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

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Antiferromagnetic interactions

in TM materials

Chapter 4

112 Chapter 4: Antiferromagnetic interactions in TM materials

4.1 INTRODUCTION

Atoms with partly filled shells can possess a net spin moment, characterizedby the quantum number S. The interaction between the spin moments ondifferent centres is known as magnetic interaction. For a two-electron system it iseasily derived [1, 2] that the interaction can be described with an effective spinHamiltonian. By defining a parameter J as E(S=0) - E(S=1) a relation between theenergies of the two spin couplings can be constructed which for particular valuesof k gives the energies of the individual spin states:

E = 14

( E(S=0) - 3 E(S=1) ) - 12

k ( E(S=0) - E(S=1) ) = Eav - 12

k J (4.1)

for k = -3/2 the energy of the S=0 state is obtained and for k = 1/2 the energy of theS=1 state.

The total spin operator S is defined as s1 + s2, and hence

2 s 1 ⋅s2 = S2 - s12 - s2

2 = S2 - 32

(4.2)

Eigenvalues of 2 s 1 ⋅s2 are -3/2 for S=0 and 1/2 for S=1. These numbers coincidewith those values for k for which (4.1) results in the energies of the S=0 and S=1states. This means that we can obtain the energies of (4.1) by the operator

H = Eav - J s 1 ⋅s2 (4.3)

Equation (4.3) indicates that J parametrizes the magnitude of the spin-spininteraction consistent with the usual understanding that J > 0 leads toferromagnetic states: a parallel alignment of the spin moments with E(S=1) <E(S=0), and J < 0 to antiferromagnetic states: antiparallel alignment of the spinmoments with E(S=1) > E(S=0). Usually this specific two-electron case isgeneralized to many-electron systems and to more than two centres yielding thespin Hamiltonian as proposed by Heisenberg [3] for the description of spin-spincoupling.

H = Jij S i ⋅Sj∑i > j

(4.4)

This generalization is discussed in great detail by Herring [4]. He showed that thegeneralization is not exact (as will be numerically illustrated later in this chapter)

4.1 Introduction 113

and that higher order terms emerge in a more elaborate description. However, itwas also shown that these higher order terms —contributions to the energies thatare not linear in S(S+1)— are rather small and therefore they are usually notincluded in the effective spin Hamiltonian. Herring also pointed out that thevalue of Jij is expected to decrease rapidly with the distance between centres. Thisis why it is practically always assumed that the summation in equation (4.4) canbe restricted to the nearest spin-carrying neighbours only (the sum denoted by Σ').

H = Jij S i ⋅Sj∑'i, j

(4.5)

Eigenfunctions of the isotropic Heisenberg Hamiltonian are necessarilyeigenfunctions of S2 and Sz. A simpler anisotropic description of the spin-spininteraction is given by the Ising model [5-7], which only considers the z-component of the total spin moment.

H = Jij S z i⋅Szj∑'

i, j(4.6)

The spin functions are no longer eigenfunctions of S2 but only of Sz . As aconsequence the spin moment at each centre can be viewed as a classical degree offreedom; either 'up' or 'down', 'α ' or 'β', '↑ ' or '↓ ' and no spin symmetry adaptedlinear combinations are constructed. This leads to a very appealing and simplemodel for spin-spin interactions, in which the antiferromagnetic state is a statewith all spin moments ordered alternatingly '↑ ' and '↓ '. For solid-statecompounds this state is known as the Néel state. The description of the spin-spincoupling by approximating the spin moments as classical vectors is widely usedin solid-state physics [8, 9] and very successful in explaining phenomena asneutron magnetic scattering, order-disorder transitions, magnons, and magneticdomains. However, the connection between the spin adapted antiferromagneticground state of the Heisenberg Hamiltonian (4.5) and the more approximate Néelstate, which is the antiferromagnetic ground state of the Ising Hamiltonian (4.6),is not immediately clear. Imagine a system consisting of 4 centres each with S=1/2. The spin part of the wave function of the Néel state is represented by: α(1)β(2) α(3) β(4) or, equally valid with all spin reversed: β(1) α(2) β(3) α(4). The spin-adapted antiferromagnetic ground state is a linear combination of all possiblespin functions with a net Ms value of zero (ααββ ; ββαα ; αββα ; βααβ ; ββαα;αβαβ; βαβα), coupled in such a way as to give a total function with S=0. Theconnection between the Néel state and the spin adapted antiferromagnetic


ground state could be that for large N the Néel state is the leading term in thespin adapted function. However, except for some symmetry relations there seemsto be no way to predict their relative importance. The explicit calculation of thespin adapted antiferromagnetic wave function in a series of systems withincreasing N can serve as a numerical basis for the connection. Unfortunately,because of the extremely rapid increase of the size of the problem it is impossibleto go much further than N=13 at present. This restricts the analysis to one-dimensional systems only, for which however long range spin order does notexist. A detailed and instructive account of this subject, going far beyond the scopeof this dissertation, is given by Caspers [10].

