin-orbit pseudopotential.

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A one-electron one-centre semi-empirical spin-orbit operator acting on node-less pseudoorbitals has the form [25]:

HSO,semi−empirical =α2

2 ∑k,ik

Ze f fk

r3kik

lkik.sik , (1)

whereα is the fine-structure constant,k indexes the nuclei andik indexes theelectrons on centrek. The value ofZe f f

k is determined at the non-correlated(Hartree-Fock) level to reproduce the relevant spin-orbit constantζk of a givenconfiguration. To compensate for the loss of the nodal structure in pseudoor-bitals, the values obtained forZe f f are considerably larger than the correspond-ing ones obtained with the usual orbitals. This semi-empirical method should beof good accuracy as long as the physical character of the valence orbitals doesnot change drastically from the reference atomic state forZe f f

k [26, 27, 28, 29,30]. Another approach defines a fine-structure semi-empirical operator as [31]:

Vsemi−empiricall j (r) = (al j +bl j r +cl j r

2)exp(−βl j r p)

rq , (2)

whereal j , bl j , cl j andβl j are parameters to be fitted from experimental data,and defining the spin-operator as (see equation (8))

HSO=∞

∑l=1

l+1/2

∑j=l−1/2

j

∑mj=− j

Vsemi−empiricall j (r)|l jm j〉〈l jm j | , (3)

where |l jm j〉 is a two-component angular function. However, as in an all-electron scheme, semi-empirical methods suffer from several drawbacks:i) iftheZe f f-method is applied to multiconfigurational wave functions and possiblyat a correlated level, errors may arise because theZe f f parameter, though ex-tracted at the non-correlated level, was adjusted to experimental values. It thusalready intrinsically contains various electronic effects including Breit interac-tions (see section2.1.3) that might be counted twice;ii) typically it is not trans-ferable to others atomic configurations;iii) as a consequence it is not able tocorrectly account for difficult situations like avoided crossings between poten-tial curves;iv) it contains an operator≈ r−3 which acts in the inner core regionand is not reliable for pseudopotential schemes involving nodeless orbitals. Forthese reasons, such methods are of a limited reliability for molecular calcula-tions and are presently rarely employed, and all the more because the followingpseudopotential method does not suffer from the mentioned drawbacks.

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Another unusual method to obtain spin-orbit integrals in a SOCI treatmentfollowing an AREP calculation at the SCF step, is to import in the pseudopo-tential SOCI valence all-electron spin-orbit integrals computed with atomic or-bitals basis [32]. This requires to use a general contracted basis set in orderto be able to do a one-to-one correspondence between the atomic all-electronorbitals and the atomic pseudoorbitals. The integrals are separately evaluatedfrom atomic all-electron calculations. The AMFI code [33] provides spin-orbitintegrals on the atomic orbitals basis in a mean-field approach [34]. Due to itsregularisation properties, the no-pair spin-orbit operator is used instead of theBreit-Pauli one [2]. In this approach the multi-centre integrals, that are usu-ally negligible are discarded [2,34], and an effective (Fock-type) one-electronspin-orbit operator is generated. It is worthwhile to note that, unlike the pre-viousZe f f-method or the following pseudopotential one, this method is totallyindependent of the AREP used in the SCF step. Although this method is notvery comfortable for the user since it obliges to deal with general contractedbasis sets, it is reliable and can be used to substitute a SO pseudopotential if notavailable, or even to check the validity of computed pseudopotential spin-orbitintegrals.

The third alternative to compute the spin-orbit integrals is to define a specificspin-orbit pseudopotential (SOREP) acting on the scalar relativistic pseudoor-bitals. Its function is tosimulatethe action of the true SO operator on a trueorbital. In other words, the pseudopotential SO integrals have in principle toreproduce the corresponding valence all-electron SO integrals. Even thoughthis condition is not directly imposed in the current SOREP derivations [7], it isnevertheless approximately fulfilled. Indeed, in the usual shape-consistent pseu-dopotential methods where the SOREP is directly related to two-componentrelativistic effective potentials, the solutions have to reproduce the all-electronorbital fine-structure [35,36]. It has also been recently shown that one can derivefor eachl -symmetry a shape-consistent SOREPUSO

l , that fulfils the followingcondition on the averaged relativistic pseudoorbital|φl〉:

〈φl |USOl |φl〉= ζl , (4)

whereζl is the SO parameter to be reproduced [37]. It is less obvious whetherthis condition is also fulfilled in energy consistent technique where the SOREPis derived from several atomic ground and excited states energies of the neutralatom and the low-charged ions. However it amounts to the same thing, as theSOREP first acts on averaged pseudoorbitals to givein fine the correct atomicexcited states energies. For a review of the various extraction techniques, the

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reader is invited to read Dolg’s article [7]. The modern SOREPs, properly ex-tracted and employed, give very accurate results, as it can be verified in the wellknown difficult case of the thallium spin-orbit splitting in references [10, 11].However let us draw the attention on the fact that in principle each SOREP isadapted to a given AREP, and for a specific AREP, using any SOREP other thanthe one with which it is paired in the couple (UAREP,USOREP), gives no guar-anty on the result. This comes from the fact that the SOREP acts essentiallyin the arbitrary nodeless core part of the pseudoorbital [38], as it is shown inFigure1, this part being determined by the AREP.

Figure 1. Iodine atomic data in atomic units from reference [37]. The scalar-relativistic all-electron 5p orbital in dotted line, the corresponding 5p pseudoor-bital in dashed line, and the 5p-spin-orbit pseudopotential in plain line.

2.1.2. Spin-orbit pseudopotentialsAs there is neither a unique definition nor a unique representation of a

SOREP, it is worthwhile to remind the current definitions and to derive theirrelationship. This is of prime importance if one wants to work with a different

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SOREP definition than the one already (eventually) implemented in the usedSOCI code.

Let us first discuss the usual spin-orbit pseudopotentials, which can be de-fined in a general way via relativistic two-component pseudopotentialsUREP

l j (r).They originate from the definition of a Schrodinger-like valence model Hamil-tonian in a two-component form shown here for an atom

HPP = ∑i

[−∆i

2−Q

r i+

∞

∑l=0

l+1/2

∑j=l−1/2

UREPl j (r i)

j

∑mj=− j

|l jm j〉〈l jm j |

]

+∑i< j

1r i j− α

2(∑

i

r i

r3i

[1−exp(−δ r2i )])

2 , (5)

where|l jm j〉 is the two-component angular function [7]. In the second line, thefirst term is the usual electron-electron Coulomb interaction while the secondterm — sometimes neglected — is the core-polarisation potential taking into ac-count correlation terms originating from the core orbitals. In this equation thesummation overl is in practise limited to a finite valuelmax (see equation (14)).The two-component relativistic effective potentialsUREP

l j (r) can be recombined

in order to define the couple (UAREP,USO) of averaged and spin-orbit pseudopo-tentials (see equations (17), (18), (19) for USO). The Lande’s averaged AREPfor a givenl is defined as [9]:

UAREPl =

12l +1

[lU REPl ,l−1/2(r)+(l +1)UREP

l ,l+1/2(r)] . (6)

The Lande’s interval rule underlies the definition of the spin-orbit operatorHSOwhich first acts on atomic one-electron functions. Indeed, a single-electronspin-orbit interaction obeys this rule, which means that the Lande’s average ofa multiplet coincides with the non-split term [39]. For a givenl symmetry,taking the non-split term as a reference, one defines a zero-diagonal operator inrelation to the reference

lU REPl ,l−1/2(r)+(l +1)UREP

l ,l+1/2(r) = 0 , (7)

whereUREPl j (r) is thel j th two-component relativistic effective potential (REP).

This is consistent with the usual definition of a Lande’s averaged AREP (cf.equa-tion (6)). The spin-orbit operator can then be defined as:

HSO=∞

∑l=1

l+1/2

∑j=l−1/2

j

∑mj=− j

UREPl j (r)|l jm j〉〈l jm j | . (8)

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From equation (8) we obtain by summation overj

HSO =∞

∑l=1

l−1/2

∑mj=−(l−1/2)

UREPl ,l−1/2(r)|l , l −1/2,mj〉〈l , l −1/2,mj |

+l+1/2

∑mj=−(l+1/2)

UREPl ,l+1/2(r)|l , l +1/2,mj〉〈l , l +1/2,mj |

, (9)

and introducing the two projectors

Pl ,l±1/2 =(l±1/2)

∑mj=−(l±1/2)

|l , l ± 12,mj〉〈l , l ±

12,mj | , (10)

we obtain

HSO=∞

∑l=1

UREPl ,l−1/2(r)Pl ,l−1/2+UREP

l ,l+1/2(r)Pl ,l+1/2 . (11)

Then, defining

∆UREPl (r) = UREP

l ,l+1/2(r)−UREPl ,l−1/2(r) , (12)

and using equation (7)

∆UREPl (r) =

2l +1l

UREPl ,l+1/2(r) =−2l +1

l +1UREP

l ,l−1/2(r) , (13)

equation (11) takes the useful form

HSO=L−1

∑l=1

∆UREPl (r)

2l +1lPl ,l+1/2− (l +1)Pl ,l−1/2 . (14)

In this expression the summation overl is limited to a finite valueL− 1, as-suming that radial parts of REPs are the same for all spinors having angularquantum numbersl ≥ L and implying∆UREP

l≥L (r) = 0 [40].In the strategy using the couple (UAREP,USOREP), it is considerably easier to

deal with projection operatorsPl = ∑lml=−l |Yl ,ml 〉〈Yl ,ml | that only act on spatial

coordinates, as was successively proposed with different approaches, first byHafneret al [41], then by Teichteilet al [35] and finally by Pitzeret al [42].Pitzeret al reformulated the SO Hamiltonian using the relation

lPl ,l+1/2− (l +1)Pl ,l−1/2 = 2 Pl~l ·~s Pl , (15)

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so that

HSO=L−1

∑l

2∆UREPl (r)

2l +1Pl

~l ·~s Pl . (16)

Teichteil et al developed the angular part (spherical harmonic spinors) of thePauli spinors as functions of the spatial spherical harmonics [43]:

j = l + 12 : |l jm j〉=

√j +mj

2l +1|Yl ,mj−1

2〉|α〉+

√j−mj

2l +1|Yl ,mj+1

2〉|β 〉

j = l − 12 : |l jm j〉=−

√j−mj +1

2l +1|Yl ,mj−1

2〉|α〉+

√j +mj +1

2l +1|Yl ,mj+1

2〉|β 〉

The projectors|l jm j〉〈l jm j | and their sumsPl ,l±1/2 are then reduced to com-binations of usual spin-free projectorsPl onto spherical harmonics. Due to thespin projectors|α〉〈α|, |β 〉〈β |, |α〉〈β | and|β 〉〈α|, the computation of spin-orbitintegrals is carried out for separate(α,α) and(α,β ) spin-orbit matrices on thebasis of atomic pseudo-spin-orbitals; it can be shown that(β ,β ) = (α,α)∗ and(β ,α) =−(α,β )∗. Of course both approaches give the same results.

To determineHSO one has to define a procedure to obtain∆UREPl (r) for

eachl symmetry used in the pseudopotential calculation. Let us first discussthe importance of the nature of the relativistic Hamiltonian to derive spin-orbitpseudopotentials. Wanget alanalysed the influence of neglecting the core spin-orbit splitting on the atomic valence orbital energy with a frozen core approx-imation [44]. They found significant errors for the 6-block elements attributedto: i) the change of the core electron density due to the spin-orbit splittingof the core orbitals, thus changing the Coulomb and exchange-correlation in-teraction between the core and valence electrons;ii) the change of the Paulirepulsion between the core electrons and the valence electrons through orthog-onalisation. These effects (among others) are fully taken into account in anall-electron 4-component Dirac-Hartree-Fock (DHF) atomic calculation, and allthe recently published pseudopotentials are extracted from a full relativistic ref-erence atomic calculation [7]. Moreover, to account for core-polarisation andcorrelation effects, shape-consistent and energy-consistent pseudopotentials areextracted from multiconfigurational Dirac-Fock atomic reference data [45,46].It can be shown however, that core-polarisation effects, although having signifi-cant effects on orbital properties, have negligible effects on spin-orbit splittingsof atomic multiplets, but sizable effects on averaged energies [47].

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Let us now return to the derivation of∆UREPl (r). Using the form (16), and fol-

lowing the technique proposed by the Berkeley group [48] after Hafneret al[41],Pacioset al [36] extracted a SOREP (called hereUSO,Berkeley

l ) within a shape-consistent procedure using

USO,Berkeleyl = ∆UREP

l /(2l +1) . (17)

The Petersburg group proposed a generalisation based on the previous methodto extract the couple (UAREP,USOREP), but used the same definition [49]. TheToulouse group [35], working with a different shape-consistent method, definedthe SOREP as

USO,Toulousel =

l2l +1

∆UREPl = UREP

l ,l+1/2 . (18)

The Stuttgart group [50] used an energy-consistent procedure, by taking forHSO the form (16), and defined

USO,Stuttgartl = ∆UREP

l . (19)

The connection between the different SOREPs currently used therefore is

USO,Berkeleyl =

1l

USO,Toulousel =

12l +1

USO,Stuttgartl . (20)

We note in passing thatUSO,Stuttgartl gives a Lande’s averaged orbital level which

does not correspond to the non-split energy level, since the extraction data aretotal atomic energies which do not obey in general to the Lande’s interval rule;a shift correction might then be applied to satisfy this condition [10]. The im-plementation ofHSO in a SOCI code takes into account the definition of theSOREP used to compute the spin-orbit integrals. The relations (20) allow theuse in a given SOCI code of the different SOREPs, multiplying their coefficients(see equation (21)) by a pre-factor deduced from (20).

Finally, all the SOREPs are published within a semi-local Gaussian-typefunction, and for a givenl symmetry [51]

USOl (r) = ∑

iCli rnli e−αli r2

, (21)

whereCli , nli , αli are parameters to be fitted.USO,Toulousel andUSO,Stuttgart

lare directly extracted in a very compact and reliable semi-local form whichnever exceeds two terms, whileUSO,Berkeley

l is first extracted in a numerical

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form and then fitted in a less compact (more than 6 Gaussian terms) semi-localform. However, Stevenset al using the numerical extraction procedure for aUSO,Berkeley

l -type SOREP along with the optimisation method for the parame-ters proposed by the Toulouse group [52], obtained a more compact form [53].In most SOCI codes, either included in packages such that COLUMBUS [54],MOLPRO [55] or independent of any standard packages like EPCISO (thepresent version runs after HONDO9 [56], MOLCAS [57] or MOLPRO [55]),the spin-orbit integrals are computed using directly the semi-local form (21),which is comfortable for the user which intends to use any kind of SOREPs.Few other codes can use an expansion of the semi-local SOREP in a non-localrepresentation using Fourier techniques [58] (delocalisation step), which wasinitially intended to facilitate gradient SOCI calculations. However the expan-sion in a non-local form is not trivial and such an operator can only be dis-tributed on a library file directly accessible for the user. This is a disadvantagefor the SOCI codes which can only use the non-local form of the spin-orbitpseudopotentials, since no other SOREPs than the ones already delocalised canbe used.

2.1.3. Correlation effects on spin-orbit splittingTo conclude this discussion on spin-orbit effective core potentials, it is in-

teresting to discuss how correlation effects (namely the core-core, core-valenceand valence correlations), influence SO-coupling with respect to core-valenceseparation.

First we discuss the importance of correlation effects involving the core or-bitals, which are definitively absent in the pseudopotential calculation. Amongthese effects, care has to be taken on the influence of core-valence interac-tions on the atomic spin-orbit splitting already carefully scrutinised by Lind-grenet al [59]. Although this analysis was done within an all-electron scheme,it gives valuable insight for valence-only calculations. Few pseudopotentialcalibration studies have been made to test the influence of the core-valence cor-relation on the spin-orbit splitting. In a calibration study of valence-electroncorrelation effects on the fine-structure splitting in the Pb isoelectronic series,Shuklaet al [60] performed pseudopotential calculations with and without aCPP correction [61]. This CPP is extracted from experimental data (the core po-larisability) and then accounts for all effects not present in the pseudopotential,especially for core polarisation, core-core and core-valence correlations effects.The neutral Pb atom in its3P ground state presents a zero-field splitting withthe J = 0 component for the lower energy (E(3P0) < E(3P1) < E(3P2)) [39].

