University of Groningen Local excitations and magnetism in ...electron model. In the periodic...

University of Groningen

Local excitations and magnetism in late transition metal oxidesde Graaf, Cornelis

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Publication date:1998

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General introduction and

theoretical framework

Chapter 1

12 Chapter 1: General introduction and theoretical framework

1.1 INTRODUCTION AND OUTLINE OF THE DISSERTATION

A very fundamental and successful concept in both physics and chemistry isthe ordering of the chemical elements in the periodic table; virtually any textbookon elementary physics or chemistry contains a periodic table. In the 1860s theconcept was developed independently by Mendeleyev and Meyer. The knownelements were ordered by increasing atomic weight in such a way that elementswith similar properties were placed in the same column. The power of thisconcept is illustrated by the prediction by Mendeleyev of the properties ofscandium (Sc; Z=21) and germanium (Ge; Z=32); at that time these elements hadnever been observed in an isolated form, but Mendeleyev was able to predictcorrectly many of their properties by simply looking at the properties of the otherelements in the corresponding column. The idea of ordering the elements hasnot changed very much since then, but the number of elements in the table hasincreased considerably, including even some artificially made elements. Onegroup of elements in the periodic table comprises the so-called transition metals.These elements share the feature of a progressive filling of shells of d-orbitals.Three such series of transition metals can be distinguished, depending on the d-shell being filled. In this dissertation attention is focused on the electronicproperties of some solid-state compounds containing elements from the firstseries of transition metals, which runs from scandium (Sc; Z=21) to zinc (Zn;Z=30). In addition, the compounds studied in this dissertation contain one ormore other elements from the periodic table, e.g. a halogen (F, Cl, ....) or achalcogen (O, S, ...). This kind of compounds —here referred to as transitionmetal materials— shows a wide range of properties depending on thecomposition. Some transition metal (TM) materials show a very intense colour,due to a strong absorption of visible light of certain wavelengths, some behave asstrong magnets. Furthermore, the TM materials show extremes in theconduction of electric current, from true insulators to metals. Some of the TMmaterials even possess the very intriguing property of superconductivity below acertain temperature, i.e. their resistance drops to zero.

This dissertation deals with the description of the electronic structure of TMmaterials, because many of the properties characteristic to TM materials aremainly connected to the behaviour of the electrons. TM materials often can beconsidered as ionic compounds. This means that in a first approximation theycan be thought of as constructed from ions with an integral number of electrons.

1.1 Introduction and outline of the dissertation 13

Notwithstanding its simplicity, the ionic model provides a basis for many usefulpredictions about the electronic structure of TM materials.

In this first chapter a framework is given for the theoretical investigation ofthe electronic structure. The local cluster approach is introduced as animprovement on the simple ionic model. Furthermore, the concept of electroncorrelation is described and it is briefly sketched how electron correlation can beaccounted for in TM materials. In Chapter 2 all concepts and methods introducedin Chapter 1 will be investigated and tested for the model system nickel oxide(NiO), as this material has been studied very extensively both experimentally andtheoretically for a long time. Therefore, many properties of the electronicstructure are well known, although some controversial points still remain (for areview of the properties of TM oxides see [1]). The findings for bulk NiO will becompared to those for the NiO (100) surface, about which much less is known,although interest has recently been growing. Chapter 3 is devoted to the study ofTM materials other than NiO, namely CoO, La2CuO4 and La2NiO4. Questions areaddressed concerning the interpretation and assignment of so-called d-dtransitions. In Chapter 4 magnetic properties are investigated. Again bulk NiOserves as a model system for testing the quantum chemical approaches outlinedin Chapter 4. An estimate for the size of the magnetic coupling is given and themechanisms determining the magnetic interaction are discussed. A prediction ispresented for the magnetic coupling in a (100) surface layer of NiO, for which noexperimental data exist yet. Finally, in the last chapter we investigate the Ni-3sphotoelectron spectra of NiO.

1.2 MOTIVATION FOR A LOCAL APPROACH

A crystal is a macroscopic object built from a very large number of atoms.Neglecting impurities, defects and other irregularities, a crystal can be thought ofas constructed from a unique building block, which is repeated many times in allthree directions. This building block is known as the unit cell. Properties ofinterest in solid-state physics and chemistry are not only ground state properties,such as equilibrium geometries and cohesive energies, but also the electronicspectra arising from excited or ionized states. Because the immense number ofatoms in a crystal makes an exact theoretical or computational treatment of theelectronic structure of the crystal as a whole impossible, it is inevitable to


introduce simplifications. To a very good approximation all symmetry equivalentatoms in the bulk of the crystal experience the same influence of theirsurroundings. Therefore a very natural simplification is the introduction oftranslational symmetry by imposing periodic boundary conditions, thus leadingto the band theory of electrons moving in a periodic potential. A differentapproach is to model the crystal by a small number of atoms at positions as theyoccur in the real crystal and to embed this collection of atoms in a potential thataccounts for the part of the crystal not explicitly treated. This way of representingthe crystal is generally known as the embedded cluster approach and will beapplied in this dissertation to study local excitations and magnetic interactions inTM materials.

The most straightforward way of treating the electronic structure is the one-electron model. In the periodic approach this leads to band theory (see standardtextbooks on solid-state physics or chemistry, e.g. [2-4]) and within the localapproach to the molecular orbital method. For the electronic ground state of ionicmaterials built from closed-shell ions, e.g. NaF or CuCl, it is known that the twoapproaches lead to identical results in the one-electron model. The wavefunctions obtained from a sufficiently large (embedded) cluster calculation areequivalent to so-called Wannier functions, i.e. crystal wave functions that arepeaked at the lattice positions. These Wannier functions can be transformed intodelocalized Bloch functions by a unitary transformation, hence connecting theembedded cluster approach and the band theory [5]. Therefore, ground stateproperties can often be described satisfactorily with approaches based on bandtheory, computationally implemented either on the basis of density functionaltheory (DFT) [6, 7] or the Hartree-Fock scheme [8, 9]. The situation changes foropen-shell systems, both for the electronic ground state, and for excited states.

The dissimilarity between band theory and the cluster approach can beillustrated for the ionisation from a Cu-3d level in CuCl [5]. As argued above forthe un-ionized system the band approach and the cluster approach are equivalentin the limit of large clusters. In the delocalized band description an electron isremoved from a Bloch state at the top of the filled band with mainly Cu-3dcharacter. Because this state is delocalized over all copper sites, the change in theelectronic density at each site is infinitely small and no relaxation effects occur.On the other hand, in the local description an electron is removed from any ofthe Cu-3d Wannier states, which are strongly peaked around the Cu sites. Hence,locally a large change in the density occurs and the electronic wave functionresponds to this. The relaxation is rather similar to the relaxation effect when anisolated Cu+(3d10) ion is ionized, i.e. a decrease in energy of about 6 eV. In

1.2 Motivation for a local approach 15

general, when local relaxation effects are large compared to delocalizationinteractions, which lead to the band width: a wide band indicating largedelocalization effects and a narrow band highly localized electrons, the lowestenergy is obtained by first accounting for the local relaxation and after thatrestoring the translational symmetry. This is particularly the case in the study ofproperties connected with the 3d states of the TM atoms in TM materials, hencethe local approach is the more natural basis for describing these properties.