15

It was found that the3P1−3 P0 splitting at the correlated level is increased by281 cm−1 to 7254 cm−1 when the CPP is added, accounting for all effects in-volving core orbitals. However, one of the most striking and also one of themost studied case, is the2P ground state splitting of the thallium atom. Onecould expect for this atom, having ap shell less than half-filled, large core-valence correlation effects to the spin-orbit splitting, due to the large spatialextent between the components of the 6p valence spinor. Leiningeret al [10]addressed this problem using alarge-coreenergy-consistent pseudopotential(leaving three electrons in the valence shell, configuration 6s26p) [62] with andwithout a CPP correction. Although the choice of the SOCI procedure used tocompute the spin-orbit splitting plays an important role in this case (see refer-ences [11,63]), we do not discuss such details here as they are discussed laterin section2.2. We rather focus on the influence of core-valence correlation.Without the CPP, using a large uncontracted basis set to avoid basis-set arti-facts (even-tempered 17s20p12d) along with a single and double substitutionsvalence CI (SDCI), they found a spin-orbit splitting of 7311 cm−1, comparedto the experimental value 7792.7 cm−1 [64]. Adding the CPP, the spin-orbitsplitting amounts to 7808 cm−1, very close to the experimental result. Consid-ering now the core-valence interactions in a fullab initio way, they extracted asemi-corepseudopotential leaving 13 electrons in the valence shell (5d106s26p1

configuration) and an optimised 12s13p7d4 f basis set. A SDCI valence corre-lation with the(n−1)d shell within the valence space gives without a CPP aSO splitting of 6891 cm−1, and with a CPP and a valence correlation restrictedto the 3 outer electrons, 7578 cm−1. Finally, removing the 5s, 5p electronsfrom the core with asmall-corepseudopotential leaving 21 electrons in thevalence space, they eliminate the deficiency of the semi-core pseudopotentialwithout a CPP and obtained 7810 cm−1 (7298 cm−1 with a CPP and only the 3outer valence electrons correlated). They attributed the deficiency of the semi-core pseudopotential to the nodeless shape of the 6p orbital, and thus explainedwhy the small-core pseudopotential is able to give accurate results providedit is used with a suitably flexible AO basis and a high correlation treatment.Buenkeret al [63] also investigated the accuracy of the pseudopotential approx-imation using different SOCI methods and a shape-consistent semi-core pseu-dopotential [65], and found a similar discrepancy: the SCF value 7424 cm−1

is lowered by more than 400 cm−1 when 13 valence electrons are correlated,and if only the 6s,6p electrons are correlated this value increases only slightlyto 7438 cm−1. However all the studies involving a semi-empirical CPP correc-tion cannot distinguish between the different core orbital contributions, namely

16

the core polarisation (single excitations from the core to virtual orbitals), thecore-core correlation (double excitations from the core to virtual orbitals) andthe core-valence correlation (single core + single valence excitations to vir-tual orbitals) effects. Rakowitzet al [66] using an all-electron Hamiltonianand a no-pair spin-orbit operator [67] checked the influence of the sub-valence(n−1)spd excitations, but unfortunately did not separate the different contri-butions in their MRD-CI treatment (multireference single and double excitationCI) [68]. However, they found an excellent agreement with experimental results(7796 cm−1) correlating the 19 electrons 5s25p65d106s26p1, but worsened theresults while diminishing the CI space: (7672 cm−1) correlating the 13 electrons5d106s26p1, and (7519 cm−1) correlating the three 6s26p1 valence electrons .Wahlgrenet al [69] redid these calculations using an all-electron mean-fieldspin-orbit operator [34], correlating first three valence electrons and then, in-cluding the sub-valence 5d shell into the CI space, 13 electrons. Describingfirst thed shell polarisation effects including only single excitations from thed shell, the spin-orbit splitting of the2P ground state is enhanced by 249 cm−1.Taking partially core-core correlation effects into account by adding double ex-citations from thed shell, lowers the previous splitting by 132 cm−1 resultingin a final spin-orbit splitting of 7720 cm−1. Neither the 5s and 5p electrons northe core-valence correlation effects were considered in their CI. Nevertheless,one can conclude that core-core correlation does not cancel core-polarisationeffects, so that the semi-empirical CPP correction accounts for all the corre-lation effects originating from the core orbitals, and not only the core-valencecorrelation.

Let us now consider another atom taken from the right-hand side of the maingroup elements, for which the core polarisation should be less pronounced.Dolg extracted an energy-adjusted large-core pseudopotential for neutral iodinefrom Dirac-Hartree-Fock reference calculations, and computed the fine struc-ture splitting of the2P ground state using various valence basis sets within aKramers-restricted Hartree-Fock scheme followed by multi-reference CI, withand without a CPP [70]. With the smaller basis set (7s7p) and without theCPP he obtained a splitting of 7745 cm−1. Adding a CPP the splitting became7950 cm−1 compared to the experimental result of 7603 cm−1. We should no-tice that even in the case of a ”hard core” atom like iodine, the CPP increasesthe spin-orbit splitting by as much as 205 cm−1. The splitting was progres-sively lowered while the AO basis set was enriched, and with a CPP and themost extended AO basis set ((7s7p)/[3s3p]+3d1 f ) Dolg was able to obtain aspin-orbit splitting of 7620 cm−1 in good agreement with experiment. This ex-

17

ample highlights the important role of the basis set for the correct descriptionof spin-orbit splitting: with the CPP and the same multi-reference CI treatment,increasing the size of the AO basis set lowers the splitting by 330 cm−1. An-other quite different role of the AO basis set is to bring, in an economical way,into the SOCI treatment some correlation effects via the use of Atomic Natu-ral Orbitals (ANOs) originating from a previous electronic correlation spin-freetreatment [67]. In the same way, in pseudopotential SOCI methods, either MC-SCF orbitals (see for instance references [71,11] or an ANO basis set (see forinstance reference [72]) are often used to improve the convergence of the SOCItreatment.

In discussing the role of core-valence interaction on the spin-orbit splitting,we have pointed out the dependence of valence correlation on the size of thebasis set. These conclusions are not limited to pseudopotential studies. Thereis another effect, not directly related to the pseudopotential theory, namely theinfluence of singly-excited valence configurations on spin-orbit splitting. Asanalysed in an all-electron framework [59], these single excitations are mostlyresponsible for the valence spinor relaxation and tend to enhance spin-orbitsplitting. Indeed, coming back to the thallium atom which shows a tremen-dous spatial relaxation of the valence spinors, Valletet al [11] obtained for theground state2P splitting an increase of 450 cm−1 when all single excitationsfrom a (6s26p, 6p3) reference space were considered (using orbitals obtainedfrom RHF calculations and an extended basis set). We refer to the review ofHeß et al [2] for an in-depth discussion of the spin-orbit splitting of the2Pground state of the thallium atom.

Although this does not enter into the discussion of correlation effects, wepoint out the role of higher-order relativistic effects, such as the Breit interac-tion, on the spin-orbit splitting, which are not explicitly included. For the neu-tral Pb atom, the Breit interaction estimated by a four-component all-electroncalculation using first order perturbation theory lowers the SCF spin-orbit split-ting by 166 cm−1, thus compensating partially the increase due to core-core andcore-valence interactions [60].

To summarise, the pure valence correlation (double excitations) tends tolower the spin-orbit splitting while the valence spinors relaxation (single exci-tations) tends to increase the splitting. Concerning the role of the core orbitals,the whole core polarisation, core-core and core-valence correlations taken intoaccount via a semi-empirical CPP tend to enhance this splitting.

18

2.2. SOCI methods2.2.1. Spin-orbit CI methods versus the full two-component treatments

Let us start this section with a brief discussion on the ability of the SOCImethod to efficiently handle the calculation of excited states of molecules con-taining very heavy elements. A SOCI method uses a spin-orbit operator associ-ated to a prior scalar relativistic SCF calculation (to an AREP when pseudopo-tentials are used). The CI process and the spin-orbit operator deal with usualorbitals (or pseudoorbitals) instead of spinors (or pseudospinors), and this is ob-viously perfectly adapted to systems which contain atoms having a weak spin-orbit splitting. However whenever heavy atoms intervene, one would expect alarge spatial separation between the two components of the valence spinors. Inthese cases, it may seem that the best approach is to work with two-componentspinors (or pseudospinors) obtained at the SCF step. Actually in an atomiccorrelation treatment with the usual scalar orbitals, singly-excited configura-tions mostly bring the differential relaxation of the corresponding spinor com-ponents [59]. It has been shown that for heavy atoms orbital relaxation effectsfrom spin-orbit interactions could become sizable for valence spinors, and spin-orbit CI calculations employing LS configurations as a basis could suffer fromslow convergence [18]. Choi et al [19] compared a SOCI process using rela-tivistic scalar molecular pseudoorbitals where the correlation and the spin-orbitcoupling are treated on an equal footing, with a CI process based on the use oftwo-component molecular pseudospinors. They chose the most unfavourableexamples for the SOCI convergence, the TlH and (113)H molecules where theheavy atoms have among the most pronounced spatial separation for the va-lencep-spinor components observed in main group elements. The SOCI spacebuilt on 41 references was not sufficient to describe the orbital relaxation ef-fects for the (113)H molecule, while it was easier to reach the convergencewith two-component CI where the number of the leading configurations wasmuch smaller than in the SOCI process. In a two-component scheme, the dif-ferential relaxation of the spinor components is mostly accounted for in thetwo-component SCF process. In other words, in the SCF step the physicalcontributions arising from single excitations are directly accounted for, whilea SOCI method needs a large number of single-excitation in the CI expansionto relax the spinor components from the initial valence orbitals. It is worth-while to note that it is possible to improve considerably the convergence ofa SOCI scheme by bringing into play an effective Hamiltonian theory [20].It was also shown that within this theoretical framework, one can account formost of both orbital relaxation and dynamical correlation within separate treat-

19

ments, and therefore improve the SOCI convergence [11]. Indeed if the CIprocess is carried out independently of spin-orbit coupling on can make use ofthe best available CI treatments in non relativistic symmetries, but reversely ifSO coupling is done on fixed (contracted) correlated wave functions, one doesnot properly account for the differential correlation of the spinor components.

The treatment of the correlation is crucial. Apart from the extreme exam-ples mentioned above, and even without the use of the effective Hamiltoniantheory, it is well known that the use of two-component spinors compared tothe use of scalar orbitals considerably burdens the CI treatment, when corre-lating a given number of LS configurations. For molecules containingd- andf -elements strong mixing of molecular states are expected, and one has to usemore sophisticated and very extended CI treatments. Molecular electronic ex-cited states applications are mainly based on the use of scalar relativistic ap-proximations at the SCF step, followed by a SOCI treatment. The main problemfor all SOCI methods is the interplay between correlation and spin-orbit cou-pling, but unfortunately the full account of this interplay increases substantiallythe computational cost. The removal of core electrons helps, but additionalapproximations are required to treat complex chemical systems. The differentavailable SOCI codes can be classified according to their numerical efficiencyand to their ability to accurately treat both electronic correlation and spin-orbitcoupling, leading to four categories of methods obtained by combining the fol-lowing approximations:

1. whether or not the correlation treatment and SO coupling are carried outseparately

2. whether or not the effective Hamiltonian technique is used

A large number of spectroscopic applications using spin-orbit CI methodsare nowadays performed using effective core potentials, especially when heavyatoms are involved. However, by nature all SOCI methods can in principle beused either in an all-electron or in an ECP scheme, provided the code can com-pute the appropriate integrals (see section2.1.1). Although we are dealing inthis article with pseudopotential methods, we do not limit this review of theSOCI methods to the exclusive use of pseudopotentials. Moreover the physi-cal content of the valence calculations using different SOCI methods is largelyindependent whether an all-electron or pseudopotential method is applied, pro-vided that in the pseudopotential calculation the effects on valence orbitals com-ing from core orbitals are taken into account either via a CPP correction or byusing a small-core pseudopotential as discussed in section2.1.3.

20

2.2.2. Contracted SOCI methods (CI/SO)If spin-orbit coupling is considered as a perturbation, the total Hamiltonian

decouples into a spin-free (referred asHsr for a scalar relativistic Hamiltonian)1

and a spin-orbit part:H = Hsr + HSO. The electrostatic and spin-orbit inter-actions are in general computed independently. The correlation treatment iscarried out in a scalar relativistic scheme within a Schrodinger-like formalism,and takes advantage of non relativistic symmetries. In this scheme the CI ma-trix without spin-orbit interaction is diagonalised in afirst step, providing wavefunctions expanded over Slater determinants. The correlated wave functionsform a basis set on which the spin-orbit Hamiltonian is built in asecond step.These procedures were originally calledtwo-step methods, but are now betterdescribed asconventional two-step methodsor more properly -in contrast of theuncontracted DGCI methods- ascontracted SOCI methods[63,11].2 In the fol-lowing these methods will be called CI/SO to stress the separate treatment ofelectrostatic and spin-orbit interactions.

The simplest way to compute spin-orbit interactions between spin-free cor-related states is to employ correlated wave functions as zero-order basis func-tions to compute the spin-orbit splitting to first order within degenerate pertur-bation theory (FOPT). The spin-orbit matrix is built on the set of degeneratewave functions originating from a given multiplet, and its diagonalisation giveseigenvalues which are simply added to the correlated scalar relativistic ener-gies computed in the first step. One could expect that a FOPT treatment cangive sufficient accuracy for light atoms for which〈Hsr〉 〈HSO〉, but becomesquestionable when large spin-orbit splitting or even when significant contribu-tions from others states intervene. Indeed, Rakowitzet al tested this methodwithin an all-electron scheme in the critical case of the2P ground state splittingof the thallium atom (see section2.1.3) [66]. The FOTP splitting varies in therange between 6221 cm−1 and 6406 cm−1 depending of the MRD-CI expansionand on the number of correlated electrons (3, 13 or 19), giving a large error, upto 20%, with respect to the experimental value of 7793 cm−1. They concludedthat the large difference in spatial extension of thej j -coupledp3/2 and p1/2components of the atomic spinor of thallium is responsible for the failure of

1 The atoms are the simplest examples which are chosen to illustrate the ability of the SOCI methods to obtainaccurate results. In this case the notationHsr stands for the usualHLS notation whereL andS are good atomicquantum numbers.2 The contracted labelling describes the fact that the expansion coefficients of the correlated wave functions usedas basis set in the second-step are kept frozen in the diagonalisation of the spin-orbit matrix, making reference tothe contraction coefficients of atomic basis sets used in the SCF step.

21

this perturbation theory approach. Buenkeret al repeated these calculationsusing large-core and semi-core pseudopotentials (see section2.1.3) and founda slightly better FOPT splitting of 6888 cm−1, with an error of about 12%.However the single valence excitations (6p → p∗i ) play a crucial role in therelaxation of the valence spinors, and simply adding the second order correc-tions of perturbation theory (SOPT) from the two lowest configurations 6s2p∗iincreases the splitting by 651 cm−1, leading to an improved value 7539 cm−1,still underestimated by 254 cm−1 with respect to experiment (3% error). Notehowever that only one particular state is corrected by low order perturbationtheory by simply adding to the zero-order results corrections originating fromother configurations.

A straightforward generalisation of the above perturbation CI/SO treatmentis to use the correlated scalar relativistic eigenfunctions|Φsr

m〉 of the scalarHamiltonianHsr as a truncated set ofcontractedmany-electron basis functionsfor the total Hamiltonian. Introducing the subscriptim for a given|Φsr

m〉 wavefunctions to mark out the spatial and spin degenerate components of this multi-plet, the matrix representation of the Hamiltonian writes

〈Φsrm,im|H|Φ

srn, jn〉 = 〈Φsr

m,im|Hsr +HSO|Φsr

n, jn〉= δm,nδim, jnEm+(1−δik, jk)〈Φ

srk,ik|H

SO|Φsrk, jk〉 , (22)

wherem,n∈ [1,N] andk = m,n. N is the number of correlated multiplet statescoupled by the spin-orbit interaction, and the total number of states in the matrixrepresentation isNt = ∑m=1,N Nm whereNm is the degeneracy of them-th multi-plet (im∈ [1,Nm]). The block-diagonal elementsEmδm,nδim, jn containNm identi-cal Em values. The off-diagonal spin-orbit elements(1−δik, jk)〈Φsr

kik|HSO|Φsr

k jk〉

(wherek takes the valuem or n) allows the coupling between components ofeither a given multiplet or even different multiplets.3 Clearly whenN = 1 thebasis functions in equation (22) are just the degenerate components of a multi-plet and give the above perturbative treatment. The diagonalisation of the totalHamiltonian is split into two reduced diagonalisations: the first one forHsr con-cerns the SCF and the CI treatments not burdened by the spin-orbit interaction,while the diagonalisation ofHSO benefits from the small number of basis cor-related functions. Typically the dimension of theHSO matrix varies from 6 (asingle2P atomic state has 6im components) to approximately 100; for instancethe set of valence states of a di-halogen molecule which dissociates into two

3 For instance thex andy components of an atomic2P state are coupled〈2Px|HSO|2Py〉 6= 0, as well as for two

different atomic2P and2P′multiplets:〈2Px|HSO|2P

′

y〉 6= 0.

22

2P atomic states has the dimension 36, but the dimension of theHSO matrix islowered by making use of the double degeneracy of states having a non-zeroΩvalue [71].