Another important advantage of the cluster model approach is the ability toextend the description of the electronic structure beyond the one-electron modelin a systematic and controlled manner. An ab initio many-body description of theelectronic wave function can be obtained with the use of standard quantumchemical techniques (see section 1.4), which are also applied in molecularquantum mechanics for correlated wave functions. There also exists an ab initio

implementation to treat electron correlation in band calculations on the basis ofthe Hartree-Fock scheme [10, 11], however, this approach is limited to groundstate properties only. In the density functional approach many-body effects areincorporated without losing the appealing picture of the one-electron model.Several functionals exist nowadays that improve the simple local densityapproximation considerably, but a systematically converging scheme does notexist and the functionals tend to become rather complex. Furthermore, in generalDFT is limited to describe ground state properties. Only recently has it becomepossible to calculate excitation energies and oscillator strengths through timedependent DFT [12].

In semi-empirical cluster approaches on the basis of model Hamiltonians asproposed by Anderson [13] and Hubbard [14] the competitive delocalization andelectron interaction effects are incorporated in parameters with values obtainedby fitting to experiment. This leads to a simple physical picture that provides apowerful approach to rationalize the electronic excitations occurring in, forexample, TM materials. Nevertheless, problems can arise when parameters areneeded for new materials or situations for which experimental data are lacking.An example of this will be given in Chapter 4, in which the contradicting ab

initio and semi-empirical predictions for the magnetic interaction in NiO (100)are discussed.


1.3 THE EMBEDDED CLUSTER MODEL APPROACH

The static interaction of the cluster with the surroundings can be dividedinto a long-range electrostatic part and a short-range repulsion. The latter isprimarily due to the Born or Pauli repulsion supplemented by Coulomb andexchange effects brought about by the charge distributions of the neighbouringcentres. The first interaction is easily incorporated into the material model bycalculating the Madelung field of the ions not included in the cluster and addingthe potential to the cluster Hamiltonian. To this purpose first the exact Madelungfield is calculated by an Ewald summation [15] with formal point charges at theexperimentally determined lattice positions. From this total potential thecontribution of the cluster ions is subtracted. The choice of using formal chargesis consistent with the assignment of an integer number of electrons to the cluster.For additional information on this subject the reader is referred to [16].

To include this external potential essentially three methods can be applied.The first one is not very appropriate and consists of just putting formal charges atall lattice positions and summing their contributions to the potential. It is wellknown that the convergence of the Madelung potential with distance is veryslow. Hence, a very extensive array of point charges is needed to obtain sufficientaccuracy. Better is the Evjen method [17], which ensures a much fasterconvergence with the number of point charges by requiring overall chargeneutrality. This is achieved by putting formal charges at all lattice positions exceptfor the charges in the outermost shell: half of the formal charge is used for thefaces; a quarter for the edges; and one eighth for the vertices. The third method isfitting a small set of point charges at lattice positions within a certain radius of thecluster to the exact value of the potential at a large number of grid points in thecluster region.

Because of its quantum mechanical nature, the short-range electrostaticrepulsion cannot be accounted for by simple schemes based on classicalmechanics. There are different approaches to account for these interactions [18-35], two of these methods will be discussed here. The first approach is a methodbased on the Adams-Gilbert equations, which transform the extended Hartree-Fock orbitals for the complete crystal into localized orbitals modified toincorporate the effect of the environment of this local region. The second methodis discussed somewhat more extensively, since it is applied in this dissertation. Itis based on the theory of electron separability and differs from the Adams-Gilbert

1.3 The embedded cluster approach 17

approach in the requirement that the orbitals belonging to different subsystemsare orthogonal.

Adams-Gilbert approach

One method to include the short-range interactions of a quantum mechanicallytreated region with its surroundings is based on the building block approachformulated by Adams [36-38] and, in more general terms, by Gilbert [39-41]. In thisapproach the Hartree-Fock equations for a large system, for example a completecrystal, are transformed to a smaller set of self-consistent equations by localizingthe orbitals into different subsystems. The Adams-Gilbert equations are defined asfollows:

( F - ρLρ ) ϕ iocc = βi ϕ i

occ (1.1)

where ϕ iocc are the modified HF orbitals, related to the standard HF orbitals,

ψiocc , by a linear transformation and ρ is the one electron density operator:

ρ = ψiocc ψi

occ ∑i=1

n

(1.2)

Choosing an appropriate form of the operator L, which is added to the standardFock operator F, the delocalized HF orbitals for the complete crystal transforminto localized orbitals mainly centred in smaller subsystems. Orbitals belonging todifferent subsystems are not necessarily orthogonal to each other.

Kunz and Klein incorporated the interaction of a cluster A with its sur-roundings by choosing the following form of the Adams-Gilbert equations [18]:

FA

+ VLA

+ VSA

- ρA VSA

ρA ϕiA = βi ϕi

A (1.3)

where FA

is the cluster Fock operator, and VLA

and VSA

account for the long andshort-range interaction of A with its environment, respectively. In the limit ofself-consistency VS

A and ρA VS

A ρA cancel, which leads to

FA

+ VLA

ϕiA = βi ϕi

A (1.4)


-ρA VSA

ρA is called the Kunz-Klein localizing potential, it selects a set of occupiedcluster orbitals, ϕi

A , having incorporated the effect of the rest of the crystal as aweak perturbation.

Theory of electron separability

An alternative approach to include the short-range interaction of a clusterwith its environment is based on the theory of electron separability (TES) [42-46].This results in a method that is very appropriate to combine with the standardquantum chemical approaches developed for isolated systems. The crystal isdivided in groups {A, B, ..., R, ....} each with {NA, NB, ....., NR, ....} electrons. Eachgroup is described by an antisymmetrized wave function ΦR(x1, xR) built fromorthogonal orbitals. The total wave function for the crystal is written as a properlyantisymmetrized product of group functions:

Φ(x1, xN) = MOa{ Φ

A(x1, xN

A) ΦB(xN

A+ 1, xN

A+ N

B)....} (1.5)

where M denotes the normalization factor and Oa the antisymmetry operator.

TES assumes the condition of strong orthogonality [42] between the differentgroup functions

ΦA

ΦexB

dτex = 0 (1.6)

in which ΦexB

represents what is obtained from ΦB by exchanging one or more of

its electrons with electrons of ΦA

. The integration is over the coordinates of theexchanged electrons [42]. Under the condition of strong orthogonality the totalenergy of the system can be written as a sum of the internal energies of all groupsplus the interaction energies between the different groups

E = HR∑R

+ 12

∑R

JRS - KRS∑S≠R

(1.7)

with

HR = Φ R

h(i)∑i∈ R

ΦR

+ 12

ΦR

g(i,j)∑i≠j∈ R

ΦR

(1.8)


JRS = ΦRΦ

S ∑i∈ R

1rij

∑j∈ S

ΦRΦ

S(1.9)

and

KRS = ΦRΦ

S ∑i∈ R

P(i, j)rij

∑j∈ S

ΦRΦ

S(1.10)

P(i, j) interchanges electron i in ΦR with electron j in ΦS and rij is the distancebetween electron i and electron j. Equation (1.7) implies that the total wavefunction can be optimized by optimizing the wave function of each groupsequentially. Instead of solving one N-electron problem a set of smaller equationshave to be solved repeatedly until self-consistency is obtained. By introducing theCoulomb and exchange group operators J

S(i) and K

S(i)

JS(i) = Φ

S 1rij

ΦS∑

j∈ S

and KS(i) = Φ

S P(i, j)

rij Φ

S∑j∈ S

(1.11)

the total energy for a group R in the presence of all other groups can be written as

ER = ΦR

h(i)∑i∈ R

+ JS(i) - K

S(i)∑

S≠R

∑i∈ R

ΦR

+ 12

ΦR

g(i, j)∑i≠j∈ R

ΦR

(1.12)

and the optimization of the wave function for group R is now reduced to anessentially NR-electron problem with the influence of the electrons of the othergroups absorbed into the one-electron Hamiltonian. In other words, the staticinteraction of a quantum mechanical cluster (group R) with its surroundings canbe accounted for by adding specific one-electron matrix elements to the standardenergy expression of the total cluster energy. Notice that Φκ does not need to be asingle determinant (Hartree-Fock) wave function; electron correlation within thegroups can be included by means of different methods which will be commentedon later in this chapter.