One of the first conventional two-step method was proposed by Heßet al [73]in an all-electron scheme where the spin-orbit integrals were derived froma Breit-Pauli operator. This method was then applied by Gleichmann andHeß [74] for the calculation of the excited states of LiHg, using for the firsttime a spin-orbit operator derived from a Douglas-Kroll-transformed no-pairHamiltonian [2]. In the first step the usual extrapolated MRD-CI correlationtreatment provides the diagonal elements of the Hamiltonian (22), and the spin-orbit matrix was split into two matrices for two fine-structure systems follow-ing the criteria of the energy separation of the states and the magnitude of thespin-orbit coupling elements. Within the context of pseudopotentials, similarCI/SO calculations can be done, but they were from the beginning coupled withan effective Hamiltonian technique as described in next section. One shouldnote however the work of Buenkeret al [63] on atoms (where the CI/SO methodis called LSC-SO-CI), and the calculation by Alekseyevet al [75] for potentialcurves and radiative lifetimes of low-lying states of BiN. In this work the au-thors used shape-consistent pseudopotentials for both atoms (a semi-core pseu-dopotential for the bismuth), in connection with the extrapolated MRD-CI cor-relation method. An accurate description of the excited electronic states wasachieved, but required the addition of some higher states to the low-lyingΛ–Sstates of interest in the Hamiltonian representation (cf.equation (22)). It is note-worthy that for an accurate treatment of the upper states of interest, it is oftennecessary to include in thesecond stepsome supplementary states in the basisset used, that constitute a “buffer”. We briefly detail this point.

A common feature of all CI/SO methods, is to couple only the wave functionstaken as basis functions to represent the Hamiltonian. No other configurationsthan the one already present in the considered wave functions are coupled byspin-orbit interaction. This means that some “external” configurations, close inenergy to the ones of interest, may have large spin-orbit interactions with thestates of interest. Neglecting them leads obviously to an uncontrolled loss ofaccuracy. A way to solve this problem is simply to consider the states based onthese configurations. For a linear molecule for instance, the spin-orbit matrix〈Φsr

m,im|HSO|Φsr

n, jn〉 is split into sub-matrices of the sameΩ quantum number(states belonging to differentΩ values are not coupled by spin-orbit interac-tion, and in general only fewΩ values are involved in such ”external” spin-orbit interactions). As a consequence, not all the non-relativistic symmetries

23

are concerned by the buffer of higher states but only the ones which have alarge ”external” spin-orbit interaction. In practise, the criterion used to selectthe buffer space of states lies on the energy separation between the wanted statesand the external ones, disregarding their contribution to spin-orbit coupling. Ofcourse the added ”external” states do not require an accurate CI treatment, butare nevertheless requested to yield reasonable energies. Note however that Gle-ichmannet al [74] used this energy criterion in a very different way: due tothe more costly all-electron treatment they had to split the spin-orbit matrix,neglecting the coupling between two sub-systems of states.

Finally, one could think that CI/SO methods which postpone the spin-orbittreatment after the CI treatment, are not able to provide accurate results forsystems with a very large spin-orbit splitting. For an idealised infinite basisset, the present two-step CI/SO method must give the same result as a one-step method treating correlation and spin-orbit coupling on an equal footing(see section2.2.4). Indeed the problem comes from the truncated basis set ofstates. Coming back to the example of the thallium atom (see section2.1.3),Vallet et al [11] showed that the CI/SO method succeeds for the2P groundstate spin-orbit splitting provided a large number of excited2P states are in-volved. They used a large-core energy-consistent pseudopotential and a CPPcorrection, and with five2P′ states each of them built from five configurations6s2mp(m≥ 6), they obtained 7795 cm−1 compared to the experimental split-ting 7792.7 cm−1. We shall remark, however, that this example is a simple casestudy, and such a large buffer of states cannot be used in practise for applicationsin molecular spectroscopy. Other methods are then preferable.

2.2.3. Effective Hamiltonian-based contracted SOCI methods (CIeff/SO)The introduction of the effective Hamiltonian methods into the SOCI treat-

ment comes from the fact that the spin-orbit interaction is governed by two fac-tors, namely the size of the spin-orbit coupling itself between the|Φsr

m〉 states,and their energy differences. Concerning precisely the last point, the underly-ing idea of the CI/SO methods is to exploit non relativistic symmetries as longas possible, and beside the savings in computational costs, the main advantageof the CI/SO methods lies in the use of sophisticated non-relativistic CI meth-ods, allowing the largest CI treatments and a free choice of the CI code, espe-cially endowing with size extensivity and size consistency properties. However,despite the small size of the spin-orbit matrix representation on the basis ofcorrelated wave functions, the computation of the matrix elements themselvesbecomes rapidly cumbersome when the expansion of the correlated wave func-tions increases. Indeed, the spin-orbit matrix elements are first computed on

24

the basis of atomic orbitals and are complex numbers, then transformed to themolecular orbital basis set, and later to the determinants which are finally com-bined to build the matrix elements on the basis of correlated wave functions.Fortunately, most determinants of the wave function do not give significantcontributions to spin-orbit matrix elements, but the convergence of these el-ements in relation to the expansion of the wave function is very slow. Hencethere is no need in keeping all determinants, except the ones contributing mostlyto the wave function required to evaluate the off-diagonal spin-orbit elements〈Φsr

m,im|HSO|Φsr

n, jn〉. If so, the scalar part of equation (22) can receive a poor cor-relation treatment. This deficiency can be lifted by recovering the missing cor-relation effects via any effective Hamiltonian technique. We note by CIeff/SO acontracted SOCI method where the missing correlation effects inHsr are takeninto account using an effective scalar Hamiltonian.

Let us summarise in the effective Hamiltonian language [20] the general fea-tures of a CIeff/SO method, in its simple Bloch-type version. In a first step,the scalar relativistic secular equations for states under interest are solved, andextensive CI calculations define a determinanttarget spaceLT (dimLT = NT)providing accurate energiesEm and the corresponding multiconfigurationalstates of interest|Φm〉 (m∈ [1,NSi ]). In a second step, a determinantinter-mediate model subspaceLD (dimLD = ND) is defined in order to have a suit-able reduced representation|Ψ0

m〉 = ∑I∈LD|I〉 of the states|Φm〉. LD ⊂ LT

is much smaller thanLT (ND NT) but of sufficient extent to give a cor-rect physical description of the states of interest. An orthonormal basis ofscalar relativistic wave functions expanded in the subspaceLD is generatedby adding to theNSi previous multiconfigurational contracted (with fixed coeffi-cients) states|Ψ0

m〉 all their spatial and spin degenerate components marked outby the subscriptim, thus defining themain model spaceof statesLS such that|Ψ0

mim〉 ∈LS (dimLS= NSand NS> NSi). An obvious advantage of the choiceof this representation of the full Hamiltonian is its compactness. Themainmodel spaceLS is limited to states of interest, so thatNS ND allowing at theend of this second step a fast standard diagonalisation of a small complex Her-mitian matrix for the full Hamiltonian. Let us note that, in contradiction withthe commonly used state-universal (Bloch-Brandow) effective Hamiltonian ap-proach [76], the dimension of theintermediate model spaceexceeds the numberof states of interest. Nevertheless, in order to keep this discussion as simple aspossible, we do not want to enter the more appropriate formalism of interme-diate effective Hamiltonians, and only refer in the following to the Bloch-typeeffective Hamiltonian technique. Theintermediate model spacerepresentation

25

for the wave functions only allows a quite poor correlation treatment

Hsr |Ψ0m〉= E0

m |Ψ0m〉 . (23)

In order to take into account the correlated results obtained in the first step, aspin-free effective Bloch-type Hamiltonian is defined

Hsr = Hsr +∑k

|Ψ0k〉(Ek−E0

k)〈Ψ0k| . (24)

In the SOCI methods, the eigenfunctions|Ψk〉 of Hsr

Hsr |Ψk〉= Ek |Ψk〉 (25)

are identical to|Ψ0m〉 computed in the intermediate model space. The total ef-

fective Hamiltonian incorporates the spin-orbit coupling

H = Hsr +PHSOP , (26)

whereP projects onto themain model space. As a matter of fact, the spin-freeenergiesE0

k are not used in a CIeff/SO method, as the spin-free matrix elementare simply defined as

〈Ψ0m|Hsr|Ψ0

n〉 = 〈Ψ0m|Hsr|Ψ0

n〉+∑k

〈Ψ0m|Ψ0

k〉(Ek−E0k)〈Ψ0

k|Ψ0n〉

= Em δmn . (27)

The matrix elements of equation (22) are now written for the full effectiveHamiltonian in themain model spaceLS of dimensionNS as

〈Ψ0m,im|H|Ψ

0n, jn〉 = 〈Ψ0

m,im|Hsr +HSO|Ψ0

n, jn〉= δm,nδim, jnEm+(1−δik, jk)〈Ψ

0k,ik|H

SO|Ψ0k, jk〉 . (28)

A Bloch-type effective Hamiltonian technique in a CIeff/SO method simplyamounts to replace the diagonal energyE0

m of the intermediate model spacebythe full correlated one,Em, coming from thetarget spaceof the first step cal-culation, without knowing — at least in principle — the corresponding wavefunctions. As a consequence one can use whatever sophisticated CI code in thefirst step to obtainEm. However, we note that the Bloch-definition of the effec-tive Hamiltonian is not the best one as it is based on the exclusive correctionof the energy. Indeed a choice of a too smallintermediate model spaceLDleads to a poor physical content of the wave function|Ψ0

m〉 which is crucial for

26

the accurate calculation of observables like transition moments for instance. Toremedy this drawback, one can either raise the dimension ofLD and burdenthe computation of the SO matrix elements, or use a more sophisticated inter-mediate Hamiltonian technique preserving a reasonable size ofLS, but givingrise to model wave functions|Ψm〉 that contain the proper physical informa-tion [20,11,77].

The first ab initio effective Hamiltonian based SOCI method (namelyCIeff/SO) was the CIPSO algorithm (CI with perturbation including spin-orbit coupling) introduced by Teichteilet al within the pseudopotential frame-work [35]. This method was then implemented in other SOCI pseudopoten-tial codes [78], in AIMP SOCI treatments [79], or even in codes working inall-electron schemes [80]. It was also adapted, within the usual Bloch formu-lation of effective Hamiltonians, to more sophisticated DGCI codes (see sec-tion 2.2.5). This methodology was sometimes presented in the literature as asimple shifted-energy technique, ignoring the effective Hamiltonian context andat the same time masking its potentialities, especially in a CIeff/SO framework,and its capacity to improve the physical content of a reduced model wave func-tion as already proposed by Valletet al [11]. In the CIPSO code, the main modelspaceLS is automatically constructed from a single spin component of one ofthe spatially degenerate states for each scalar relativistic state of interest. Inmost applications a multireference calculation MCSCF/MRCI (Multiconfigura-tions self consistent field/multireference configuration interaction) provides theintermediate model spaceLD in which the correlated wave functions are repre-sented, as well as a starting reference determinant space for a larger CI or PT2treatment in the target space [71]. In others applications [78,80] a very similarMRD-CI technique (selected single and double substitutions from a referencedeterminant space) is used in the target space with an extrapolation correctionfor the discarded configurations. An efficient size-consistent multipartitionningperturbation theory (MUPA code) was implemented into the CIPSO code [35]by Zaitsevskiiet al, and proved to be very accurate for relativistic transitionprobabilities calculations of di-halogen molecules [81].

Finally, Malmqvistet al recently proposed to extend the CIeff/SO methodto the use of a non orthogonal set of wave functions for the main model spaceLS [82]. In this algorithm the molecular orbitals can be state specific, and theintermediate model spaceLD is the union of intermediate subspaces spannedby specific determinants built on specific orbitals. To correct the energies fordynamic correlation, they run either CASPT2 or MRCI calculations in the target

27

space, and replace the equation (27) by:

〈Ψ0m|Hsr |Ψ0

n〉= 0.5(∆Em+∆En)〈Ψ0m|Ψ0

n〉 , (29)

where∆Em and∆En are the corrections for statesm andn. As each correlatedstate can be computed with his proper orbital basis set, this algorithm allows inprinciple an easier convergence for excited states. Reliable differences in theenergies of the multiplets coupled byHSOare crucial for an accurate evaluationof the spin-orbit interaction. However in the present version this code is adaptedto all-electron calculations in the MOLCAS5 software [57]. Extensions to theAIMPs or pseudopotentials are under development.

2.2.4. Uncontracted SOCI methods (DGCI)For molecules containing heavy elements, the large spin-orbit splitting im-

plies the presence of a certain number of excited configurations to bring the re-quired flexibility into the wave function. Such configurations are present in thecontracted correlated wave function, but their contribution to spin-polarisationeffects is weighted by their coefficient in the CI expansion and are thus under-estimated. In a CI/SO or CIeff/SO method, the only way to consider these con-tributions is to include in the main model space containing the states of interesta larger number of excited states defined on such configurations, even when weare not interested in these higher states. Moreover whend- and f -valence shellsintervene, large numbers of unpaired electrons possibly arise, giving rise to alarge number of configurations within a small energetic window. In additionthe coupling scheme for such heavy systems is intermediate between the pureLS- or j j - atomic schemes, and the simplest way of treating intermediate cou-pling cases without favouring electrostatic or spin-orbit interactions, is to treatthem simultaneously. In this case neither spatial symmetry operators nor spinangular momentum commute with the Hamiltonian. The symmetry operatorswhich operates on both spin and space coordinates may still commute with theHamiltonian, which is taken into account within double group symmetries [17].Such SOCI methods are called in this article Double Group CI (DGCI) methods.

One of the first DGCI codes was developed by Christiansenet al [83], butwas limited to diatomic molecules and about 5000 determinants. This code wasgeneralised later to polyatomic molecules independently by Pitzeret al [42] andBalasubramanian [72]. Pitzeret al used Configuration Spin Functions (CSFs)in their CIDBG program [42] as a basis for the SOCI treatment in a typical one-step method, and kept all the single and double substitutions from the chosenreference configurations without any selection. The usual expansion length forthe CSFs was 5000 to 10000 and occasionally up to 70000 [84]. The CIDBG

28

code was recently improved with the implementation of the Graphical UnitaryGroup Approach (GUGA) [85] and included as a module in the COLUMBUSsuite of programs [54]. The possibilities of the CI treatment are then consid-erably enlarged: for instance an accurate treatment of the excited states of theneptunyl ion used 8× 106 determinants [86]. Balasubramanian instead basedhis DGCI treatment on a kind of effective Hamiltonian two-step approach pre-sented in the next section (2.2.5).

DiLabio et al improved the initial DGCI code in a different way, includinga selection procedure for both correlation and spin-orbit coupling, defining theSelected Intermediate Coupling CI (SICCI) procedure [87]. They incorporatedin the previous DGCI code of the COLUMBUS suite a selection threshold forsingle and double promotions from a set of reference configurations. Typically,the reference set includes all configurations contributing by more than 1% tothe final wave function. The SICCI code remains a pure one-step algorithm, andone of the main advantages of such a double selection of the contributing config-urations, is to allow a better correlation treatment than a non selected procedurecan do for a given number of CSFs. Another advantage of the generalisation ofthe selection procedure to both electrostatic and magnetic interactions is to offera better control to account for the actually contributing configurations. Indeed,they illustrated in that article the importance of the spin-orbit criterion in theselection procedure, comparing their results on the low-lying excited states ofbismuth hydride with those obtained by Alekseyevet al [88] using a differentDGCI procedure which only keeps an electrostatic criterion. The spin-orbit se-lection brings into play some important configurations for spin-orbit coupling,which are essential for an accurate description of the charge transfer avoidedcrossing in the 0+(V) state; an electrostatic criterion alone misses such configu-rations and a correct qualitative description of this state. Adding additional rootsin Alekseyev’s calculation, allowed a correct description of the 0+(V) potentialcurve. Thanks to this example, we note the importance of an energy buffer nec-essary for all SOCI methods, as it was already noted for the CI/SO methods(section2.2.2).

In the framework of all-electron schemes, a direct DGCI program based onSlater determinants was initially written by Esser [89]. In this category ofcodes, Sjøvollet al proposed a generalisation of the direct CI code LUCIA(Lund CI Approach) to the SOCI treatment [90]. LUCIA is a non selecteddeterminant-based method, with single and double excitations from a RestrictedActive Space (RAS) as a reference set of determinants. As the number of CSFs

29

and Slater determinants are the same in SOCI methods,4 the determinant ba-sis is preferred for its simplicity. Preliminary RASSCF-CI calculations pro-vide starting super-vectors for the DGCI process. The starting super-vectorsare constructed with all the wanted vectors in non relativistic symmetries andmultiplicities which are possibly coupled by spin-orbit interaction. A Davidsondiagonalisation is performed for some roots, and this method gives in the firstiteration a usual SO/CI contracted solution with coefficients of the determinantsfixed on the starting states. More recently, Fleiget al proposed a generalisationof the previous procedure, using now the concept of Generalised Active Space(GAS) to build the LUCIAREL code [91]. Briefly, the (GAS)-CI method gener-alises the (RAS)-CI in the sense that an arbitrary number of active spaces witharbitrary occupations constraints are used to determine the CI wave function.We note that the relativistic (GAS)-CI allows for the use of two-component rel-ativistic operators in the fully variational optimisation in configuration space.