The total wave function Ψ is further improved by accounting for the mixingof different products of group functions. In analogy with configuration inter-action, this is known as group state interaction.

Ψ = cκ∑κ

Φκ (1.13)


where κ selects for all groups a particular state. Up to first order, only matrixelements need to be considered that differ in at most two group functions.

κ and κ ' differ in group R only: κ selects MOaΦa

AΦb

B...Φr

R...

κ ' selects MOaΦa

AΦb

B...Φr'

R...

Φκ H Φκ' = H(rr') + [JRS(rr',ss) - KRS(rr',ss)]∑S≠R

(1.14)

κ and κ '' differ in group R and S: κ selects MOaΦa

AΦb

B...Φr

RΦs

S. . .

κ '' selects MOaΦa

AΦb

B...Φr'

RΦs'

S. . .

Φκ H Φκ'' = JRS(rr',ss') - KRS(rr',ss') (1.15)

The energy expression, which can be derived from variation theory with the useof these matrix elements, has an important implication in the form of the groupfunction analogue of the Brillouin theorem:

HeffR (rr') = Φr

R Heff

R Φr'

R = 0 (1.16)

HeffR is the effective Hamilton operator for group R in the field of the other groups

as used in equation (1.12). This implies that in the limit of self-consistency thecontribution of terms in (1.13) connected to single replacements in groupfunctions vanishes. Since the admixture of single excitations polarizes the groupfunction owing to changes in the charge distribution of other groups, theconclusion can be drawn that the polarization of each group by the presence ofother groups has been fully accounted for by optimizing the one-group stateapproximation of equation (1.5).

TES applied to a cluster interacting with a lattice

The above considerations are completely general and can be applied to anysystem that can be considered to consist of well-separated groups of electrons. Thetheory can be used to treat intermolecular forces by dividing the system intoelectrons belonging to molecule A and to molecule B, or more generally to


handle the effects of a solvent on a solute [47-50]. It can also be applied to derivemodel potentials for atoms to represent the core electrons, as they are wellseparated from the valence electrons [51-57]. Furthermore, the theory of electronseparability was used to derive the energy expression of a cluster interacting withthe rest of the crystal [22, 26]. The crystal is divided into groups {A, B, ..., R, ....},each with {NA, NB, ....., NR, ....} electrons and {νA, νB, ...., νR, ...} nuclei. Group R isthe cluster described with either a Hartree-Fock wave function or any type ofwave function which accounts for electron correlation. For the moment, all othergroups are represented by frozen charge clouds. In analogy to equation (1.12) thetotal energy of the cluster in the field of the other groups can be written as a sumof the internal cluster energy ER plus the interaction energy with the other groupsERS.

ER = ΦR

Ti∑i∈ R

- ∑i∈ R

Zαriα

∑α∈ R

+ 1rij

∑i>j∈ R

ΦR

+ ZαZβrαβ

∑α>β∈ R

(1.17)

ERS = ΦR

-∑i∈ R

Zβriβ

∑β∈ S

ΦR

+ ΦR

[JS(i) - K

S(i)] Φ

R∑i∈ R

+ ∑α∈ R

ZαZβrαβ

∑β∈ S

+ ΦS -∑

j∈ S

Zαrjα

∑α∈ R

ΦS

(1.18)

In these expressions T(i) is the kinetic energy operator, Z is the nuclear charge,and J

S and K

S are the Coulomb and exchange operators as defined in equation

(1.11). All indices in Greek characters refer to nuclei. After some reordering of theterms the effective total cluster energy can be written as:

EeffR = Φ

R Heff

RΦ

R + Φ

S -∑

j∈ S

Zαrjα

∑α∈ R

ΦS

+ ∑α∈ R

ZαZβrαβ

∑β∈ R

∑S≠R

(1.19)

HeffR

= T(i) - Zαriα

+ VeffS

(i) + PS(i)∑

S≠R∑α∈ R

∑i∈ R

+ 1rij

∑i>j∈ R

+ ZαZβrαβ

∑α>β∈ R

(1.20)

The first term in (1.19) shows that the interaction of the cluster electronswith the other electrons in the crystal is absorbed into an effective 1-electronHamiltonian for the cluster group only. The operator Veff

S in (1.20) is the


potential energy operator that takes into account the interaction of electron i ofgroup R with the frozen group S. This operator can be replaced by fixedcontributions to the 1-electron integrals, provided sufficient knowledge exists ofthe frozen group S, as will be discussed further in the next section. P

S is a shift

operator and is added to the expression to ensure that the wave function of groupR remains strongly orthogonal to the other group functions during theoptimization process by shifting upwards all occupied orbitals of group S.

PS(i) = ϕg

S -xgS εg

S ϕgS∑

g∈ S

occ

(1.21)

xgS is a projection factor and εg

S is the orbital energy of orbital g in group S. Thevalue of the projection factor xg

S has been subject of discussion [58-61]. It has beenshown by Francisco et al. [59] that for the case of closed-shell groups described atthe HF level a rigorous derivation of the energy expression leads to a value of 2.However, in some applications deviating values were used to correct for the lackof orthogonality between the cluster and the rest of the crystal when no functionsare present at the groups around the cluster [61].

The second term in (1.19) contains the interaction of the nuclei of the clusterwith the nuclei and electrons of the environment. It contains Coulomb andnuclear interactions only, which are constant as long as the groups are keptfrozen and the positions of the nuclei remain unchanged.

Practical implementations

The only approximations to the electronic structure of the crystal made sofar are the usual Born-Oppenheimer approximation, which separates themovement of the nuclei and the electrons, and the approximation that the crystalcan be considered to consist of groups of electrons with negligible mutualoverlap. The latter is only true in systems where no excitations from one group toanother occur. Therefore, charge transfer effects between different groups cannotbe included in the expansion of the total wave function of equation (1.13), becausethey violate the strong orthogonality condition. For this reason the method isonly applicable to rather ionic systems and cannot be used to study electronictransport properties. It is very suitable, however, to improve on the point chargeapproximation for the embedded cluster model approach to study local electronic


excitations. Furthermore, it can be used to study structural properties of ionicsystems.

Two slightly different implementations of the ideas of the theory of electronseparability for solid-state problems exist. The first method, the ab initio

embedding model potential (AIEMP) method [22] is an extension of the modelpotential method developed by Huzinaga and co-workers [51-57] to represent coreelectrons. The influence of the environment on a particular group of electrons isaccounted for by an adaptation of equation (1.19), in which the Coulomb operatorJS(i) is approximated by an analytical function and the non-local exchange

operator KS(i) by its spectral representation in the function space used for the ions

in the determination of the model potential.