2.2.5. Effective Hamiltonian based uncontracted SOCI methods (DGCIeff)

The DGCI methods are clearly burdened in the CI process by the loss of thenon relativistic symmetries. The double group symmetries multiply severely thenumber of determinants arising from a given spatial configuration, leading todrastic limitations of the CI expansions as compared to the corresponding nonrelativistic cases. However the electronic correlation has slower convergenceproperties than the spin-orbit interaction, and for any method, high-accuracycalculations need to handle efficiently the electron repulsion aspect. It was al-ready suggested by Dolget al in their review on the rare earths that in accu-rate relativistic calculations, the bottleneck is more connected to the correlationproblem rather than the form of the relativistic operators used [92]. More cost-effective methods alternative to the expensive genuine DGCI ones are needed,and once again the effective Hamiltonian technique can be used to enforce thecorrelation treatment.

One of the first two-step DGCI treatment was implemented in the RCI (Rela-tivistic CI) code proposed by Balasubramanian [72]. Although his Hamiltoniancannot be written in a simple Bloch-effective Hamiltonian form, it has neverthe-less certain similarities with the effective Hamiltonian approach. In a first stepMulticonfiguration self-consistent-field (MCSCF) calculations are carried out ina scalar relativistic scheme using AREPs, followed by large-scale CIs. Natural

4 In non-relativistic symmetry, the number of CSFs can be considerably smaller than the number of determinants, asthey are spin-adapted for the state under interest. As spin-orbit coupling mixes all the spatially and spin-degeneratecomponents, the number of determinants and CSFs are equal.

30

orbitals (NO) are generated from the MCSCF/CI calculations to account for theaccurate correlation treatment in non relativistic symmetries in this first step. Inthe second step a limited CI matrix in double group symmetries is constructedas follows. First, a set of reference configurations is built from the leading con-figurations of the appropriate states, and all possible states of different spatialand spin symmetries that are close in energy and mix by spin-orbit interactionare included as reference configurations. TheH0 Hamiltonian is redefined onthe basis of NOs, adding now the spin-orbit integrals to the one-electron termsof the Hamiltonian. The one-electron spin-orbit integrals are first computed onthe basis of canonical orbitals using SOREPs, and then transformed to the basisof NOs. The final SOCI matrix is built with non selected single and doublesubstitutions from the reference set of determinants. The RCI code is approx-imatively limited to 25000 configurations, but the NOs basis set improves theconvergence of the CI process. In this DGCIeff code a significant part of thecorrelation is brought into the DGCI treatment via the NOs, but this is not ex-actly the same amount of correlation than using conventional Bloch-effectiveHamiltonians. Let us remark that, due to the absence of a selection criterionon spin-orbit interaction, the final RCI matrix has to be sufficiently large not tomiss any important configurations and accurately treat the higher-lying excitedstates.

All others DGCIeff methods are based on a direct energy correction throughconfiguration selection to account for the missing dynamic correlation due tothe truncation of the CI space. One of the first in this category is the SOCIEXalgorithm proposed by Rakowitz and Marian in an all-electron scheme [80].Unlike usual effective Hamiltonian approaches which use the first step to eval-uate the best correlated energy, this method uses the resulting eigenvectors toset up the subsequent SOCI treatment. In order to limit the dimension of theSOCI matrix, an energy-based selection procedure of configurations is used inthe first scalar relativistic MRD-CI calculations (single and double substitutionsfrom a reference space). States of various spatial symmetries, and all the fine-structure components of the multiplets in their truncated representation, are usedto form a CSFs basis for the SOCI matrix. All configurations which describespin-polarisation or static correlation effects are included in the variational treat-ment. A Davidson diagonalisation procedure is used, and the final energies arecorrected following a generalisation of a well-known extrapolation technique.The total lowering due to the discarded configurationsmi is approximated by:disc.

∑mi

|〈Ψ|Hsr +HSO|mi〉|2

E(Ψ)−E(mi). (30)

31

Test calculations on excited states of thallium hydride correlating 14 electrons,showed that this technique can accurately describe potential energy curves, butthe CI with single and double substitutions is only treated at the DGCI leveland may become heavy if one wants to enlarge the reference space for a bettercorrelation treatment.

Alternatively, the Bloch-type effective Hamiltonian technique includes theeffects of the correlation into the SOCI matrix representation, and these correc-tions influence the spin-orbit splitting in the diagonalisation process. The firststep is identical to the one of the CIeff/SO methods, and provides well corre-lated total energiesEk for all statesk of interest in a scalar relativistic approxi-mation. Depending on the method, this first step could also provide a tractablelimited representation of the wave function, then completed with all spatial andspin degenerate components. As in a DGCI algorithm the matrix representationof the Hamiltonian uses a determinant (or CSFs) basis|i〉 instead of states|Ψ0

k〉. The relation (24) defining the scalar energy in a Bloch-type effec-tive Hamiltonian formulation, is modified by introducing the identity resolution∑k

|Ψ0k〉〈Ψ

0k|= 1:

Hsr = Hsr + ∑k,i, j

〈i|Ψ0k〉〈Ψ

0k| j〉|i〉(Ek−E0

k)〈 j| . (31)

Schimmelpfenniget al transformed the DGCI code of Sjøvollet al [90] to in-clude the above correction [93]. The code remains an all-electron determinant-based non selected direct CI, with single and double substitutions from a RAS-type structure of the active space. The single and double CI is corrected em-ploying energies obtained in the first step with methods which take into accountthe size-consistency corrections. Test calculations on thallium hydride, usingan ACPF (Average Coupled Pair Functional) correlation treatment in the firststep and correlating at the SOCI level the four outermost electrons, showed abetter agreement with experiment than the above SOCIEX results [80] where14 electrons were correlated at the SOCI level.

Within the Bloch-type effective Hamiltonian DGCIeff schemes, Valletet alproposed a very different method, the so-called EPCISO algorithm [11] (Ef-fective and Polarised CISO) with the aim to enhance the possibilities of theDGCIeff treatments. As usual, sophisticated correlation calculations are car-ried out in a scalar relativistic approximation in the first step for all states ofinterest, in order to provide high-correlated energies used into the Bloch-typeeffective Hamiltonian in the second step. Eventually this first step could alsoserve as a guide to build a limited representation of the wave functions, and if

32

so a determinant|i〉 is kept if its weight in the wave function|Φk〉 of the targetspace exceeds a user-defined thresholdτsr (|Ck,i| > τsr). A reference space ofdeterminantsLr is automatically created, adding to the previous ones all deter-minants coming from all spatial and spin degenerate components of the statesunder interest. Alternatively the code can however construct, independently ofthe first calculation step, a reference CAS or MR spaceLr of determinants.In the second step a determinant model spaceLD is then built, adding to theprevious reference spaceLr (selected from an electrostatic-based criterion) alldeterminants| j〉 which have a spin-orbit interaction with a determinant|i〉 ∈Lrexceeding a user-defined thresholdτSO: 〈i|HSO| j〉

E(i)−E( j)

≥ τSO . (32)

The ”external” determinants| j〉 are generated from single excitations from thereference spaceLr , then accounting for spin-orbit polarisation of the referenceconfigurations by all ”external” important configurations.5 The matrix repre-sentation of the total Bloch-type effective HamiltonianHsr + HSO is built onthe model spaceLD accounting for the high-level correlation energy via equa-tion (31), and diagonalised using a complex Davidson procedure. Valletet alproposed a final perturbative improvement for the discarded configurations sim-ilar to the extrapolation technique for the whole Hamiltonian of Rakowitz andMarian [80] (see equation (30)), but this turned out to be unnecessary for theexamples treated so far. Apart from the highly correlated calculations of the firststep, all others processes are integrated into an unique code making the EPCISOalgorithm an actual two-step DGCI method. This algorithm uses a double selec-tion procedure on both electrostatic and magnetic interactions and then ensuresto take into account all important configurations for any state under interest,like the DiLabio one (see section2.2.4), but taking profit here of cost-effectiveextensive CI calculations via an effective Hamiltonian method. The accuracy aswell as the economical aspects of the computational procedure DGCIeff for thecalculation of relativistic electronic structure was first demonstrated on the sim-ple but challenging example of the spin-orbit splitting of the thallium atom [11].It was also very recently illustrated in a spectroscopic study of an heavy centreembedded into a crystal structure (see section3.2.2).

Recently, Stollet al [94] used a very similar approach to EPCISO. One minordifference is the use of the DGCI Pitzer’s code which works with CSFs basis5 In pseudopotential calculations,HSO is a one-electronic operator, and only single substitutions can couple twodeterminants. However one has to note that single excitations are the most important ones to take into account thespin-orbit polarisation of states under interest (see reference [59]).

33

functions instead of determinants. Apparently another difference here is theabsence of a selection process of the spin-orbit matrix elements. In this studysmall-core and large-core energy-consistent pseudopotentials were combinedfor the calculation of spectroscopic constants of lead and bismuth compounds(BiH, BiO, PbX, BiX, (X=F, Cl, Br, I)).

In the framework of the ab initio model potential technique, Llusaret al [79]proposed a two-step DGCI treatment where the spin-orbit effects are computedwithin two stages using the Pitzer code [42]. In a first step, usual large-scaleCI calculations are carried out in order to provide highly correlated energiesEkin a scalar relativistic approximation. In the second step spin-orbit effects aredescribed via a limited DGCI treatment using a multi-reference CI with sin-gle and double excitations. The SOCI matrix expressed on a basis of CSFsis diagonalised with and without the spin-orbit operator. Without the spin-orbit operator, one gets the approximate spin-free energiesE0

k and the corre-sponding restricted representations of the wave functions on the intermediatemodel space|Ψ0

k〉 ∈LS. In the last stage the full correlation treatment is takeninto account using a spin-free Bloch-type effective Hamiltonian (equation (24))slightly modified in order to correct the transition energies∆E0

k obtained with-out the spin-orbit operator in the second step, by the well-correlated ones∆Ekcoming from the first step

Hsr = Hsr +∑k

|Ψ0k〉(∆Ek−∆E0

k)〈Ψ0k| . (33)

Taking∆Ek instead ofEk only amounts to choose the ground state as origin forthe energy. Llusaret alshowed by calculations of a MgO-embedded (NiO6)10−

cluster, that the effective Hamiltonian technique applied on the spin-free part ofthe Hamiltonian, significantly improves the spin-orbit splitting.

Finally, Zaitsevskiiet al [95] proposed a DGCIeff method differing mainlyfrom the previous ones by the scalar effective Hamiltonian used and by the con-tent of the spin-orbit interaction. In this work a dressed intermediate Hamilto-nian is constructed using the spin-adapted many-body multipartitionning theory(MPPT) up to the second order. The MPPT theory is based on the simultane-ous use of several quasi-one-electron zero-order Hamiltonians (see for detailsref. [96] and references therein). The intermediate effective Hamiltonian matrixis defined in the intermediate model determinant spaceLD by

〈i|Hsr| j〉= 〈i|Hsr| j〉+ 12 ∑|k〉/∈LD

〈i|Hsr|k〉(

1∆( j → k)

+1

∆(i → k)

)〈k|Hsr| j〉 , (34)

34

where the energy denominators∆( j → k) are given by

∆( j → k) = ∑r:Nk

r <Nir

(Nir −Nk

r )ε⊕r − ∑s:Nk

s>Nis

(Nks −Ni

s)εr . (35)

The numbersNir andNk

r denote the occupancy of therth orbital in the inter-mediate model space determinant|i〉 and in the outer space determinant|k〉.The energiesε⊕r andεr are the non relaxed orbital ionisation potentials andelectron affinities with opposite signs, defined with respect to the intermedi-ate model space approximation. A major advantage of the MPPT correlationmethod is to join the quasi-size-consistency property with an efficient cost-effective treatment. However, up to now the off-diagonal spin-orbit matrix el-ements are computed between the determinants belonging toLD, without anyselection of outer-space determinants having a possible strong spin-orbit inter-action betweenLD and its orthogonal complement. This method was appliedto the description of electronic transitions involved in the radiative decay of theA0+

u , B0+u and B1u states of Te2 [95]. A good agreement was found with ex-

periment for the theoretical radiative lifetimes estimates for several low-lyingrovibrational levels of the states under interest.

3. MOLECULAR APPLICATIONS

3.1. Molecules in the gas phase3.1.1. Excited states for molecules containing main group elements

As one of the main difficulties of the SOCI methods lies in the electroniccorrelation treatment, we choose an example with a large number of valenceelectrons to correlate, namely a di-halogen molecule, in order to illustrate howmain-group pseudopotentials perform for spectroscopic constants. Moreoverto test the ability of the various SOCI methods to handle accurately very largespin-orbit splittings, we deal with the heaviest experimentally known, the iodinemolecule.

Let us first consider ground state spectroscopic properties. Schwerdtfegeret altested relativistic and correlation effects using a large-core (seven valence elec-trons) energy-consistent pseudopotential, with (UAREP, USO) or without (UNR)relativistic corrections, and with and without a semi-empirical CPP correc-tion [97]. They used a 9s6p2d basis set and a CI with single and double exci-tations. Table1 presents the calculated equilibrium distancere and dissociationenergy De.

We first consider relativistic effects at the SCF step. The first two entries ofTable1 show the influence of the scalar relativity (UAREP compared toUNR),

35

Table 1Equilibrium distancesre and dissociation energies De for the ground state of molecular iodine,from Ref. [97].Method Level re (A) De (eV)UNR (SCF) 2.702 0.94UAREP (SCF) 2.689 0.87UAREP+USO (SCF) 2.704 0.39UAREP+USO + CPP (SCF) 2.674 0.45UAREP+USO + CPP (CISD) 2.702 0.87UAREP+USO + CPP (CISD+SC) 2.719 1.17Exp. 2.666 1.556

namely a shortening ofre by 0.013A and a reduction of De by 0.07 eV. Addingthe spin-orbit couplingUSO to UAREP lengthensre by 0.015A and consider-ably reduces De by 0.48 eV, since at the equilibrium distance the valence closedshell experiences a tiny second order spin-orbit lowering of the energy, whileat dissociation the atomic open-shell structure presents a large spin-orbit low-ering of the energy of 0.62 eV. Concerning correlation effects, the core-valenceseparation due to the use of pseudopotential implies to divide correlation ef-fects into the ones originating from the core electrons and the others from theexplicitly treated valence electrons. Taking first into account the correlationeffects coming from the core electrons by adding the CPP correction to the pre-vious calculation (withUAREP+USO), reverses the previous trends at the SCFlevel, shorteningre by 0.013A and increasing De by 0.06 eV. The correlationof the 14 valence electrons (and keeping the CPP correction) worsens therevalue, which is lengthened by 0.028A, while the De value is significantly im-proved by 0.42 eV. Finally the size-consistency correction (SC) added to theprevious correlated results enhances this trends, increasingre by 0.013A andDe by 0.30 eV. In brief, the correlation effects (CPP, CISD, SC) altogether im-prove the dissociation energy, and the remaining discrepancy of 0.38 eV canbe interpreted as a lack of correlation, if we suppose a correct behaviour of thepseudopotentials used. A last point concerns the spin-orbit second-order effectswhich are responsible for a small lowering of the ground state energy at theequilibrium distance. Teichteilet al [71] using shape-consistent pseudopoten-tials and the CIPSO code [35] found a lowering of 0.01 eV, while Schwerdt-fegeret al found 0.14 eV. This effect was checked by van Lentheet al [98] inan all-electron calculation using the zero-order regular approximation (ZORA)of the Dirac equation and a local density functional; they reported an effect of0.11 eV close to Schwerdtfeger’s result. It seems that the too weak value found

36

by Teichteilet alwas due to not only the lack of a CPP correction potential, butalso to the use of a CIeff/SO method without a sufficient buffer of higher statesable to ”polarise” the ground state by the spin-orbit interaction. Of course theDGCI treatment of Schwerdtfegeret al could contain the determinants neces-sary to handle this small effect.