VCS (i) =

AjS ri

njS exp -αj

S ri2

ri∑

j(1.22)

VExS (i) = a AS(a,b) b∑

a,b

with:

AS(a,b) = -S-1KS-1, where Spq = p q and Kpq = p KS(i) ∑

i∈ S

q (1.23)

The AIEMPs are obtained by optimizing the wave function for each type of atomof the crystal in the field of the other ions until self-consistency is obtained. In thecase of open-shell atoms, e.g. Ni2+ ions, a spherical charge distribution isassumed. Once the model potentials are obtained a cluster can be constructed thattypically consists of a central ion with the first shell of neighbours. The short-range interaction of the cluster with the crystal is then included by placing theappropriate AIEMP at the lattice positions; usually about three or four layersaround the cluster are used. After also incorporating the long-range Coulombinteraction, the cluster wave function can be optimized with any quantumchemical method available. In the applications discussed in subsequent chaptersthis AIEMP method will be applied to improve on the point chargeapproximation for embedding the cluster.

The second approach, the ab initio perturbed ion (AIPI) approach [25, 27], isaimed at structural and other global crystal properties. The same scheme isfollowed as in the construction of the AIEMPs. Each ion is optimized in the fieldof the other ions in an iterative process until self-consistency is obtained. Afterthat no cluster is constructed, but global properties of the crystal are extracted


from the results. The main difference with the AIEMP implementation is thatthe AIPI method is not a molecular orbital method and therefore it is not themost appropriate method to study localized electronic excitations for whichcharge transfer effects of neighbouring ions are expected to be of importance. Onthe other hand, the method goes beyond static embedding; because the wavefunctions of all groups are determined variationally, the polarization of eachgroup due to change in the other groups is fully accounted for. Anotheradvantage is that very large clusters (a collection of individually optimized ions)can be used. For example, in the study of the local geometry distortion due to aCu+ impurity in NaF the complete relaxation of the position and the electronicwave function was considered of the first 179 ions surrounding the impurity [62].

Long-range polarization

Although the methods described above do have the possibility to includesome polarization effects of the rest of the crystal due to changes in the electrondistribution in the cluster region [23], it is almost impossible to account for thelong-range polarization accompanying ionization processes such as occur forexample in core level spectroscopy. An immense set of group functions needs tobe optimized in order to obtain a reasonable estimate for the long-rangepolarization effects, creating a demand on computer resources which is clearlyout of proportion. To estimate the long-range polarization one needs to employmore approximate methods. One way to include the effect is to embed the clusterin a set of point polarizabilities, and solving the complete system self-consistentlyby means of the (Direct) Reaction Field method [63-65]. This approach gives verysatisfying results for the interaction of solute and solvent molecules [47-50], andwas also successfully applied to describe large biomolecules [66]. One problem inapplying this method to embed clusters representing semi-conductingcompounds is to uniquely define a value for the polarizability of the ions in thecrystal. An estimation of the long-range polarization effects was given by Janssenand Nieuwpoort in their study of the band gap in NiO [16, 67]. The polarizationenergy was calculated by using the classical Mott-Littleton method and by the DRFapproach, both yielding essentially the same result. For an NiO6 cluster thecorrection to the ionization or electron addition process on a Ni2+ site wasestimated to be 3-4 eV. The effect on a local charge transfer process was estimatedto be smaller (about 1 eV) which can be rationalized by the fact that the overall


charge in the cluster is not changed so that at larger distances from the cluster theelectrons do not experience a different potential.

A somewhat different approach to incorporate the long-range polarizationeffects is the so-called shell model [68-70]. The embedding ions are represented bya core of charge X and a shell of charge Y, with X + Y equal the formal charge. Theshell and the core are elastically coupled with an empirical force constant k. Thelong-range Coulomb interaction is fully accounted for and the short-rangerepulsion between ions and the remaining dispersion interaction are modelled bythe Buckingham potential

V(r) = B exp(-r/ρ) - C/r6 (1.24)

where r is the ion-ion distance and B, ρ and C are constants to be optimized forthe system under investigation. Because this method is often combined with themethods formulated on the basis of the Adams-Gilbert approach [36-41] thisclassical approximation of the short-range interaction is not as severe as it seemsbecause in that approach the cluster orbitals already incorporate part of the effectof the environment. Pascual and Seijo coupled this shell model to the AIEMPmethod [71] as an alternative for the very elaborate way described in Ref. [23, 24] ofaccounting for polarization effects via group state interaction.

1.4 THE N-ELECTRON WAVE FUNCTION

Once the material model is chosen, it remains to find approximate solutionsof the Schrödinger equation for the N electrons in the system underinvestigation. A common approach is to expand the N-electron wave function inantisymmetrized products Φk of one-electron functions, ϕi.

Ψ = CkΦk∑k

(1.25)

Φk = Oa

ϕp(1)ϕq(2)......ϕs(N) (1.26)

Φk are normally referred to as Slater determinants. The spinorbitals ϕ i depend onfour coordinates, the three space coordinates and one spin coordinate. Since spin-orbit interaction is neglected, the ϕi can be written as the product of an orbital (the


spatial part) and a spin function α or β. In principle the exact wave function isgiven as a complete (i.e. infinite) expansion of all Slater determinants that can begenerated from a complete one-electron space. This is (of course) not possible, andtherefore approximations have to be made in all practical applications. A fewmethods will be briefly summarized here; for a more complete description of thewide range of quantum chemical methods to approximate the N-electron wavefunction the reader is referred to for example [44, 72-74].

Hartree-Fock

The simplest approximation is to limit the N-particle space to just one Slaterdeterminant. The best wave function of this type can be found by applyingvariation theory to minimize the expectation value of the Hamiltonian. Thisleads to the Hartree-Fock (HF) or Self-Consistent Field (SCF) equations:

Fϕi = ϕjεij∑j

occ

(1.27)

F = h(1) + J(1) + K(1)

= h(1) + ϕj(2) g(1,2) ϕj(2)∑j

occ

+ ϕj(2) g(1,2)P12 ϕj(2)∑j

occ

(1.28)

The Coulomb operator J and the exchange operator K depend on the set of one-electron functions ϕ i and hence the equations need to be solved in an iterativescheme until self-consistency is obtained. In the Unrestricted Hartree-Fock (UHF)approach no additional restrictions on the one-electron functions are imposed.As a consequence in the case of open-shell systems, the wave function is not aneigenfunction of the S2 operator. This problem is remedied in the RestrictedHartree-Fock (RHF) approach, formulated by Roothaan [75, 76]. In RHF the sets oforbitals for α and β spin are restricted to be identical. For open-shell systems asingle Slater determinant is not always sufficient for obtaining eigenfunctions ofS2 and it is necessary to use linear combinations of Slater determinants.Moreover, it is customary that the wave functions are constructed so as totransform following one of the irreducible representations of the point group ofthe cluster model, thus leading to the spatial and spin Restricted Hartree-Fockapproach. The linear combinations of Slater determinants that fulfil the spin and

1.4 The N-electron wave function 27

spatial symmetry requirements are usually referred to as configuration statefunctions (CSF).

In principle, the one-particle space can consist of numerical functionsobtained by numerical integration of equation (1.27). Since in practice this canonly be applied for atoms, the one-particle space is almost always spanned byanalytical functions that are expressed as expansions in a set of basis functions[75].

ϕi = χpcpi∑p

(1.29)

This expansion of the one-electron functions into a discrete set of basis functionsturns the Hartree-Fock equations into a matrix equation, the so-called Roothaanequation [75, 76].

F C = S C E (1.30)

Here F is the Fock-matrix, C the collection of column vectors ci, S the overlapmatrix of the basis functions χ and E the eigenvalue matrix, which can bediagonalized by a unitary transformation of the orbitals to yield the orbitalenergies.

Multiconfigurational SCF

For many systems in the electronic ground state and not too far from theirequilibrium geometry the RHF method gives a rather good description of theelectronic structure and often useful information can be extracted from theresults. However, the method is known to have some drawbacks. For example itis unable to correctly describe dissociation or bond breaking processes inmolecules. Furthermore, in many cases it is not possible to approximate theelectronic wave function by a single configuration because some configurationsare (almost) degenerate and need to be included in the wave function from thebeginning. In the Multiconfigurational SCF (MCSCF) method the N-electronwave function is approximated by a linear combination of CSFs.

Ψ = C1Φ1 + C2Φ2 + .... CiΦi + .... CmΦm (1.31)


The variational parameters are both the orbital expansion parameters of equation(1.29) and the CSF expansion parameters Ci. One method to determine the bestset of parameters is based on a Taylor expansion of the energy expression.

E(p) = E(0) + ∂E

∂pi 0

pi∑i

+ 12

pi ∂E

∂pi∂pj 0

pj∑i,j

(1.32)

Here, pk are both kinds of expansion coefficients. The best parameters at theexpansion point can be found with a Newton Raphson algorithm. These newparameters define a new expansion point, leading to an iterative scheme inwhich the optimization of the orbital and CSF expansion parameters is done atthe same time. Among various other alternatives, the method can be simplifiedsomewhat by leaving out the mixed second-order terms in equation (1.32) inwhich pi refers to orbital expansion coefficients and pj to the CSF coefficients. Inthis way a two-step formalism is obtained in which the parameters are optimizedin turn. Another method, the so-called super-CI approach [77], also performs theoptimization of the parameters in a two-step formalism, but has a differentstrategy to find the optimal parameters. First the CI problem is solved, i.e. for thepresent set of orbitals the best CSF coefficients are calculated. The optimization ofthe orbitals is performed by the construction of a super-CI wave function of theform:

Φ = Ψ + Cia Ψi

a∑i,a

(1.33)

in which Ψia denotes the configurations connected to single excitations with

respect to the present MCSCF wave function. Now the super-CI problem is solvedand the resulting CI coefficients are used to express a new set of orbitals ϕ .

ϕp = cpq ϕq∑q

(1.34)

This cycle is repeated until self-consistency is obtained.Both methods have advantages and disadvantages. The Newton-Raphson

methods usually converge faster given a reasonable start; the super-CI iscomputationally more complicated but more stable and will always converge to aminimum, whereas in the Newton-Raphson methods special precautions mustbe taken to avoid convergence to saddle points or maxima. For a more completediscussion see Ref. [78, 79] and references therein.


Within the MCSCF formalism different kinds of wave functions can beconstructed. The most common one is based on the Complete Active Space SCF(CASSCF) method [77, 78]. The orbital space is divided in three subspaces: in theinactive space the orbitals remain doubly occupied in all configurations in thewave function expansion; in the secondary or virtual space the orbitals remainunoccupied; and within the active space a fixed number of electrons is distributedover the active orbitals in all spin and symmetry allowed ways. Performing acomplete CI in the active space has the advantage that the orbital rotations amongactive orbitals are redundant and do not need to be considered. A more generalMCSCF method is the Restricted Active Space SCF (RASSCF) method [80]. In thisapproach the active orbital space is divided in three subspaces, normally referredto as RAS1, RAS2 and RAS3. Restrictions are imposed on the number ofelectrons or holes in the different subspaces. In RAS1 a maximum number ofholes is allowed, in RAS3 a maximum on the number of electrons is imposedand in RAS2 a complete CI expansion is constructed. This has the advantageabove the CASSCF method that in general more active orbitals can be considered;the disadvantage is that the active - active orbital rotations between the threeactive spaces are no longer redundant and need to be included.

Configuration Interaction

In the Hartree-Fock wave function the electron-electron repulsion isaccounted for in a mean-field way. The most straightforward way to improve onthis approximation is to include more terms in the configuration expansion andto variationally optimize the configuration expansion coefficients withoutchanging the orbitals as is done in the MCSCF method. For systems where theHartree-Fock wave function is a good starting point, the inclusion of configu-rations connected to single and double excitations usually recovers a large part ofthe electron correlation effects. This method is referred to as the singles doublesCI (SDCI):

ΨSDCI = C0ΦHF + Ci

a Φia∑

i,a + Cij

ab Φijab∑

ij,ab(1.35)

The first sum in equation (1.35) depicts all configurations connected to theHartree-Fock reference state by single excitations and the second sum thoseconnected by double excitations (i and j are used as indices for orbitals which are


occupied and a and b for orbitals which are unoccupied in the HF wave function).The optimal set of coefficients is obtained from the secular equations:

Hpq - ESpq ∑q

Cq = 0 (1.36)

where p and q run over all configurations in equation (1.35). Normally, theoverlap matrix S is diagonal (the non-orthogonal CI discussed in the followingsection is an exception to this rule). Because of the Brillouin theorem, the initialHamilton matrix H contains zeros for all matrix elements between the HFconfiguration and singly excited configurations. The matrix elements between thelatter and the doubly excited configurations are non-zero, however, and thereforesingly excited configurations can have significant contributions to the SDCI wavefunction as well.

In cases where the HF wave function gives a poor description of the N-electron wave function, SDCI is not the most appropriate method to includeelectron correlation. Then an MCSCF wave function which contains all near-degenerate configurations is a much better reference wave function. Theresulting multireference CI (MRCI) wave function is most easily written byordering the external configurations by their number of external electrons:

ΨMRCI = CmΦm∑m

+ Cma Φm

a∑m,a

+ CmabΦm

ab∑m,ab

(1.37)

where m indexes all configurations in the reference wave function. This methodprovides a very potent scheme to include both non-dynamical (molecular) anddynamical (atomic) electron correlation in the N-electron wave function. A widevariety of very efficient implementations exists nowadays and very large CIexpansions can be handled. Approximations are made in some schemes to reducethe number of parameters in the calculation under the restriction that theaccuracy is not significantly affected. This can be done either by selecting (withperturbation theory) those configurations that are expected to have the largestcontribution to the wave function, or by imposing additional restrictions on theconfiguration expansion coefficients. In the internally contracted multireferenceCI (ICMRCI) scheme [81, 82] the configurations are generated by applyingexcitation operators to the reference wave function as a whole. This results in aconfiguration space that has non-orthogonal and even linear dependentcontributions which need to be removed. The evaluation of the matrix elements


is rather complicated but the advantage is that the number of parameters isgreatly reduced without any significant loss of accuracy.

Non-Orthogonal CI

Non-Orthogonal CI (NOCI) [83, 84] is a somewhat different approach todescribe the N-electron wave function. The central idea is that the number ofconfigurations in the CI expansion can be greatly reduced when eachconfiguration in the expansion is expressed in its own optimal orbital set insteadof one common orbital set. This leads to a wave function expansion consisting ofmutually non-orthogonal configurations and therefore the secular equationscannot be solved with the standard schemes as applied in the conventional CIcalculations. Based on the factorized cofactor method [85] an approach wasdeveloped [83, 84] and implemented in the computer code GNOME [86] thatenables the routine evaluation of the matrix elements between configurationsexpressed in mutually non-orthogonal orbitals.