Later on, taking profit of hardware and software progress, Dolg checkedthe accuracy of several pseudopotential approximations on halogen atoms andmolecules and compared the results to all-electron calculations [99]. The to-tal valence energies were computed at the CCSD(T) level using several pseu-dopotentials (one energy-consistent denoted as PP1, and two shape-consistentpseudopotentials denoted as PP2 and PP3, see references therein), and for theall-electron case (denoted as AE), using a 25s20p14d4 f 3g basis set. Enlargingthe basis set fromspd to spd f gleads to an increase of the total valence energyboth in the AE case (by 0.109 au) and in the pseudopotential one (0.120 au,0.121 au, and 0.122 au, for PP1, PP2, PP3, respectively). With the largest ba-sis set, the total valence energy is slightly overestimated in the pseudopotentialcase as compared to the AE results (by 0.025 au, 0.032 au, and 0.036 for PP1,PP2, PP3, respectively). This overestimation of the valence correlation energywas early analysed by Teichteilet al [100] and Pittelet al [101] as being dueto the loss of the core nodes of the valence orbitals. However, this effect is sosmall that it does not weaken the conclusions drawn by Schwerdtfegeret al. Ina second article Dolg [102] repeated the comparisons of the ground state spec-troscopic constants of I2, using the same energy-adjusted pseudopotential, thesame basis set and the same correlation treatment as previously, and obtainedthe results summarised in Table2. As expected, a large basis set with high an-

Table 2Equilibrium distancesre, vibrational constantsωe and dissociation energies De for the groundstate of molecular iodine, from Ref. [102]. All-electron (AE) calculations are obtained fromeither the scalar-relativistic Douglas-Kroll-Heß (DKH) approximation or 4-component Dirac-Hartree-Fock (DHF) correlated calculation taken from Ref. [103].Method Level re (A) ωe (cm−1) De (eV)AE-DKH (SCF) 2.671 232 0.92PP (SCF) 2.669 238 0.95PP + CPP (SCF) 2.639 241 1.04AE-DHF (CCSD[T]) 2.717 206 1.28PP + CPP (CCSD[T]) 2.657 227 2.06PP + CPP + SO (CCSD[T]) 2.668 215 1.57Exp. 2.666 215 1.556

37

gular momentum values and an extended CI give for the best pseudopotentialcalculation (PP + CPP + SO [CCSD(T)]) highly accurate values compared toexperiment, especially for the dissociation energy which requires a particularlyhigh level of correlation treatment. The most sophisticated 4-component all-electron values (AE-DHF [CCSD(T)]) were obtained with a polarised valencetriple-zeta basis set, and cannot be easily improved as 4-component correlatedtreatments are quite tedious. Valence only calculations using a couple of pseu-dopotentialsUAREP,USO, are not only easier than fully relativistic all-electroncalculations, but can also be more accurate since they allow the use of largevalence basis sets along with the most extended valence CI treatments.

For a long time the excited states of I2 were a challenge to theoretical chem-istry methods. The first reliable theoretical estimates of the excited states of I2was done by Mulliken in 1940 [104], using a semi-empirical treatment with thehelp of experimental data. One of the firstab initio spectroscopic calculationswas made by Liet al [105] using large-coreUAREP andUSO shape-consistentpseudopotentials with a 4s4p2d basis set. They employed the RCI method thatbelongs to the class of DGCIeff method (see section2.2.5), beginning by corre-lated calculations in non-relativistic symmetries in order to construct a basis ofnatural orbitals for the relativistic valence Hamiltonian. The final DGCI calcula-tions includes up to 105,004 configurations for 29 electronic states, although thenumber of fine-structure valence states is 36, making the reliability of the spin-orbit coupling calculation somewhat questionable. They found a very differentpicture of the excited states than that of Mulliken which reproduced experimen-tal knowledge. This disquieting situation questioned either the pseudopotentialsthemselves, or the SOCI method used, or the correlation treatment, or even thephysical content of the coupled states. Teichteilet al [71] tackled this prob-lem with their own shape-consistent pseudopotentials, the CIPSO code [35]which is of CIeff/SO type (see section2.2.3), and a 5s5p2d basis set qualityalong with a selected MRSDCI algorithm for the correlation treatment in non-relativistic symmetries and a Davidson correction for size consistency. Theycoupled the whole set ofΛ–Σ states in the spin-orbit treatment dissociating intotwo 5s25p5 2P iodine atoms, giving rise to 36 fine-structure valence states. Theyfound for the ground state a poor dissociation energy (0.76 eV), and indeed theanalysis in the previous paragraph has shown that a very extended basis set isneeded along with a sophisticated CI treatment and a semi-empirical CPP cor-rection, to reach a better accuracy. Considering the lack of correlation at a giveninter-nuclear distance to be approximately the same for all the valence states,

38

they estimated the missing correlation contribution by taking the difference be-tween the computed ground state potential curve and the experimental RKRone, and adding this estimate to the energy of the excited states. The computedpotential energy curves are shown in Figure2, and the vertical transition ener-gies corresponding to the experimentally known valence states are displayed inTable3, along with the theoretical results discussed here.

Table 3Vertical transition energies (in eV) at the equilibrium distancere = 2.667 A oftheX 1Σ+

g ground state.

Λ−Σ states Ω−ω states Exp.1 Semi-emp.2 PP13 PP24 All-electron5

X 1Σ+g X 0+

g 0 0 0 0 0

3Πu A’ 2u 1.69 1.66 1.43 1.65 1.75A 1u 1.84 1.79 1.55 1.82 1.91B’ 0−u 2.13 2.34 2.60 2.18 2.30B 0+

u 2.37 2.37 2.05 2.34 2.43

1Πu B” 1u 2.49 2.38 3.19 2.57 2.62

3Πg a 1g - 3.4 3.46 3.56 3.69a’ 0+

g - 4.1 (3.93) 4.09 4.17

1 3Σ+u C 1u 4.57 4.57 4.03 4.58 4.67

1 The references for experimental values are given in Ref. [71]2 From Mulliken, reference [104]3 Pseudopotential from the Berkeley group (see section2.1) and RCI calcula-

tion (see text), reference [105]4 Pseudopotential from the Toulouse group (see section2.1) and CIPSO cal-

culation (see text), reference [71]5 Four-component Dirac-Hartree-Fock and MR-CISD, reference [106]

The two pseudopotential calculations (PP1 from Liet al and PP2 from Te-ichteil et al) lead to very different results. The vertical transition energies to thefine-structure components B’ 0−u and B” 1u are calculated at 2.6 and 3.19 eVwith PP1, while PP2 places them at 2.18 and 2.57 eV, in closer agreementwith the experimental results (2.13 and 2.49 eV). Moreover the PP1 calcula-tions place the B’ 0−u state above the B 0+u state whereas it should be belowsince the former dissociates to a lower atomic limit than the latter. For the3Πgstate, the component(1) 2g (non observed experimentally and not reported in

39

Figure 2.RelativisticΩ−ω potential energy curves of molecular iodine from reference [71].

Table3) is found below a 1g with PP2, in agreement with Mulliken’s predic-tions and as it could be expected for an inverted spin-orbit coupling of halo-

40

gens, while this order is inverted by PP1. In this configuration the a’ 0+g state

is assigned by PP1 to the3Σ−g which does lie above the3Πg state, while atequilibrium distance PP2 assigns a’ to the3Πg. The vertical transition ener-gies calculated by Teichteilet al support the assignment of the 1u (3Σ+

u ) as theC state, in agreement with Mulliken, while Liet al invoked the 1u componentof the 3∆u state, which is found to be 1.66 eV higher by the PP2 calculation.The 1u (3Σ+

u ) state of Liet alcrosses the fluorescent state B 0+u (3Πu), while it

should be the B” 1u (1Πu) state; indeed a unique 1u state does cross the B stateand the B” 1u (1Πu), which dissociates to the2P3/2+2 P3/2 lowest atomic limitis the unique possibility. Finally if one wants to explain the predissociationof the B state, all dissociative curve crossings have to be carefully computed.Li et al found four states crossing the B state, while Mulliken and Teichteilet alfound six dissociative states possibly implicated in the predissociation process(see Figure2). The high quality of these computed excited states allowed Te-ichteil et al to discuss and confirm some experimental hypothesis about thenatural, hyperfine, and induced collisional, electric and magnetic field predisso-ciation process of the B state (see for details reference [71]). Due to the quitedifferent qualitative behaviour found by Liet al and Teichteilet al for the ex-cited states of the iodine molecule, de Jonget al [106] checked these resultsusing an all-electron 4-component correlated calculation. They corroborated allthe conclusions drawn by Teichteilet al, as can be seen in Table3. However,due to the extreme difficulty to carry out a sufficient correlation treatment ina 4-component calculation, naturally discrepancy between their results and theexperimental data is larger than that obtained with the PP2 treatment.

Once again the quality of the valence correlation treatment is crucial, mak-ing the large-core pseudopotential calculation (even without a CPP correction inthis case) more accurate than a fully relativistic all-electron calculation. Anotherremark is relative to the comparison of the two pseudopotential SOCI calcula-tions. A DGCI treatment, which in principle is better than a CI/SO one whenlarge spin-orbit interaction is involved, does not guaranty by itself an accurateSOCI calculation, since in this example a CIeff/SO method gave much more re-liable and accurate results. If we suppose that the pseudopotentials are accurate,at least two tasks have to be fulfilled:i) a sufficient valence correlation treatmenthas to be done (eventually with the help of an effective Hamiltonian method),and if necessary along with a CPP correction;ii) a prior physical analysis of thechemical system is essential in order to have in whatever SOCI treatment themain physical content of the system.

41

3.1.2. Excited states for molecules containingd elementsThe investigation of physical and chemical properties of transition metal

complexes is an extremely active experimental field and is the subject of nu-merous theoretical studies going back to 1980. One fascinating aspect of thed-block elements is the important role of relativistic effects that can induce sig-nificant and surprising changes in the properties of a series of isoelectronic ele-ments going from the lightest to the heaviest element [107]. Therefore specialattention has to be taken for the inclusion of relativistic effects (scalar and spin-orbit), especially for the second- and third-row transition metals. Dyall [108]has examined the importance of both scalar relativistic and spin-orbit effectsin the study of the ground state of PtH2 and several low-lying states of PtH+

and PtH at the all-electron self-consistent-field level of theory. His study whichneglects correlation effects, clearly shows the significant influence of spin-orbitinteractions on the properties and energetics of these molecules, both in theground and excited states. It reflects the participation of thed orbitals in thebonding and the well-known fact that the modifications of valence atomic lev-els induced by relativity are seen in the molecular properties. For PtH Viss-cheret al [109] included on top of four-component Dirac-Hartree-Fock calcula-tions correlation effects in a relativistic configuration interaction scheme. Elec-tron correlation does not strongly affect the excitation energies of the spin-orbitmixed states but has a significant impact (especially the dynamic correlation inthed shell) on the spectroscopic properties: namely the equilibrium bond lengthis shortened by about 0.04A, and the corresponding harmonic frequencies arehigher by several hundreds of cm−1. These fully-relativistic calculations pro-vide benchmark calculations against which the accuracy of efficient and lessexpensive approaches such as pseudopotentials can be tested. We will illustratethis point later on in this section.

Beyond the description of relativistic effects, systems containing transitionmetals are quite demanding with respect to the treatment of electron correla-tion. This is particularly true when there is a large covalent contribution tothe bonding. Correlation is also important in cases where mixing of atomicasymptotes with differentd occupations occurs. It poses tremendous demandon the theoretical treatment to correctly account for the large differential effectsbetween states with different numbers ofd electrons. This rules out the useof single-reference correlation methods (single-reference CI, Coupled-Clustermethods,...) and enforces the use of sophisticated multi-reference approaches,that include spin-orbit coupling, at an early or later stage. Whereas the num-ber of theoretical studies for electronic ground states never stopped increasing

42

in the past fifteen years, investigations in electronically excited states are byfar less common due to difficulties in accurately treating at the same level thedense manifolds of electronic states. For an extensive list of theoretical studiesof transition metal complexes, we forward the reader to the following refer-ences [110,111,112,113,114,115,116].

The spectroscopy of gas-phase monohalides of metals of group I-B (cop-per, silver and gold) exhibits interesting properties that have stimulated a largenumber of experimental and theoretical studies (see Ref. [117,118] and refer-ences therein). From the theoretical point of view, they represent a paradigm ofchemical bonding in transition metal-containing molecules. The following dis-cussion of the recent series of paper by Guichemerreet al [117] and Ramırez-Solıset al [118,119,120,121] will allow us to contrast the theoretical difficultiesone has to face when dealing with transition metal systems. The first problemand perhaps the easiest one is the treatment of scalar relativistic effects withinthe pseudopotential approach. In this context, several aspects have to be care-fully considered.

1. Large-core pseudopotentials are well known to lead to sizable errors [122,123,124,125]; we thus recommend the use of small-core ECP’s that in-clude semi-core orbitals in the valence space [120].

2. The accuracy of the pseudopotential fit procedure also plays an importantrole, like the number of reference states used [26,126].

3. Care should also be taken for the quality of the atomic basis set. As inthe case of all-electron calculation, an optimal accuracy is reachable bycombining high level correlated treatment and extended atomic basis set.

Schwerdtfegeret al [124] have systematically tested several pseudopotentialsfor the heaviest element of group I-B, gold atom and its hydride. The varia-tion between the results obtained from all valence electron small-core Stuttgartenergy-adjusted pseudopotentials [122,127] and all electron Douglas-Kroll cal-culations for AuH is found to be small (∆re = 0.001A, ∆De = 0.03 eV,∆ωe =9 cm−1). This demonstrates that the pseudopotential approach is a reliable andefficient alternative to all-electron calculations (see section2.1). Both Ramırez-Solıs et al (cf. Ref. [118] and references therein) and Guichemerreet al [117]used the Stuttgart small-core scalar ECPs [127,128] to study the spectroscopyof group I-B halides MX (M = Cu, Ag, Au; X = F, Cl, Br, I). To cal-culate the spin-orbit coupling, Ramırez-Solıs et al used the spin-orbit pseu-dopotentials extracted at the same level of all-electron calculations (Wood-

43

Boring) as the scalar ones, whereas Guichemerreet al used SOREPs fitted tomulti-configuration Dirac-Hartree-Fock (MCDHF) spin-orbit splittings of sev-eral atomic states [129].

Let us now discuss the calculation of the asymptotes of MX systems. Theground state of the coinage metal atoms (Cu, Ag, Au) is a2S (ns1(n−1)d10) andthe excited states are2D (ns2(n−1)d9) and a Rydberg-state2P (np1(n−1)d10).In silver atom, the Rydberg-state turns out to be lower than the2D excitedstate. In contrast to the lighter coinage metals, the lowest state of the super-heavy element 111 is2D (6d97s2), the reversal being attributed to relativisticeffects [130]. The change of the occupation number of the(n− 1)d shell in-duces a large difference in the radial extent of thed orbitals. Thus, the2D–2P separation is very sensitive to correlation effects and to the basis set ex-tension. Ramırez-Solıs et al [131] recently demonstrated for the silver atomthat a basis set including diffused functions and at least threef andg func-tions is required to get within 0.2 eV of the experimental separation. More-over, the correlation of the semi-core(n− 1)s and (n− 1)p orbitals in theMRCI treatment plays an important role in the ordering of the2P and2D states.Whereas the CASPT2 method gives satisfactory results for the atomic spec-trum of the Ni atom for example [132], it completely fails in reproducing theproper state ordering and underestimates the excitation energies by 1694 cm−1

and 7864 cm−1 for the 2P and2D states, respectively. When high accuracy isrequired, the Equation-of-motion coupled-cluster (EOM-CCSD) method per-forms better than MRCI treatments [117]. It is noteworthy to point out that theEOM-CCSD method, though extremely accurate, must be ideally used in re-gions of the potential surface where the ground-state wave function is mostlysingle-reference, that is away from avoided crossings. It cannot be applied tocompute the complete potential energy curves.

Spin-orbit coupling splits the2P and2D asymptotes. By applying the EP-CISO method [11] and using the SOREP adapted to the Stuttgart AREP,Ramırez-Solıs et al have shown that the accuracy of the atomic spin-orbitsplitting depends on three factors:i) Spin-orbit relaxation effects arising fromsingly-excited configurations increase the splitting values as already mentionedin section2.1.3. ii) It is advisable to use state-specific orbitals rather than state-averaged ones since the spin-orbit integrals are sensitive to the inner part ofthe pseudo-orbital.iii) In silver atom, the2P and2D states are close in energy(1800 cm−1) and the spin-orbit coupling constant of the 4d orbitals of Ag+ israther large,ζ4d(Ag+) = 1830 cm−1. It is therefore essential to use an effectiveHamiltonian to correct the position of spin-free states. Indeed, the interaction

44

between theJ = 3/2 components of2P and2D states influences the ordering ofthe fine-structure components.

This analysis leads us to the conclusion that the calculation of atomic spectraof transition metals is very demanding since correlation and spin-orbit interac-tions are intertwined. DGCIeff methods such as EPCISO are the most suitedSOCI methods to tackle such difficult problems. Errors in correctly position-ing the atomic states will affect the description of molecular states that dependon the mixing of atomic states. The error in the atomic spectra is one mea-sure for the accuracy of molecular MX systems. Another important aspect isthe choice of the zeroth-order reference, and the choice of the method to opti-mise the molecular orbitals. Considering that the various excited states underinterest are often based on configurations with very different electronic char-acter, the use of a common set of orbitals obtained either from SCF or state-averaged CASSCF calculations does not seem particularly suited, even thoughit is very convenient for the subsequent correlated treatments. State-specificorbitals obtained for each state individually, provide a better zero-order wave-function and improve also the quality of the correlated vertical transitions. How-ever, when looking at potential curves, or potential surfaces, the character of thewave-function may change along the internal coordinates, for instance close toavoided crossing, making the calculation of state-specific orbitals difficult, ifnot impossible. Namely, in the MX systems, the mixing between ionic Ag+–X− and neutral Ag–X configurations varies along the internuclear distances asreflected by plots of the dipole moment functions reported in Ref. [117]. Insuch a complicated case, the only way to optimise state-specific orbitals is tostart from a state-averaged calculation and progressively tune the weights of thedifferent states to focus on the wanted state. This discussion on the choice ofmolecular orbitals is also relevant for the spin-orbit interaction. The computedcoupling elements are sensitive to the shape of atomic orbitals as discussed insection2.1.3, and also to molecular orbitals as demonstrated by calculationson CuO [114].