The method is well suited to study systems for which a much lower energyis obtained with a symmetry-broken Hartree-Fock wave function, i.e. a functionthat does not transform in accordance with the full symmetry group of theHamiltonian of the system. This symmetry breaking occurs, for example, in thedescription of core level ionization processes in molecules with N symmetry-equivalent atoms. The HF wave function with a hole localized on only one of thesymmetry-equivalent atoms has a much lower total energy than the HF wavefunction in which the hole is delocalized over all N equivalent atoms. Similarly,for valence hole states symmetry breaking can occur. The N HF wave functions—each with a valence hole localized on one of the N symmetry-equivalentcentres— are obtained in separate SCF calculations. In contrast to the core holestates, these wave functions have large mutual overlap and interaction matrixelements. Therefore, a symmetry projection followed by NOCI is necessary toaccount for the overlap and interaction. Note that valence hole states not onlyoccur in valence ionization processes but also in charge transfer configurations.

Complete Active Space second-order Perturbation Theory

The possibility of improving on the Hartree Fock wave function byperturbation theory is not yet discussed. For closed-shell systems with a dominant


HF wave function a division of the Hamiltonian was proposed by Møller andPlesset [87]. In H = H

(0) + H

(1) the zeroth-order Hamiltonian H

(0) is defined as the

sum of the one-electron Fock operators:

H(0)

= F(I)∑I

N , F(I) ϕi(I) = εi ϕi(I) (1.38a)

H(0)

ΦHF = E(0) ΦHF , ΦHF H ΦHF = E(0) + E(1) (1.38b)

where I runs over all electrons. The zeroth-order energy is given by the sum ofthe occupied orbital energies εi and the first-order energy is equal to the HFenergy. Second-order perturbation theory (MP2) leads to the following estimatesof the total correlation energy and the first-order wave function:

E(2) = -∑i<j

ϕiϕj H(1)

ϕaϕb 2

εa + εb - εi - εj∑a<b

(1.39)

Ψ(1)

= ∑i<j

Cijab (1) Φij

ab∑a<b

with Cijab (1) =

ϕiϕj H(1)

ϕaϕb εa + εb - εi - εj

(1.40)

The ingredients for E(2) and Ψ(1) are easily obtained from the preceding HFcalculation, which makes MP2 a very efficient and simple method to estimate theelectron correlation effects. However, for a multireference wave function theMøller Plesset partitioning of the Hamiltonian is not possible and other schemeshave to be developed. One such scheme is the complete active space second-orderperturbation theory (CASPT2) developed by Andersson and Roos [88-90]. On thebasis of a CASSCF wave function the method calculates the second-order estimateof the remaining electron correlation energy and the first-order estimate of thewave function. H

(0) is a sum of Fock-type one-electron operators that reduces to

the Møller-Plesset operator in case of a single configurational reference wavefunction.

H(0)

= fpqEpq∑pq

with fpq = - 0 [ [ H , aq†] , ap]+ 0 (1.41)

Here 0 is the CASSCF reference wave function and Epq an excitation operatordefined as the product of the creation operator ap

† and the annihilation operator


aq. In equation (1.41) and in all following (and previous) equations the followingindices are used for the different classes of orbitals:• i, j, ... inactive orbitals• t, u, v, .... active orbitals• a, b, ..... virtual or secondary orbitals• p, q, r and s general

Contrary to the Møller-Plesset H(0)

, the one-electron operators in this zeroth-order Hamiltonian has non-zero matrix elements between the different orbitalspaces and is only diagonal within the three different subspaces. Therefore, anextra diagonalization step is needed to obtain the corrected energy and wavefunction. Analogous to the ICMRCI method, the first-order wave function Ψ

(1)

is constructed by an internally contracted expansion of all configurationsconnected to single and double excitations from the CASSCF wave function:

Ψ(1) = CpqrsEpqErs 0 ∑pqrs

(1.42)

These configurations span the so-called first-order interaction space and can beclassified into eight different classes according to the number of electrons createdin the virtual orbitals:

• Internal (no external electrons)

EtiEuv 0 (1.43a)

EtiEuj 0 (1.43b)

• Semi-internal (one external electron)

EatEuv 0 (1.43c)

EaiEtu 0 , EtiEau 0 (1.43d)

EtiEaj 0 (1.43e)

• External (two external electrons)

EatEbu 0 (1.43f)

EaiEbt 0 (1.43g)


EaiEbj 0 (1.43h)

The internally contracted character of the first-order wave function stronglyreduces the number of parameters but complicates the expressions of E(2) and thecoefficients of Ψ

(1) , because the construction of the third-order density matrix is

needed. However, as with the NOCI, once implemented [88-90] a very efficientmethod is obtained.

In all perturbational schemes which are based on Fock-type zeroth-orderHamiltonians, configurations can appear in the first-order wave function with anexpectation value of H

(0) which is very close to (or even lower than) the

expectation value of the reference wave function. This leads to very small (oreven negative) energy denominators in the expressions for E(2) and C(1), therebycausing a breakdown of the perturbation theory. In fact, TM materials are rathersusceptible to this phenomenon. As a consequence of the competition betweenone-electron energy terms (kinetic energy and nuclear attraction) and two-electron terms (electron-electron repulsion), orbital energies do not strictly followthe aufbau principle in these compounds. The orbital energies of the partly filledTM-3d orbitals are in general not as high as the orbital energy of the ligand-2porbitals. Hence, the expectation value of H

(0) for a configuration connected to

excitations from the ligand-2p to the TM-3d orbitals is often smaller than theexpectation value of H

(0) for 0 .

In the case of CASPT2, which has a more complex form of H(0)

, the situationis not as simple as sketched above, but intruder states may appear as well andspecial action needs to be undertaken in such cases. A large interaction matrixelement between such an intruder state and the reference wave functionindicates that it is inevitable to include that state in the reference wave function.On the other hand it can also happen that the intruder state does not have a largeinteraction with the reference wave function. At the cost of losing anysignificance of the first-order wave function, the intruder state can be artificiallyshifted up in energy by adding an arbitrary value to the expectation value of H

(0)

of this intruder state. In this way the (near-) degeneracy which makes theperturbation theory divergent is avoided. Roos and Andersson [91] introduced aso-called level shift technique for CASPT2 and successfully applied it to varioussystems, e.g. to the notorious problem with the potential energy curve of Cr2.This technique shifts all configurations in the first-order interaction space (1.43a-h) up in energy by an amount µ , which leads to the following second-orderenergy and coefficients of the first-order wave function:


E(2)

= - j H

(1) 0

εj - E(0) + µ

2

∑j

(1.44)

Cj = - j H

(1) 0

εj - E(0) + µ

(1.45)

in which j is a configuration in the first-order interaction space and εj theexpectation value of H

(0) of this configuration. Under the assumption that εj - E

(0)

» µ, the second-order energy thus obtained can be corrected for the applied shift by

E(2) - E(2) = µ 1 - 1ω

(1.46)

Here ω is the weight of the CASSCF wave function in Ψ(1) obtained with theapplied level shift µ. In principle, this weight reaches 1 at very large values of µ.The correction of the second-order energy, however, is not complete (as will beillustrated in the next chapter) and in order to obtain meaningful results it isessential to apply the same level shift in the CASPT2 for the state which is used asthe relative zero of energy. Equation (1.46) shows that as soon as ω for the stateaffected by an intruder state and the state at zero energy coincide, the correctionfor both states is the same and does not need to be considered anymore. Thismakes the relative energies of the two states only weakly dependent on the size ofthe applied level shift, provided that the intruder state has a small interactionmatrix element and µ does not get extremely large. For all clarity, it is alwayspreferable to solve the problem of the intruder states by extending the referencewave function. Sometimes, however, such an extension is not possible becausethe wave function expansion would become too long. In these cases the levelshift can be helpful to repair the divergence of the second-order energy estimate.

1.5 ELECTRON CORRELATION IN TM MATERIALS

The energy difference between the (hypothetical) exact non-relativisticenergy and the Hartree-Fock energy is generally called the correlation energy.Various attempts have been made to divide the electron correlation energy intodifferent contributions. We will follow the scheme which divides the correlation


energy in dynamical (mainly atomic) electron correlation, and non-dynamicalelectron correlation.

Dynamical correlation effects arise from the instantaneous electron-electronrepulsion. Electrons avoid each other more effectively than predicted by theHartree-Fock method. Non-dynamical electron correlation effects are caused bythe inability to describe the N-electron wave function with a single CSF only;configurations that lie low in energy often strongly mix with the Hartree-FockCSF. For example, a closed-shell molecule near its equilibrium distance usuallydo not show large non-dynamical electron correlation effects and the correlationenergy is largely caused by dynamical correlation only. On the contrary, if bondsare formed or broken, important effects will arise from non-dynamicalcorrelation. As with all other divisions, the division into dynamical and non-dynamical electron correlation is not entirely unambiguous. A well-knownexample is the Be atom. The ground state of this atom can be described by a 1s22s2

configuration. Large contributions to the wave function are observed fromconfigurations connected to double excitations from the 2s orbital to the 2porbital. This can be considered to be a non-dynamical electron correlation effect.However, it is equally valid to ascribe it to dynamical electron correlation, becausethe spatial extent of a 2p orbital is larger than of a 2s orbital and hence theelectron-electron repulsion is reduced.

In many TM materials the electron correlation effects are very important.The localized nature of the 3d electrons in the ionic insulators causes largedynamical correlation effects and the open-shell character of the TM ions forcesone to go beyond the Hartree-Fock description of the N-electron wave function toaccount for the non-dynamical correlation. Apart from the multiconfigurationalcharacter of the wave function due to the appearance of more than one d8 CSF,the reference wave function may contain large contributions of configurationsthat arise from excitations representing ligand-p to TM-d charge transfer. In thefollowing sections some methods will be discussed to calculate and analyze theimportance of the different dynamical and non-dynamical correlation effects onthe electronic structure of this class of materials.

FOCI + CEC

It is rather straightforward to account for the non-dynamical electroncorrelation that arises from the fact that in general the 3dnLm states in TMmaterials cannot be described accurately with a single CSF. The different 3dnLm

1.5 Electron correlation in TM materials 37

CSFs can be either included in a small MCSCF or in a CI using the orbitalsoptimized for an average of the different 3dnLm states.

The First-Order Configuration Interaction (FOCI) method as applied byJanssen and Nieuwpoort [16, 67, 92, 93] provides a way to include the non-dynamical electron correlation effects arising from the 3dn+1Lm-1 charge transfer(CT) configurations. A reference wave function is constructed by considering allinternal replacements within the TM-d shell and the ligand-p shell

Ψref = C1 | ....3dnLm | + C2 | ....3dn+1Lm-1 | + C3 | ....3dn+2Lm-2 | + ... (1.47)

In this expression the terms that are schematically denoted by Slater determinantsrepresent linear combinations of all Slater determinants of a particular type,rather than individual Slater determinants. In first approximation the coeffi-cients c2 of the single CT configurations are expected to be very small becausethere is almost no direct interaction with the 3dnLm configurations.

In the FOCI approach the reference wave function (1.47) is extended withconfigurations connected to single and double excitations under the restriction ofat most one external electron. In analogy to the terminology used in equation(1.43) for CASPT2, these configurations will be referred to as semi-internalconfigurations.

ΨFOCI = Ψref + Cl | ....3dn-1Lma | + Cl+1 | ....3dnLm-1a | +

Cl+2 | ....3dn+1Lm-2a | + ... (1.48)

In these configurations 'a' refers to an orbital in the virtual space. Solving thesecular equations leads to a drastic increase of C2 in the wave function withrespect to its value in the reference wave function. The main source of thisincrease lies in the large interaction matrix elements between the CT and thesemi-internal configurations [16, 92, 93]. The effect of the interaction with thesesemi-internal configurations can be regarded as relaxing the CT configurations,bringing them down in energy.

The FOCI approach is well suited for studying the non-dynamical corre-lation effects due to the CT configurations. Dynamical correlation effects, on theother hand, cannot be treated because these effects are connected to configurationswith at least two external electrons. In fact, in any CI approach the treatment ofdynamical correlation is rather cumbersome for TM materials. An accuratetreatment of the effect can only be achieved by configuration expansions whichare too large to be handled by the current computer resources in almost all cases.


A semi-empirical approach was proposed by Pueyo and Richardson [94]. Theircorrelation energy correction (CEC) approach is based on the assumption that thedynamical correlation for transition metal ions surrounded by other ions (eitherin a molecule or in a solid) can be deduced from that in the isolated TM ion. Thecorrection is determined by comparing the excitation energies of the calculatedHF atomic 3dn states to the experimental ones. The corrections can then be addedto the FOCI energies as was done by Janssen and Nieuwpoort in their study of theelectronic structure of bulk NiO [67, 93].

CASSCF / CASPT2

The CASSCF / CASPT2 approach is a very suitable method to include bothnon-dynamical and dynamical electron correlation in the description of theelectronic structure of TM materials. The non-dynamical correlation is accountedfor by a CASSCF wave function that is constructed so as to include all near-degenerate configurations; the remaining, mostly dynamical, electron correlationis estimated by CASPT2. Contrary to the FOCI+CEC method, there is no need forempirical parameters in this approach. However, a drawback is that the inclusionof the different non-dynamical correlation effects is not as straightforward as it isin the FOCI+CEC method. Similarly difficult is the analysis of the importance ofthese effects.

Obviously, a CASSCF with an active orbital space consisting of five TM-dorbitals and n 3d electrons can only account for the non-dynamical correlationarising from the different 3dnLm configurations. The largest part of the configu-rations treated in the subsequent CASPT2 account for the dynamical correlationof the electrons. Nevertheless, there are some configurations which are con-nected with charge transfer processes and hence some non-dynamical correlationis included in the CASPT2, be it not completely because the interaction matrixelements between excited configurations are not considered.

The matrix schematically depicted in Figure 1.1 shows which matrixelements are treated in the FOCI approach and in the CASSCF/CASPT2 based ona reference function with 3dnLm configurations only. The elements in the firstrow of the matrix are taken into account in the CASSCF/CASPT2 method. InFOCI all matrix elements are accounted for and in this way the interactionbetween CT configurations and semi-internal configurations is included. Asstated in the previous section this causes a drastic increase of the charge transferconfigurations in the wave function. Because there is no direct interaction


FIGURE 1.1 Interaction matrix elements between configurations treated in FOCI

or CASPT2.