To obtain consistently accurate results, it is necessary to correlate the valenceelectrons of the ligand, namely then′p electrons of the halogen, as well asthe metal valence(n− 1)d and ns. In silver halide systems, one must alsoinclude thenpπ orbitals along with thenpσ to get the proper description ofthe all excited states that dissociate to the lowest limit Ag(2P)+ X(2P). Thisleads to an active space including 12 active orbitals and 16 active electrons.The correlation of inner(n−1)sand(n−1)p shells has almost no effect on the

45

calculated atomic transition energies [131]. One can therefore safely discardthem from the correlated treatment and at the same time keep the CI-treatmentto an affordable size, though not too small (about 3×107 uncontracted CSFs).

Table 4Spectroscopic data for AgCl and AgI. MRCI, CC, CC+SO and experimental data aretaken from Ref. [117] and references therein, MRCI2 from Refs. [118, 121] and all-electron ZORA calculations from Ref. [133].

AgCl AgIState Method Te re ωe µe Te re ωe µe

(eV) (A) (cm−1) (D) (eV) (A) (cm−1) (D)X1Σ+ MRCI 0 2.31 327.2 6.74

CC 0 2.30 318.7 -CC+SO 0 2.30 328.8 -MRCI2 0 2.33 359.7 6.38 0.0 2.65 204.7 6.11ZORA 0 2.31 324 -Exp. 0 2.28 343 5.70 0.0 - 206.2 -

21Σ+ MRCI 3.98 2.30 1.36CC 4.05 2.32 -CC+SO 4.13(0+) 2.34 -MRCI2 3.89 2.30 324.1 1.41 2.90 2.53 532 1.41

(4.29)1

ZORA - - - -Exp. 3.92 2.32 280.0 - 2.96 2.69 131 -

(3.41)1

13Π MRCI 3.62 -1.64CC 3.49 -CC+SO 3.34(0+) -MRCI2 3.41 - 2.40 3.92 26 0.90ZORA - -Exp. 3.63 - - - - -

1 The value in parentheses refers to the second minimum at longer distances.

One of the key question in the spectroscopy of silver halides is the natureof the experimentally observed excited states B, C and D, but we refrain our-selves from discussing this problem and refer to the work of Guichemerreet aland Ramırez-Solıs et al for further details. In Table4, we only report thespectroscopic constants of the ground state and two low-lying1Σ+ and3Π ex-cited states. As far as the ground state is concerned, we observe a satisfactoryagreement, on one hand between pseudopotential calculations and all-electron

46

ZORA results, and on the other hand between computed and experimental data.There are strong mixtures of the ionic Ag+(1S)+X−(1S) and Ag(2P)+X(2P)or Ag(2D)+X(2P) neutral configurations in the3Σ+,21Σ+ and 13Π states, lead-ing to neutral-ionic crossing along the potential curves. The use of state-specificmolecular orbitals for the MRCI treatment (labelled MRCI2 in Table4) leadsto a better computed dipole moment for the ground state and also slightly im-proves the quality for the excited singlet state, as compared to the data obtainedwith state-averaged orbitals (MRCI) or ground-state orbitals (EOM-CCSD).The 21Σ+ state in AgI has two minima, one at shorter distance (about 2.69A)and a shallow one at larger distances (3.41A). As can be seen from the reportedresults (Table4), it is extremely difficult to place correctly these two minima,due to the problem of ionic-neutral mixing in the wave-function.

In order to assign experimental spectra, spin-orbit coupling as to be accountedfor. Guichemerreet al calculated the coupling elements between the variousspin-free states in a contracted CI/SO approach (see section2.2.2), using anewly extracted SOREP (see Ref. [117]). The 3Π components are split byabout 0.5 eV at the equilibrium geometry, leading to strong interactions betweentheΩ = 0+ components of the3Π and1Σ+ states. In AgF, the interactions arestrong enough to make the3Π0+ bounded. This kind of phenomena is crucialwhen it comes to discuss the assignment of the experimentally observed tran-sitions. Only calculations of the complete potential curves including spin-orbitcoupling can give a solid basis to the interpretation of the spectra. One shouldalso notice that the heavier the metal M or the halide X is, the more complex thespectra are. This is a typical case where uncontracted effective SOCI methodssuch as the EPCISO, RCI or others (see section2.2.5) are ideal.

Gathering the experience of the previously discussed examples, we can saythat the theoretical study of spectroscopic properties of transition metal com-plexes is far from being simple. Relativistic pseudopotentials (AREP andSOREP) were shown to be efficient and accurate tools to tackle this problem.From the methodological point of view, recently developed effective Hamil-tonian SOCI methods that can treat correlation and spin-orbit coupling on thesame footing exist (see section2.2.5), and efforts have to be invested in applyingthem to these critical cases. However, the application of these explicitly corre-lated methods is confined to small molecules, although there is no theoreticalreason why they should not be extended to larger systems. Promising alter-natives to study larger molecules are coming from the field of Density Func-tional Theory (DFT), with the recent development of Time Dependent-DFT

47

(TD-DFT) [134] approach based on the linear response of the charge density toan applied field allowing direct computation of excitation energies and oscilla-tor strengths. Few applications on the spectra of transition metal complexes forwhich either experimental data or accurateab initio results are available, startto emerge [135,136,137,138,139,140]. TD-DFT usually performs quite wellin assigning the spectra, but tends to systematically underestimate transition en-ergies with respect to CASPT2 calculations. This is even more pronounced forMetal-to-Ligand-Charge-Transfer transitions (MLCT) excitations. However, atpresent it is not yet possible to assess the ability of current functionals at de-scribing a variety of excited states, and future methodological developmentswill certainly improve its reliability.

3.1.3. Excited states for molecules containingf elementsEven though the chemistry off -elements complexes is a very active field, the-

oretical studies of their excited states are rather scarce. From the experimentalpoint of view, the electronic spectra of lanthanide containing molecules are rel-atively well known, while actinide compounds cannot be studied so easily dueto their radioactive properties. From the theoretical point of view, complexesof f elements are challenging as, due to the number of unpairedf electrons,an important number of excited states are lying in a narrow energetic range. Inthe following, we will discuss several examples taken from the spectroscopyof lanthanide oxides, sandwich molecules such as Ce(C8H8)2 and actinyl ionssuch as NpO2+

2 or PuO2+2 .

Lanthanide molecules.

An in-depth review on lanthanide containing molecules calculations has beengiven by Dolg and Stoll [92]. Several studies carried out on CeO and YbO bydifferent groups are reported in the following. The excited states of CeO havebeen calculated at different levels of theory, namely INDO/CIS byKotzianet al[141] and ECPs SCF/CISD/CIPSO calculations by Dolget al[142].The vertical transition energies Te are presented in Table5.

As can be seen from Table5, the excited states manifold is dense: there are16 excited states within 4500 cm−1. For comparison, in the case of iodine atom,the first excited states lies at about 13000 cm−1. This multitude of low-lying ex-cited states strongly interact with each other either by Coulomb interaction orspin-orbit coupling, a fact that makes theab initio treatment difficult. Table5shows that the semi-empirical INDO results are in excellent agreement with ex-periment for all the states under consideration. Indeed the parametrisation of the

48

Table 5Vertical transition energies in cm−1 of CeO [141,142] and YbO [143,144]. In the pseudopo-tential calculations, the 4f electrons are either included in the valence shell (PPv) or frozen inthe core of the pseudopotential (PPc).

CeO YbOΩ exp. T0 INDO Te PPv Te PPc Te Ω exp. T0 INDO Te PPv Te PPc Te

2 0 0 0 0 0+ 0 31073 8410 55103 82 71 119 101 0− 910 910 910 9101 813 857 913 923 1 1015 963 1029 10112 912 918 1045 968 0− - 1014 1313 11920− 1679 1812 1396 1589 2 2408 2317 1885 17841 1875 1937 1476 1679 1 2702 2432 2125 18960+ 1925 1850 1715 1769 3 4287 3048 3310 31044 2042 2084 2139 2302 2 - 3113 3456 32853 2143 2154 2286 2487 4 - 3388 3806 47873 2618 2726 2872 3086 3 - 3504 3900 50622 2771 2794 3039 3165 0− - 10565 11208 113582 3462 3453 3386 3771 1 - 10684 11424 116021 3635 3562 3391 3766 0+ - 10570 11489 118180− 3819 4176 3476 4120 1 - 12221 12992 129591 4134 4234 3605 4249 2 - 12242 13072 130830+ 4458 4217 4234 4314

INDO method in that particular case is done in such a way to reproduce the ex-perimental results. The results obtained with pseudopotentials agree better withthe experiment when the 4f electrons are in the core than when they are explic-itly treated in the valence space. This indicates that the interaction between thecore-like 4f orbitals and the spatially well separated partially occupied valenceshell is weak and can be safely neglected. Moreover, the removal of the coreelectrons allows a better correlation in the valence shell as only singlet, tripletor quintet states are calculated. Comparing the PP with the INDO results, oneshould keep in mind that the spin-orbit treatment has been done with the CIPSOmethod [35], which computes the spin-orbit coupling between contracted wavefunctions (see section2.2.3). For these kind of problems, where the states ofinterest are not well separated in energy from the higher ones, it is necessaryto consider a buffer of higher states. In such a case a much easier treatmentof the correlated spin-orbit coupling is by using an uncontracted SOCI methodlike EPCISO or SICCI (see sections2.2.4and2.2.5), which include automati-cally all the determinants having an important spin-orbit coupling with the statesof interest. The former method needs only to correlate the states of interest ina first step, and as it uses in the second step a double selection threshold on

49

electrostatic and magnetic interactions, the size of the SOCI space (number ofdeterminants) remains under control.

Similar theoretical work has been reported for the YbO molecule. However,in that case, the situation is much more complicated and the discrepancy be-tween the INDO and the PP approaches is larger. In particular, the theoreticalcalculations do not agree on the ground state experimental assignment. Indeed,theoretical work by Dolget al [143] assigned the ground state to a 0− state aris-ing from a 4f 13 core configuration, whereas experiment rather indicates thatthe ground state is a 0+ state coming from a 4f 14 core configuration. Severalexperimental works by McDonaldet al [145] on electronic excited states, orlaser ablation experiments of Willsonet al [146], support the latter conclusion.Finally, Wanget al [147, 148] using quasi-relativistic DFT calculations foundout that the ground state is a 0+ state originating from a mixture of the two coreconfigurations. Considering this latter study, the results on the excited statesobtained by INDO/CIS [144] or by PP SCF/CISD/CIPSO [143] presented inTable5 have been shifted to agree with experiment for the 0− state.

As in the case of CeO, the INDO results are in good agreement with ex-periment but fail in predicting the ground state. Concerning the PP results, thesituation is somewhat different than in the CeO case. Indeed, for YbO, the treat-ment of the 4f electrons in the valence shell leads to a better agreement withexperiment. This is simply explained by the fact that the ground state is a mix-ing of two 4f occupations as shown by Wanget al. The results obtained withlarge-core pseudopotentials, with the 4f electrons in the core, are instructive inthe sense that some excited states are well described, meaning that the leadingconfiguration of the state is, depending on the pseudopotential chosen, the 4f 14

or the 4f 13 one. However, these large-core pseudopotentials cannot mix bothcore configurations.

Let us now consider a more representative chemical system, the ceroceneCe(COT)2 with COT = C8H8 and its excited states. The first theoretical in-vestigations by Clacket al [149] used the INDO method. Few years later, aquasi-relativistic and a nonrelativistic scattered wave Xα calculations were re-ported by Roschet al [150]. In all cases, the system was assumed to havean ideal D8h symmetry. Dolget al [151] reported a large scale state-averagePP MCSCF, MRCI, ACPF calculations, including spin-orbit interaction withthe CIDBG code [42] (see section2.2.4). According to an anionic model, theleading configuration of the cerocene corresponds to a Ce(IV) centre betweentwo C8H2−

8 cycle. However, some initial works of Neumannet al [152] have

50

shown that a single determinant calculation leads to a Ce(III) interacting withtwo COT−1.5 rings, indicating that the ground state is a triplet state3E2g. Theauthors investigated the problem using multiconfigurational techniques. In thatcase, it was found that there is a mixing between a Ce(III) and a Ce(IV) config-urations, leading to a stabilisation of the1A1g state, below the3E2g state. Thisanalysis is supported by the calculations of Dolget al. Indeed, fixing the 4foccupation to 0 or 1, the authors found the ground state to be of3E2g symmetry,the1A1g state lying 0.58 eV higher in energy. However, if one allows the con-figurations to mix, the singlet state becomes lower by 0.56 eV than the tripletstate. This picture holds even when spin-orbit coupling is accounted for: indeedit does not influence the energy of the singlet state and stabilised the triplet stateby 0.09 eV, bringing the singlet-triplet energy gap down to 0.47 eV. Howeverthe DGCI approach is very often limited by the size of the configuration interac-tion, which is of great importance when the occupation number of the 4f shellchanges.

Actinide molecules.

As mentioned in the introduction of this section, lanthanide electronic spec-troscopy is much better known experimentally than the one of actinide due to thedifficulty of handling such elements. From the theoretical point of view, actinideatoms are even more challenging than the lanthanide atoms in the sense that forearly actinides, the 5f orbitals play a role in the bonding and have to be treatedexplicitly. Moreover, the large relativistic effects and the numerous unpairedelectrons make the calculations on actinide molecules difficult, explaining thesmall numbers of theoretical studies. However the development of new SOCImethods and the increasing power of computers make now possible to studyspectroscopic properties of such compounds. For rather complete reviews oncalculations of actinide molecules, we refer to Pepper and Bursten [153], andDolg [154]. We discuss two recent theoretical studies of actinyl ions excitedstates. The first example concerns the excited states of NpO2+

2 and NpO+2 using

the CIDBG code by the Pitzer group (see section2.2.4), whereas the second oneconcerns the PuO2+

2 ion treated with the CIPSO code [35] from the Toulousegroup (see section2.2.3).

The excited states of NpO2+2 and NpO+

2 have been calculated by Mat-sikaet al [86] and we first detail the results on the isolated neptunyl ion NpO2+

2 ,as it has only one unpairedf electron. The calculated spectrum was comparedto some experiments in ionic matrices (see also section3.2.3). The authors have

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calculated the vertical transitions at the ground state geometry (re= 1.66A) andcorrelated all valence orbitals with the exception of the low-lying ones, namelythe 1σg and 2σg, the 1σu and the 1πg, resulting in a double group CI space of4 million double group configurations.

Table 6Vertical transition energies in NpO2+

2 in cm−1 taken from Ref. [86].

Ω Exp. [Cs2NpO2Cl4] Exp. [CsNpO2(NO3)3] Te

5/2u 0 0 03/2u 1000 - 4475/2u 6880 6459 55157/2u 7990 9420 65657/2u 13265 13918 126229/2u 15683 16092 154181/2u 15406 16072 156683/2u 16664 16799 1796711/2u - - 186761/2u 19375 19510 2158013/2u - - 219253/2u - - 222301/2u - - 224695/2u - - 238821/2u 17241 17843 258443/2u 20080 20816 28909

The agreement between the calculated energies reported in Table6 and ex-perimental data is satisfactory, even though one should keep in mind that thecalculations do not treat the specific interactions between the ionic lattice andthe neptunyl ion, a point that will be developed in more detail in the next sec-tion 3.2. In particular, the two highest calculated states are badly reproduced,possibly because interactions with higher states or with the ions of the latticeare missing. By analysing the nature of each fine-structure components in termsof Λ-S states, the authors assigned the first three excited states to pure atomiccombinations of thef orbitals. The fourth state involves excitations from thebonding orbitalσu to the essentially atomicf non bonding orbitals (δu, φu)and has a charge-transfer character. The presence of low-lying charge-transferexcitations is an important difference from the spectrum of the uranyl ion [155].

In the same way and using the same theoretical method, the authors studiedthe more complicated monocationic NpO+

2 which has two unpairedf electronsin the ground state. This ion is isoelectronic to PuO2+

2 , which will be discussed

52

later on in this section. In agreement with some other theoretical works [156]on the plutonyl ion, the authors found the ground state to be a3Hg state origi-nating from theφ1δ 1 configuration. As in the previous case, the authors havecalculated the vertical transitions at the ground state geometry (re = 1.72 A).As previously, the same orbitals were correlated using the CIDBG code givingrise to a configuration space of 6.5 millions. In that case, the results can not becompared with experiment. It should be noticed that in both studies the densityof states is considerably lower than the density of states calculated for the lan-thanide compounds, but higher than in the case of main group elements. Thefirst charge-transfer state is found at a vertical transition energy of 23079 cm−1,significantly higher than in the neptunyl ion, close in energy to the first exci-tation in the uranyl ion at 20861 cm−1. This simply means that the presenceof charge-transfer states is not directly related to the presence and the numberof f open-shells. According to these results it is not possible to predict the po-sition of the first charge-transfer state in the case of the isoelectronic plutonylion, PuO2+

2 . It is however clear that a charge-transfer state should appear at onepoint in the spectrum.