3dnLm 3dn+1Lm-1 3dnLm-1a 3dn+1Lm-2a 3dn-1Lma

3dnLm E1 α2 α3 α4 α5

3dn+1Lm-1 α2 E2 β3 β4 β5

3dnLm-1a α3 β3 E3 γ4 γ5

3dn+1Lm-2a α4 β4 γ4 E4 δ5

3dn-1Lma α5 β5 γ5 δ5 E5

Note that FOCI uses the matrix elements of H, and CASPT2 the matrix elements of H(0)

. Thecharacter a in the configurations denote a singly occupied virtual orbital.

between the 3dnLm and the 3dn+1Lm-1 configurations, α2 (being a matrix elementof H

(0)) is very small, implying that the CASSCF/CASPT2 approach cannot take

into account the effect in the way FOCI does.The extension of the active space with ligand-p orbitals does not help to

overcome this problem. Such an active space would result in a CASSCF wavefunction containing the same configurations as in the reference wave functionfor the FOCI (equation 1.47). For the same reason as in the FOCI approach, thecoefficient of the charge transfer configuration will be very small. Although in asubsequent CASPT2 the matrix elements of the second row of the matrix inFigure 1.1 are considered as well, hardly any change will be observed compared tothe previous CASSCF/CASPT2. This is caused by the contracted nature of thefirst-order wave function; each external configuration is weighted by thecoefficients of the CASSCF wave function (see section on CASPT2 in §1.4).Because the coefficient of the charge transfer configuration is nearly zero, theinclusion of β3, β4, and β5 with CASPT2 does not lead to an increase of theimportance of the charge transfer configurations.

Instead of adding the ligand-2p orbitals, a better choice for extending theactive space is to add a set of virtual orbitals of the same symmetry character asthe TM-3d orbitals. These virtual orbitals turn out to form a second shell of TM-dorbitals, the so-called 3d' shell [95-98]. This active space treats a large part of thed-d electron correlation effects in a variational manner, and hence avoids thelarge overestimation of the d-d correlation energy that was observed in theCASPT2 treatment of this effect [98]. No charge transfer configurations areincluded in the CASSCF wave function. Again, the corrected wave functioncontains some configurations connected to charge transfer excitations. In addition


to the first row of matrix elements treated in the CASSCF/CASPT2 with an activespace of the TM-3d orbitals, the matrix elements of the last row of the matrix inFigure 1.1 are also included. This means that at least part of the matrix elementsbetween the CT configurations and so-called relaxing configurations are included.

An important part of the charge transfer effects can be included in theCASSCF wave function by further extending the active space with occupiedligand-2p orbitals. A schematic representation of the resulting wave function is :

ΨCAS = C1 | ....3dnLm | + C2 | ....3dn+1Lm-1 | + C3 | ....3dnLm-1a | +

+ C4 | ....3dn+1Lm-2a | + C5 | ....3dn-1Lma | + ... (1.49)

This wave function is similar to the FOCI wave function (1.48). Apart from the3dnLm and 3dn-1Lm+1 states it also contains some external configurations, namelythose in which 3d' orbitals (in equation (1.49) represented by the character a) areoccupied. Two differences remain between this wave function and the FOCI wavefunction. In the first place, there is a difference in the number of virtual orbitals.In the FOCI study of the 3d8 states in bulk NiO [93] more than 70 virtual orbitalswere included, whereas only 5 virtual orbitals (the TM-3d' orbitals) are includedin this CASSCF wave function. On the other hand, contrary to FOCI, the orbitalsare optimized in the CASSCF. This not only leads to optimal occupied orbitals,but also to optimal TM-3d' orbitals, thus maximizing the effect of the externalconfigurations. The combination of these two differences makes it hard to judgeto what extent relaxation of the CT excitations is included in this CASSCF wavefunction. Nevertheless, we believe that with the CASSCF/CASPT2 approachbased on this active space all important effects are included in the description ofthe 3dnLm states.

Another strategy to reach the best choice of active space for the CASSCF andelectrons to be correlated in the CASPT2 is based on the following considerations.Four effects are important to obtain a reasonable description of the 3dnLm states.In the first place, there is the mixing of the different dnLm CSFs in the wavefunction for which an active space containing the TM-3d orbitals is sufficient.Secondly, there is the large dynamical correlation effect of the valence electrons,namely the ligand-p and the TM-3d electrons. For this purpose the active spaceshould contain the ligand-p orbitals, the TM-3d orbitals and a set of correlatingorbitals, the TM-3d' and the ligand-p' orbitals. This ensures a correct treatment ofthe largest part of the dynamical correlation due to these valence electrons; theremaining part can be estimated by CASPT2. In the third place we need to includethe electron correlation effects due to the semi-core electrons on the TM-ion, the


TM-3s and 3p electrons. A sufficiently accurate treatment of this effect can beobtained by perturbation theory. Finally, it is necessary to account for the non-dynamical correlation effects caused by the configurations connected to CTexcitations. An active space containing the TM-3d, 3d' and some ligand p-orbitalsseems to be appropriate for this.

In summary: an accurate treatment of the most important electroncorrelation effects requires a CASSCF wave function constructed from an activespace containing the ligand-p, p' and the TM-3d, 3d' orbitals and the subsequentCASPT2 should correlate the TM-3s, 3p and 3d, and the ligand-s and p electrons.However, this choice of active space leads to a computational problem which isfar too large to be handled even with the largest computers available nowadays.Hence, by reducing the number of active orbitals, the treatment of some electroncorrelation effects needs to be transferred to the CASPT2. By putting the ligand-p'in the virtual space and the largest part of the ligand-p to the inactive space—only keeping the ligand-p orbitals that are important for the configurationsconnected to CT excitations— the active space is reduced greatly at the cost oftreating the dynamical correlation of the ligand-p electrons perturbationally. Thisseems to be not a very severe approximation, because the differential effect ofcorrelating the ligand-p electrons is expected to be small for electrons in doublyoccupied ligand-p orbitals, although the total contribution is rather large.Therefore, the influence of this effect can be accurately estimated by perturbationtheory.

In short, a CASSCF calculation with an active space containing the TM-3d,3d' and some ligand-p orbitals followed by a CASPT2 that correlates the TM-3s, 3pand 3d, and the ligand-s and p electrons is believed to give a reasonabledescription of the electron correlation effects in the different 3dnLm states.

Restricted Active Space SCF

With the RASSCF method [80] a more specific choice of the configurationstreated in the multiconfigurational wave function can be made. This makes itpossible to include more orbitals in the active space and therefore allows for amore rigorous treatment and analysis of the non-dynamical correlation effectsdue to the charge transfer processes. However, no efficient computer implemen-tation exists as yet for a subsequent perturbational treatment of the remainingdynamical electron correlation effects.


To include and to analyze the importance of the charge transferconfigurations, the active space can be divided in the following way. RAS1consists of all ligand-p orbitals with the same symmetry character as the TM-dorbitals. Configurations connected to single and/or double excitations out of thisspace are included, i.e. allowing one or two holes in RAS1. The TM-d and d'orbitals are put in the RAS2 space, ensuring a proper treatment of the large d-delectron correlation. Furthermore, it accounts for the matrix elements betweencharge transfer configurations and the most important configurations that relaxthe d-orbitals for the charge transfer excitation. Finally, RAS3 is either empty, i.e.no additional dynamical correlation is included in the wave function or electronscan be included here to treat (part of) the dynamical correlation.

The division of the active space that is expected to include efficiently thecharge transfer configurations is the following: the ligand-p orbitals are treated inRAS1 including configurations connected to single and double excitations; theTM-d orbitals in RAS2; and as many virtuals as possible in RAS3 with therestriction of at most one electron. However, this restriction introduces severeconvergence problems and no reasonable results can be obtained [99]. This is notcompletely understood yet but can probably (at least partly) be ascribed to the factthat the optimization of the orbitals is based on the inclusion of single excitationsin each iteration. This problem is avoided when double excitations are treated aswell. An additional advantage of this division of the orbitals over the subspaces isthat without losing the most important part of the d-d electron correlation the CIexpansion within RAS2 is significantly reduced compared to the CI expansionobtained with the TM-d and d' orbitals in RAS2.

1.6 References 43

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