Let us now turn to an example of an actinyl ion computed with a CIeff/SOmethod. Some excited states of the plutonyl ion, PuO2+

2 , have been calculatedby Maronet al [157] using the CIPSO method [35] to account for spin-orbitcoupling. In this study, the authors have only computed the excited states aris-ing from the f 2 manifold and did not try to calculate the charge-transfer states.All the states inΛ–S symmetry coming from thef 2 manifold have been calcu-lated at the MRSDCI+Q (MRSDCI including Davidson correction) level lead-ing to 23 states before spin-orbit coupling. All these states were included in thespin-orbit calculations but only the 23 lowest fine-structure states, in the rangeof 0-37000 cm−1, were considered, as the higher part of the spectrum is notwell defined due to the lack of the charge-transfer states and some other higherstates. The authors first studied the modification of the geometry with respect toexcitation at the spin-free level. As expected, at different levels of theory MRS-DCI+Q, MR-AQCC (Multireference Averaged Quadratic Coupled-Cluster) orCASPT2, the geometry of the complex is the same in the ground state and inthe first excited states. This is explained by the fact that the considered excitedstates differ from the ground state only by differentf electronic distributions.The results are given in Table7. The authors performed the calculation at theground state geometry (re = 1.6770A), which was obtained at the best level ofcalculation (AQCC). In order to show the complexity of this calculation withthe CIPSO method, based on the use of a contracted basis set of states, we re-

53

Table 7Optimised bond lengths (given inA) for the first low-lying excited sates of PuO2+

2 with threedifferent methods of correlation, taken from Ref. [157].State AQCC CASPT2 SDCI+Q3Hg 1.6770 1.6849 1.67673Σ−g 1.6747 1.6851 1.66981Σ+

g 1.6744 1.6851 1.67051Γg 1.6576 1.6702 1.6588

port in Table8 all the states which were needed in the calculation in order to getreliable values for the lowest part of the spectrum.

The 23 firstΛ–S states are lying in the range of 86000 cm−1. As can be seenfrom Table8, all possible combinations of two electrons in seven orbitals havebeen considered. In particular, analysing the orbitals, it can be shown that thefσ and the fπ are mixed with somes, p andd orbitals of the actinide centrewhich contribute to the bonding. So, in that sense, some of the bonding orbitalshave been included in the calculation. The spin-orbit results are presented inTable8 together with the leadingΛ–S configuration of the state.

As in the case of NpO+2 , no experiment were available to compare this cal-culations with. As can be seen from Table8, the 23 lowest fine-structure statesarise from a limited number ofΛ–S states. This calculation shows that it isin principle possible for some simple actinide molecules to use a contractedCIeff/SO method. But it is worthwhile to note that it would have been mucheasier to use an adapted SOCI method, namely a DGCIeff method which in-cludes automatically all determinants coupled with the states of interest (seesection2.2.5).

To conclude this part, it should be noticed that lanthanide and actinide elec-tronic spectroscopy can now be handled byab initio methods with a sufficientaccuracy. In particular, the very recent progresses on the SOCI methods arepromising for the treatment of excited states of lanthanide and actinide com-pounds.

3.1.4. Spin-orbit effects and reactivity on the ground stateThis chapter is devoted to pseudopotential calculations of molecular excited

states, however, some words should be said about spin-orbit effects on theground state when dealing with the reactivity of open-shell systems. Eventhough, the reactivity of transition metal compounds has been extensively stud-ied (see for example Ref. [115]), spin-orbit effects are quite rarely taken into

54

Table 8Vertical transition energies in kcal.mol−1 of the PuO2+

2 molecule cal-culated at the SDCI+Q level without and with spin-orbit coupling atre = 1.6770A, taken from Ref. [157].

Without spin-orbit With spin-orbitState Configuration Te Leading state Ω value Te3Hg φδ 0.00 3Hg 4g 0.003Σ−g φ2 10.28 3Σ−g 0+

g 12.283Πg φδ 17.47 3Hg 5g 18.851Σ+

g φ2 27.92 3Σ−g 1g 20.141Hg φδ 32.36 3Πg 0−g 20.8811Γg δ 2 41.32 3Πg 0+

g 21.131Ig φ2 48.38 3Hg 6g 22.443Σ−g δ 2 49.44 1Σ+

g 0+g 26.92

1Πg φδ 57.84 3Πg 1g 36.893Γg φπ 77.29 3Πg 2g 40.513∆g φπ 80.28 1Hg 5g 48.561Σ+

g δ 2 83.88 1Γg 4g 66.021Γg φπ 91.11 1ıg 1g 77.213Φg δπ 92.21 1Ig 6g 86.503Πg δπ 101.34 3Γg 3g 94.821∆g φπ 102.58 3Γg 4g 95.261Φg δπ 116.71 3Σ−g 0+

g 95.251Πg δπ 125.28 3Φg 2g 95.463Σ−g π2 179.68 3Φg 3g 95.401∆g π2 194.30 3∆g 1g 98.701Σ+

g π2 208.61 1Σ+g 0+

g 100.673∆g δσ 228.94 3∆g 2g 102.071∆g δσ 239.71 1Πg 1g 104.941 The 0− of the3Πg state was omitted in Ref. [157].

account. This can be understood from the fact that in all the considered re-actions, such as for example inert bond activation through the classical ox-idative addition-reductive elimination mechanism, both the reactants and theproducts are closed-shell systems. In this case, spin-orbit contributions on thetotal energy are of second-order and are usually too small (see discussion insection3.1.1) to significantly change the thermodynamic data calculated at thespin-free level. However, many lanthanide and actinide systems have unpairedf electrons in their ground state and one could expect spin-orbit coupling toplay an important role. In the case of the lanthanide series, where only the ox-idation state (III) is stable, the reactions follow a concerted pathway without

55

any change of oxidation state. So, in that case, the initial and final states arethe same and one can expect that spin-orbit contributions cancel out. However,early actinides exist in several oxidation states and can participate in oxidation-reduction reactions, in which the number of unpaired electrons changes alongthe reaction path. Therefore spin-orbit coupling is likely to change the relativestability of the reactants and products and thus will modify the thermodynamicsof the reaction. This is nicely demonstrated in the works of Valletet al[158,159]on the reduction by water of the early actinyl(VI) ions MO2+

2 (M = U, Np, Pu,Am) from oxidation state VI to IV in aqueous solution. The reduction followsa two-step mechanism with an actinide(V) as an intermediate

MO2+2 (VI)+

12

H2O−→ HOMO2+(V)+14

O2 , (36)

HOMO2+(V)+12

H2O−→M(OH)2+2 (IV)+

14

O2 . (37)

Combining both steps, the reduction from oxidation state (VI) to oxidationstate (IV) reads

MO2+2 +H2O−→M(OH)2+

2 +12

O2 . (38)

In the case of uranium, the authors reported the highest influence of spin-orbiteffects on the energetics of the reaction. The calculations were done at theACPF level with small-core ECPs on the uranium atom from the Stuttgartgroup [160]. The spin-orbit calculations were performed with the CIPSOcode [35] using a mean-field-ECP combination [32]. For the uranyl ion, thetotal reaction (38) is found to be endothermic by 16 kcal.mol−1 at the spin-freelevel, but spin-orbit effects lower the reaction energy by roughly 14 kcal.mol−1,so that it becomes thermoneutral. The authors also analysed the spin-orbit in-fluence in both steps of the reaction, summarised in Figure3.

As can be seen from Figure3, the largest spin-orbit correction is on the firststep of the reduction. This is explained by the fact that uranyl ion has a closed-shell ground state which is barely influenced by spin-orbit effects whereas theoxidation state (V) is defined by one unpairedf electron leading to an open-shell system. In the second step, both initial and final states are open-shellsystems and are stabilised by spin-orbit interaction so that the contribution tothe reaction energy is smaller than in the first step. The same is true for thereactions of other actinides systems. Even if the correction is smaller than inthe uranium case, the reaction energy is lowered by spin-orbit coupling leadingto more exothermic reactions (see Figure4). To conclude this part, it should

56

Figure 3. Influence of spin-orbit interactionon the reduction of uranyl. Energies are inkcal.mol−1.

Figure 4.Reduction energetics of the actinylions in solution including spin-orbit effect.Energies are in kcal.mol−1.

be kept in mind that when studying reactions with a change in the oxidationnumber, spin-orbit effects must be accounted for. They may lead to significantchanges in the thermodynamic and kinetic data.

3.2. Spectroscopy of embedded molecules3.2.1. Modelling the spectroscopy of ionic impurities in crystal

Numerous crystals doped by ionic impurities such as main element ions orf element ions offer very attractive photophysical properties. One of the keyquestions is to identify the factors that make a host-impurity combination effi-cient. This requires the knowledge of energy levels. In many cases, the energyscheme of the electronic levels can be nicely understood by taking into accountthe proper electronic and crystal field interactions in standard crystal-field the-ory [161], that uses some “effective” parameters extracted from experimentaldata. However,ab initio methods are interesting tools as they ideally aim atstudying and understanding physical phenomena, at the atomic scale. They areonly based on first principles and fundamental constants, and can therefore,beyond the reproduction of experimental data, track down the microscopic in-teractions that are responsible for the macroscopic observations. They can alsobe used to make predictions. Thus, from a spectroscopic point of view, the keyproperties to be calculated are structural parameters, energy levels, and transi-tion probabilities.

57

From theab initio point of view, parts of the problems encountered in thecalculation of spectroscopic properties of ionic crystals doped with heavy ele-ments are identical to the ones in gas-phase spectroscopy; the dense manifoldsof states can only be accurately reproduced by methods that account for bothelectron correlation and spin-orbit contributions, as well as the interplay be-tween the two. The examples presented in the next paragraph will discuss thispoint in detail. The second methodological aspect arises from the treatment ofthe interactions between the impurity and the host. They are two classes of treat-ment [162]. The first type of methods describe highly symmetric crystals as aninfinite periodic system and exploit the point and translational symmetry withinthe framework of monoconfigurational levels of theory (HF and DFT). Rela-tivistic effects can be introduced by pseudopotentials. Richardet al [163,164],Traverseet al [165], and Crocombetteet al [166] recently demonstrated thatthis kind of approach can be successfully used to study ground-state electronicproperties of lanthanide and actinide crystals. In the present state, this typeof approaches cannot treat excited states and further methodological develop-ments are needed. The second class of methods leans on the common statementthat spectroscopic properties of doped crystals are essentially local and mainlygoverned by the geometry of a limited cluster centred on the impurity and in-volving its closest coordination shell. Theoretical schemes in which the clusteris treated at the highest possible level of theory, while the effect of the rest ofthe system (environment) is handled at a lower level, are advisable, with specialcare taken to represent both the cluster and the environment. Within this kind ofapproach, one has to work with a relativistic Hamiltonian of a relevant cluster,supplemented by a one-electron term simulating the effect of the crystalline en-vironment. The choice of the embedding scheme is particularly critical for thequality of the results. The most intuitive and most extensively used model rep-resents the lattice by bare point charges. However, it only accounts for the clas-sical long-range Coulomb interactions, ignoring important short-range quantuminteractions. Within the past 15 years, pseudopotentials [167,168] andab initiomodel potentials (AIMP) [169] that take into account the latter effects have beendeveloped and successfully applied to a wide range of problems, as shown inchapter 7 of the present book. These potentials can be applied to the completelattice or only to an intermediate layer, the outer shells being then representedby point charges until the Madelung potential in the region of the central atomconverges.

The following sections review a selection of characteristic recent applicationsof ab initio methods to the spectroscopy of ionic impurities. Our main goal is

58

to illustrate the strengths and limitations of existing approaches to treat electroncorrelation and spin-orbit effects, and we will also emphasise the importance ofcrystal effects.

3.2.2. Spectroscopy of main element impuritiesIn this section, we discuss a very recent application of the AIMP embedding

technique described in the previous paragraph and the uncontracted spin-orbitCI method EPCISO [11] to the investigation of photoluminescent crystal Y2O3doped with bismuth Bi3+ ions [170]. In this crystal, the ionic impurity can beeither in a C2 site or S6 site. The key question is to determine which site, if notboth, is responsible for the intense band appearing around 3.3 eV, or 3.7 eV andof the weaker one at 4.6 eV [171]. From the theoretical point of view, the inter-est of this system is twofold. First of all, bismuth is among the heaviest main el-ements (Z = 83), and the first excited state (configuration 6s6p) of the free Bi3+

exhibits an extremely large spin-orbit splitting of about 3.15 eV. This is an ex-cellent test case for the EPCISO method. The second aspect is to determine howthe ligand-field and the crystal field compete with spin-orbit coupling. In orderto discriminate the relative role played by spin-orbit, crystal field and molecularbonding interactions, Schampset alsubsequently studied the free Bi3+ ion, theembedded single Bi3+ ion in a S6 site of the Y2O3, and an embedded molecu-lar cluster containing the six closest oxygen neighbours around the Bi3+. Allelectrons belonging to the inner atomic shells of the bismuth atom are replacedby large-core ECPs, with correction for core-polarisation effects by means of acore-polarisation potential [62]. As explained in Ref. [170,172], electron corre-lation and spin-orbit coupling were treated within the uncontracted spin-orbit CIapproach EPCISO, detailed in section2.2.5, using the CIPSI [173] and EPCISOcodes.

The free Bi3+ ion has a1S0 (6s2 configuration) ground state and the first ex-cited states3P0, 3P1, 3P2 and1P0 correspond to the 6s6p configuration, withhigh transition energies, 8.80 eV, 9.41 eV, 11.95 eV and 14.20 eV, respectively.These are slightly overestimated, by 0.6 eV, by the theoretical calculations.More importantly, the total splitting of the triplet state, 3.15 eV, is reproducedwithin 7% of the spectroscopic value. If the Bi3+ is embedded in a S6 site, thecrystal field alone (without spin-orbit coupling) lifts the degeneracy betweenthe 6pz orbital and the 6px,y, and splits the triplet and singlet P states into Auand Eu components, separated by 1.51 eV and 1.37 eV, respectively. The addi-tion of the spin-orbit interaction essentially mixes the crystal-field symmetriesAu and Eu, but preserves the spin-quantum number since the singlet and tripletmanifolds are separated by more than 3 eV in the free ion. The splitting arising

59

from both crystal-field and spin-orbit does not lower the first transitions to theenergy window observed experimentally (∼ 4 eV).

The inclusion of the explicit first coordination sphere of six oxygen atomsdrastically changes the picture. The somewhat covalent Bi–O bond stronglystabilises, by more than 6 eV, the Au (6s6pz configuration) states with respectto the ground state 6s2, and shifts the Eu components up. As thepx/py arenot degenerate anymore with thepz, the spin-orbit splitting of the3Au state isalmost completely quenched to less than 0.1 eV, as compared to the 3.15 eVobserved in the free Bi3+ ion. These calculations assign the3Au state and1Aucalculated at 4.3 eV and 4.6 eV respectively to the first intense transitions ob-served experimentally. The higher calculated vertical transitions, above 6.9 eV,are associated with charge-transfer excitations from the oxygen ligands to thebismuth ion. One should point out at this stage the importance of including theligand orbitals into the active set of orbitals to describe charge-transfer states.Even though the case of bismuth might be exceptional, there are other systems,such as actinide impurities discussed in the following subsection, in which thecovalency of the metal-ligand bond plays a crucial role. Thus, restricting theactive space to the orbitals of the ionic impurity alone, though less computa-tionally demanding, may lead to inaccurate, if not wrong, assignments of ex-perimental spectra. Beyond the role of charge-transfer states, this study revealsthe crucial importance of the molecular bonds between the impurity and itsligands. This ligand-field is responsible for the relative stabilisation of somecrystal-field components, and quenches the spin-orbit coupling. Thus, electroncorrelation and the interactions with the crystalline environment are the mostimportant factors that influence the quality of the calculated spectra. Spin-orbitcoupling comes into play as a perturbation to the two other interactions. Thepresent state-of-the-art methods, that combine AIMP embedding techniques forthe environment and the EPCISO spin-orbit CI, provide reliable tools to dealwith the spectroscopy of impurities.

3.2.3. Spectroscopy of lanthanide and actinide impuritiesThe dense manifold of excited states of the partially filledn fN andn fN−1(n+

1)d configurations in lanthanide and actinide ions is responsible for the inter-esting spectroscopic and optical properties. In lanthanides, the electrons in the4 f shell are shielded from the surroundings by the filled 5s and 5p shells, andtherefore do not play a great role in the chemical bonding between the lan-thanide ion and the surrounding ligands. As a consequence, the influence of theligands on the optical transitions within the 4f shell is small, resulting in sharp-line spectra resembling those of the free ions in the range 0-42000 cm−1 [174].

60

In various hosts, the spectra can be understood by simple crystal-field mod-els [175]. Most of the studies only consider the 4f N levels up to 40000 cm−1.Reports on 4f N levels above 50000 cm−1, which is called the vacuum ultravio-let spectral region (VUV;λ < 200 nm), are very scarce, and yet for the majorityof the trivalent lanthanides (from Nd3+ (4 f 3) to Tm3+ (4 f 12)) the 4f N config-uration is predicted to extend into the VUV region, up to 150,000 cm−1 for theions in the middle of the series [176]. In this range of energy, numerous transi-tions to the 4f N−15d configurations come also into play. In lanthanide crystals,the 5d → 4 f broad emission bands are involved in various applications suchas phosphors, scintillators, visible-UV solid state laser materials, or infrared tovisible light up-conversion materials [177]. The 5f N−16d configurations in ac-tinides lie lower in energy than in lanthanides and therefore hamper a systematicassignment of the spectra. Only recently [178], these crystal transitions couldbe modelled using an extension of the standard crystal field theory. However,for such complicated spectra, the number of parameters that define the crystal-field Hamiltonian becomes easily large, and it might often become impossibleneither to fit them to experimental data nor to assign reasonable values. Thisstrongly limits the accuracy of such an approach and its ability to predict newresults. Any refined analysis of the experimental data needs tools that are moreprecise. In this context, the contribution ofab initio calculations is of great help.

In a recent series of articles [179,180,181], Seijo and Barandiaran have inves-tigated the spectroscopy of several actinide impurities (Pa4+, U4+ and U3+) incrystal environments. In particular, they discuss the relative position of the 5f N

and 5f N−16d1 manifolds. All calculations use relativistic large-core AIMPs onthe actinide centres and on the chlorine ligands. The transferability of thesefrozen core potentials from the neutralf elements to their cation has been dis-cussed in Ref. [182]. The crystal environment is described by the AIMP embed-ding cluster method. Electron and spin-orbit interactions are treated simultane-ously by the three-step spin-free-state-shifted method detailed in section2.2.5,using either MRCI or CASSCF/MS-CASPT2 methods in the spin-free step. Theactive space includes the 5f and 6d orbitals of the actinide centre, as well as the7s orbitals in order to avoid the problem of intruder-states in the MS-CASPT2treatment. Since the Wood-Boring spin-orbit coupling operators systematicallyoverestimate the atomic coupling constants by about 10% [183], an empiricalcorrection factor of 0.9 is applied in all calculations. The authors always reportthe calculated spectra of both the free ion and embedded cluster (ion with its firstcoordination shell). The assignment of the spin-orbit wave functions in terms of

61

the spin-free eigenstates helps to analyse the results. Let us discuss the simplestion, protactinium, which has a 5f 1 ground-state configuration. The analysis ofits spectrum allows to extract some general features that are also present in thespectra of uranium cations, though more difficult to distinguish because of thediffuseness of the manifolds. The total splitting of the2F ground state (5f 1) and2D (6d1) excited state is large, ca. 7000 cm−1. However, they cannot mix as thecentres of mass are separated by almost 50000 cm−1. Interestingly, the split-ting induced by the crystal and ligand fields is significantly more pronounced(ca. 30000 cm−1) on the 6d1 manifold than on the 5f 1 one (ca. 3000 cm−1).A calculation of the embedded protactinium ion, without its coordination shellwould have made it possible to discriminate between the relative roles of thecrystal and ligand fields. As in the case of the bismuth ion, the crystalline en-vironment lifts atomic degeneracies and quenches the spin-orbit coupling. Asan example, the2T2g crystal-field component of the2D excited state is split byabout 3000 cm−1, half of the atomic splitting (7000 cm−1). This is especiallytrue for the 6d manifold. The 5f orbitals being shielded by the 6s and 6p or-bitals, the crystal field splitting in the 5f manifold is small, and spin-orbit cou-pling dominates. In all three ions, Pa4+, U4+ and U3+, the 5f n−16d1 states havea shorter An-Cl bond distance as significant ligand-to-metal charge-transfer ex-citations contribute. Electron correlation yields a significant lowering of the6d manifold (about 10000 cm−1 in Pa4+).

Another very instructive example amongab initio studies is taken from Mat-sika and Pitzer’s work on the electronic energy levels of the uranyl ion UO2+

2and the neptunyl ion NpO2+

2 in the crystalline environment of Cs2UO2Cl4 usingthe DGCI method [184]. Relativistic large-core ECPs were used on all atoms,leaving 14 and 15 electrons in the valence shell for uranium and neptunium,respectively. Matsika and Pitzer incorporated the interactions with the ionic lat-tice within the embedding technique described above, in which the intermediatelayer (next nearest neighbours to the impurity) were described by all-electronpotentials, and outer spheres by point charges. The spectroscopy of the freeions has also been reported by the same group [86, 155] and was discussed insection3.1.3.

In both uranyl and neptunyl, the crystal environment induces a stretch of thedistance to the axial oxygen atoms (O-yl) from the actinyl unit AnO2+

2 , alongwith a weakening of the bond. The lowest excited state arise from excitationfrom the highest bondingσu orbital to the quasi-atomic 1φu and 1δu orbitalsof f character. The bond being weakened, all excited states exhibit a larger

62

equilibrium distance and a flatter potential curve. The first effect of the crys-tal field is to split the excited state levels of the uranyl ion. This is accuratelyestimated by the embedding model. The second effect is to shift the electronictransition. This can only be analysed by removing any contributions from elec-tron correlation and spin-orbit effects and comparing SCF transition energies.The φu orbital is more destabilised than theδu orbital. This, combined to thedestabilisation of theσu orbital leads to a general red-shift of the lowest partof the spectra in uranyl. In neptunyl, the same phenomenon yields a greaterstabilisation of the ground state2∆u (δ 1

u configuration) with respect to the firstexcited state2Φu (φ1

u ) state. The higher excited states are of charge-transfercharacter as they involve excitations from the bonding to the anti-bonding or-bitals. The weakening of the Np–O bond tends to lower the energy gap betweenbonding and non-bonding orbitals and therefore one observes a red shift of allcharge-transfer states, up to 6000 cm−1.

The spin-orbit splitting can be large (ca. 2000 cm−1 in the case of uranyl,cf. Table9) and substantially mixes severalΛ–S states. The pureΛ–S couplingscheme breaks down and is replaced by an intermediate coupling scheme. Dueto the presence of the O-yl on axis of the molecule, the ligand-field is confinedto the equatorial plan and does not split the 1φu and 1δu orbitals appreciably.Unlike in bismuth ion (cf. section3.2.2), the spin-orbit coupling is not par-ticularly quenched and has to be treated simultaneously along with equatorialligand interactions [185].

The comparison of the calculated spectra of the free ions and the ones in thecrystal is not straightforward. Indeed, in the crystal, the presence of the firstcoordination shell increases the number of electrons and basis functions in thecalculations, resulting in a blow-up of the CI expansion, mainly due to the gen-erated doubly-excited configurations. One should bare in mind that this increaseis about six time as fast in double group symmetries as in the non-relativisticsymmetry. In a non effective Hamiltonian method, the only way to keep the sizeof the DGCI matrix to an affordable size of few million configurations, is to cutdown the number of correlated electrons. This may essentially deteriorate thequality of electron correlation as the contributions of the spin-orbit interactionarise from the less numerous singly-excited configurations.

The results reported in Table9 illustrate the effect of electron correlation.The comparison of the second and third columns for the free uranyl ion showsthat the correlation of the low-lying orbitals (1σg, 2σg, 1σu, 1πg) has almostno effects on the transition energies; the differences do not exceed 1000 cm−1.Therefore, the reduction of the number of correlated electrons for the calcula-

63

Table 9Calculated adiabatic excitation energies∆E and experimental values in cm−1 taken fromRefs. [185,155] for the free uranyl and neptunyl ions and impurities in Cs2UO2Cl4 crystal.

Uranyl ion Neptunyl ionState ∆Efree

1 ∆Efree2 ∆Eimpur. Exp. State ∆Efree

3 ∆Efree4 ∆Eimpur. Exp.

0+g 0 0 0 0 1E1/2u 0 0 0 0

1g 20719 20366 20363 200962E1/2u 447 573 1663 10002g 21421 20930 21425 208613E1/2u 5515 5092 5775 68803g 22628 22105 22819 220514E1/2u 6565 6221 8463 79902g 23902 23154 24699 225785E1/2u

5 1222 17992 18236 132653g 26118 25448 26817 262226E1/2u 25844 24012 18367 172414g 27983 27196 29157 277387E1/2u

5 15668 21156 20150 154063g 31710 30573 32001 - 8E1/2u 28909 26983 20575 200812g - 34705 33834 29412 9E1/2u

5 15418 20593 20839 1568310E1/2u

5 16664 22145 21115 1678011E1/2u

5 18676 26559 23912 -12E1/2u

5 21580 26948 26862 193751 24 correlated electrons2 14 correlated electrons3 15 correlated electrons4 7 correlated electrons5 Charge-transfer state

tion in the crystal should not bias the results. The overall agreement is indeedsatisfactory. Neptunyl ion turns out to be different. It has one more electronthan uranyl and requires more reference configurations. Beyond the low-lyingorbitals mentioned above, the 2σu, 1πu and 3σg orbitals had to be frozen, leav-ing only highest occupied 2πu and 3σu orbitals correlated. The latter ones areof course essential to describe the charge-transfer states. The comparison of theseventh and eighth column of Table9 demonstrate that the additionally frozenorbitals play a crucial role in the correlation of the highest occupied orbitals;when frozen, allf → f transitions are moderately red-shifted, while the charge-transfer states are strongly blue-shifted, up to 5000 cm−1. This consequentlyalters the quality of the calculated transition energies in the crystal and explainsthe deviation from experimental data.

These two examples show that the relative position of the various excitedstates in lanthanide and actinide impurities is the result of a complex combina-tion of electron correlation, crystal field interactions and spin-orbit coupling thatneed to be treated accurately and simultaneously. Crystalline interactions can be

64

accurately accounted for by embedding methods and spin-orbit interactions arenicely treated at lower cost, as the main contributions arise from singly-excitedconfigurations. Our discussion emphasises that the treatment of electron cor-relation is the real bottleneck. This point is actually not specific to embeddedimpurities but more generally true for all systems in which electron correlationplays an important role. The treatment of electron correlation depends on twofactors: the number of relevant reference configurations and the number of cor-related orbitals. The expansion of the CI vectors in double group symmetriesfaces obvious limitations when these two parameters increase. This leads us tothe conclusion that working in double group symmetries from the beginning isnot advisable in such difficult cases. Methods that treat electron correlation ina separate step, using non-relativistic CI methods, and couple it to spin-orbitinteraction at a later stage in a reduced DGCI scheme (cf. section2.2.5) seemto be more adapted to the treatment of heavier elements. Finally we shouldstress that such complexab initio calculations can only be affordable within avalence-only scheme, that is with the use of reliable pseudopotentials.

3.2.4. Zero-electron pseudopotentialsIn this section we report some theoretical work on the molecular spectroscopy

in rare gas matrices. Due to the difficulty in modelling the environment, theo-retical studies for a long time discussed the results obtained from gas phasecalculations in direct comparison with experimental data in matrices. However,it has been demonstrated that the effect of the matrix is not negligible, and maystrongly depend on the rare gas, as discussed recently by Liet al [186]. Evenif relativistic pseudopotentials lead to a substantial reduction of computationalcosts, one still needs to treat explicitly a certain number of valence electrons,for instance eight electrons for each rare gas (Rg) atom. A further simplifica-tion is to consider a rare gas atom as a zero-electron system with all its electronsin the core represented by the so-calledzero-electron pseudopotentialor e-Rgpseudopotential (for further details see Groß and Spiegelmann [187, 188] andreferences therein). As the Rg anion does not exist, the pseudopotential pa-rameters are fitted to differential phase shifts of electron-Rg elastic scatteringdata for angular momenta up tol = 2. It is supplemented by a core-polarisationpotential (CPP), which allows to go beyond the frozen core approximation andaccounts for the statistical and dynamical correlation of the frozen inner core.Zero-electron pseudopotentials are an economic and often reliable approxima-tion to model matrices by large albeit finite clusters.

Groß and Spiegelmann applied this technique to the spectroscopy of the

65

NO molecule trapped in an argon matrix [188]. They simplified the prob-lem, describing the NO molecule as a system of one electron interacting with aNO+ core. The molecular NO orbitals are obtained self-consistently from theFock operator corresponding to the NO+ ion in the argon matrix

f =−12

∆i +N,O

∑A

(− 1RAi

+WA(i))+ ∑j∈NO+

(2Jj(i)− K j(i))+Ar

∑B

WB(i) , (39)

whereWA is a standard atomic pseudopotential for nitrogen and oxygen andJ and K the Coulomb and exchange operators which run over the occupiedorbitals of NO+. These orbitals already provide a good approximation for theisolated NO orbitals and can be corrected by adding a CPP centred at the centreof mass of NO and at the argon site. One then considers NO+ as a frozen coreand diagonalises the one-electron operator, sum of the Fock operator and thecore-polarisation potential within the basis of the virtual orbitals. The resultingeigenvaluesε are the one-electron levels including core polarisation. The totalenergies of the various molecular excited states of NO are then obtained as asum of the energy of the NO+ ground state, calculated at any level of theory,and these one-electron values:

Em(R) = ENO+(R)+ εm(R)+VAr−Ar +VNO+−Ar . (40)

Table 10Adiabatic transition energies (in eV) of the NO molecule isolated and trapped in argon matrix,without (Te) or with (Tpol

e ) CPP correction, taken from Ref. [188].Isolated NO NO in argon matrix

State Te Tpole Texp

e Te Tpole Texp

e3sσ 4.91 5.31 5.45 6.87 6.11 6.303pπ 5.85 6.41 6.46 - 7.41 7.293pσ 5.93 6.56 6.58 - 7.57 7.434sσ 6.97 7.58 7.52 - - -4pπ 7.30 7.99 7.99 - - -

The calculated adiabatic transition energies for the trapped molecules re-ported in Table10are in excellent agreement with experiment, especially whenthe CPP is added. It should be noticed that the interaction with the matrix leadsto an increase in the transition energies. However, the matrix relaxation effectswere not taken into account in this calculation.

66

This example demonstrates that such a spectroscopic problem is reducibleto a quasi one-electron effective Hamiltonian. Whenever spin-orbit coupling isimportant, one can take advantage of the one-electron character to easily calcu-late the fine-structure of molecular excited states, by just adding the spin-orbitcontribution to the Fock operator. An example can be found in the recent workby Spiegelmannet al on the excited states of CaAr and its cation [189]. Thezero-electron pseudopotential approachis a promising technique to calculatethe effect of the Rg-matrix on the excited states of an embedded centre in a sim-ple and efficient way, and can be applied to molecules trapped in Rg-matrices. Itcan be extended to treat other closed shell situations such as those met in MnXhalogen metal clusters [190], where negative halogen ions can also be treatedas cores bearing a negative charge [191]. This formalism can also be extendedto resonant situations such as those met in Ar∗

n and Xe∗n clusters [192] whereseveral of the rare gas atoms may be excited.

4. CONCLUDING REMARKS

Successful developments on relativistic valence-only Hamiltonians makes itmore and more feasible to obtain accurate molecular spectroscopic data. Thesedevelopments aim at accurately describing both relativistic and correlation ef-fects. However, as most of the difficulties in describing relativistic effects arenow overcome, highly correlated treatments remain the most difficult and chal-lenging task, and this remains the main problem which guides the choice ofphysically well-founded and relevant approximations. To reach maximal accu-racy as well as efficiency, the above applications show three major approxima-tions to the full relativistic Hamiltonian based on the separate treatment of bothactive/inactive electrons and physical effects, namely:

(a) separationof the inactive core electrons and the valence ones using pseu-dopotential techniques,but correctingthis approximation with a core-valence polarisation and correlation potential,

(b) separationof the fully relativistic operator into a scalar and a spin-orbitoperator,but correctingthe interplay between scalar and spin-orbit effectsusing an appropriate spin-orbit-CI method able to efficiently take intoaccount the relaxation effects of the valence spinors,

(c) separationof the extended correlation treatment in non relativistic sym-metries from the partially-correlated spin-orbit coupling in the spin-orbit-CI algorithm, but correctingthe lack of correlation in the second step

67

using effective Hamiltonian techniques.

A last remark concerns the accuracy reached on valence molecular calcula-tions by all-electron and pseudopotential calculations. If all the above recipesare carefully carried on, the pseudopotential calculations are not only easier,but can also be in better agreement with experiment than the correspondingfull-relativistic all-electron schemes, depending on the electronic complexity ofthe molecular system, due to the easier treatment of electronic correlation invalence-only schemes. The hardware and software progresses profit both theall-electron and valence-only calculations, but the latter allow the use of a moreextended valence basis set and correlation treatment, or can be applied to largersystems. The spectroscopic data obtained by pseudopotentials and correspond-ing spin-orbit-CI methods are becoming more and more reliable and accurate.It opens the way to applications on more complex systems, such as moleculescontaining heavy atoms embedded in a crystal lattice or in a solvent, or mag-netic impurities in a molecular or crystal environment. In this context, we see asa promising route the development of mixed methods treating accurately a sub-part of the molecules withab initio methods and the environment as an effectivefield calculated at the DFT level.

68

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