A way out of the apparent contradiction between theory and the interpre-tation of experiments —as noticed before by Wachters [2]— has been given byPratt [11]. First he observed that it is not at all required to assume a Néel state todescribe the cross sections of neutron magnetic scattering processes (see also Ref.[12]). Furthermore, he argued that the splitting between the antiferromagneticground state with S=0 and the states with higher S becomes infinitely small forsystems with N centres when N goes to infinity. As a consequence it is necessaryto include these states in the description of neutron scattering when therestriction of the spin symmetry is imposed and the wave function can bepresented as a linear combination of wave functions with S=0, 1, 2 etc. Remem-bering that a way of constructing the Néel state is to combine the Ms = 0 compo-nents of wave functions with different S-values, it is concluded that, althoughthe real ground state must be an eigenfunction of S2, the Néel state provides arather good description of the antiferromagnetic ground state.

The large magnetic interaction between two centres with a net spin momentS is not caused by the direct exchange only. Important numerical analyses of themagnetic interaction were published by Anderson and Nesbet [13-17]. Theyshowed that the direct exchange only contributes a very small fraction to the totalmagnetic interaction. The sum of the remaining contributions is generallyconsidered as the superexchange interaction. This chapter presents a quantumchemical approach for the calculation of the magnitude of the magneticinteraction parametrized by J. In the next section an overview will be given ofthis approach applied within the embedded cluster approximation for thematerial. After some general computational information attention will befocused on the details of the different computational methods. This discussion ofthe methods will be followed by a detailed analysis of the different mechanismsthat determine the magnitude of the magnetic interaction. As a model system wetook bulk NiO, but in section 4.6 the quantum chemical approach is applied to a

4.1 Introduction 115

model of the NiO (100) surface. The chapter will be completed by some generalremarks concerning the antiferromagnetic ground state and a short summary ofthe most important points.

4.2 THE QUANTUM CHEMICAL APPROACH

For a cluster in which only two magnetic centres are treated explicitlyquantum mechanically, the Heisenberg Hamiltonian of equation (4.5) reduces toa fairly simple expression:

H = -J S 1 ⋅S2 (4.7)

For example, two Ni2+ ions in a cluster representing NiO can be considered to bein the 3A2g ground state with two unpaired electrons coupled to a triplet spinfunction. These two triplet states can couple to quintet, triplet and singlet spinfunction for the total cluster wave function.

Q = α α α α (4.8)

T = 12

α α β β - β β α α (4.9)

S = 12 2

2 β β α α + α α β β - β α + α β β α + α β (4.10)

The eigenvalues of the Heisenberg Hamiltonian for these spin functions are -J forthe quintet, J for the triplet and 2J for the singlet spin function. Under theassumption of a common orbital part of the wave function for all three spinfunctions, the eigenvalues of the Heisenberg Hamiltonian are directly related tothe energy expectation values of the full Hamiltonian. This implies that theenergy differences between the spin states are directly connected to J.Furthermore, the mechanisms can be investigated which determine the energyseparation between the different spin states by gradually improving theapproximation to the exact solution of the Schrödinger equation and byinvestigating the effect of different material models.

The existence of antiferromagnetic ordering of the spin moments cannot beexplained by the direct exchange interaction only. The energy differences between


the spin states described by a zeroth-order wave function which is defined as theantisymmetrized product of the spin functions (4.8), (4.9) and (4.10) and theorbital part |.... d1(x2-y2) d1(z2) d2(x2-y2) d2(z2)| are exclusively determined by thedirect exchange integral. Under the assumption of zero overlap between orbitalson different centres, it is clear that the state with the highest spin multiplicity isalways lowest in energy. The maximum Ms component of this state correspondsto a parallel alignment of the spin moments, the ferromagnetic solution. A morerealistic approach is the Anderson model [13], which is commonly held to containthe essential ingredients for superexchange. The original idea of Anderson is thatthe superexchange interaction is explained by the concept that “anti-parallelelectrons can gain energy by spreading into non-orthogonal overlapping orbitals,where parallel electrons cannot” [13]. This idea can be followed by extending thewave function with all configurations connected to excitations from onetransition metal to the other, i.e. by performing a complete active spaceconfiguration interaction (CASCI) in which the active orbitals are the open-shellorbitals on the transition metal centres. The effect of adding these configurationsto the wave functions on the relative ordering of the spin states is most easilyillustrated for a system with two magnetic centres with a net spin moment of 1/2,one unpaired electron, as for example occurs in copper oxides. The zeroth-orderwave function for the singlet is extended as follows:

Ψ(1)1 = Ψ(0)1 + λ ...d1d1 + ...d2d2 (4.11)

while for the triplet spin state no contributions to the zeroth-order wave functionof this type exist. The extension of the wave function stabilizes the singlet wavefunction such that it can become lower in energy than the triplet, leading toantiferromagnetic ordering. Something similar happens in the case of the nickelcompounds. While the quintet zeroth-order wave function cannot be improvedby opening up the possibility of transition metal to transition metal excitations,the triplet and singlet function are improved by the extra configurations and inthis way the correct magnetic ordering can be described for these compounds.

The estimate for the magnitude of the magnetic interaction obtained withthe Anderson model has usually the correct sign, but its absolute value is alwaystoo small [18-32]. Several schemes exist to account for the terms not included inthe present representation of the Anderson model. Obviously, within a quantumchemical framework these terms arise from the configurations not yet includedin the wave function. One possibly important configuration is the charge transferconfiguration connected to an excitation from the bridging ligand 2p-orbital to a

4.2 The quantum chemical approach 117

TM-3d orbital. This effect was investigated by van Oosten et al. in the study ofmagnetic interactions for various cuprate compounds [26, 27]. They found thatextending the wave function in a NOCI approach with the properly relaxedcharge transfer configurations led to an increase of J by about a factor of 4, givingvalues very close to the experimental ones. Other approaches to improve on thewave function obtained from the Anderson model, based on perturbation theoryor (orthogonal) CI also show a drastic increase of J when configurations outsidethe active space are treated [19-25, 28-30, 32]. In the long list of configurationstreated with these methods many account for dynamical electron correlation. Alarge part of these configurations does not contribute to the energy differences ofthe spin states (see below). The effect of the charge transfer from the bridgingoxygen to the transition metal ion is, at least partly, present as well.

In this dissertation two approaches are applied to improve on the Andersonmodel: in the first place the CASSCF/CASPT2 approach, previously discussed,and secondly, the CIPSI method [33-35], of which a short description follows here.In the CIPSI method, an uncontracted expansion is built from a (multiconfigura-tion) reference wave function by generating all determinants, the so-calledexternal determinants, that can be obtained by single and double replacementsfrom the determinants in the reference wave function. This list of determinantscan be either treated with second-order perturbation theory or variationally in aCI calculation. Because in almost all cases the full list of determinants is too largeto be handled in a CI, a scheme has been developed that selects those deter-minants that have the largest contribution to the energy difference of the spinstates. For a system of two interacting magnetic centres with S=1/2 each, it hasbeen proven [36, 37] that for reference wave functions with equal expectationvalues of H(0) for singlet and triplet up to second-order perturbation theory onlythe determinants k that fulfil the condition:

i H(0)

k k H(0)

j E0 - Ek

≠ 0 (4.12)

(where i and j represent two different determinants in the reference space)contribute to the energy difference of the spin states. In the present case themagnetic centres have a spin moment S=1, and moreover the reference wavefunction consists of the determinants arising from all TM to TM excitations,hence the expectation values of H(0) of the spin states are not exactly equal, andtherefore condition (4.12) does not hold strictly. Nevertheless, the largestcontribution is still expected to arise from the determinants that fulfil the


condition of a non zero matrix element with at least two determinants from thereference wave function. This variational treatment of the external determinants,which are selected by second-order perturbation theory, will be referred to asDDCI-2 (Difference Dedicated CI) [37]. To estimate the error made by using aselection condition that in principle does not apply, the effect of the otherdeterminants is investigated by perturbation theory. Three different sets ofcalculations were performed in which different determinants were selected forthe perturbation theory. The external determinants can be divided into threegroups. Labelling the orbitals by i, j, ... for the inactive, t, u, .... for the active and a,b, ... for the virtual orbitals, three different groups of excitation operators can beidentified [36]:

-treated in MP2-2: EtiEuj, EtiEau and EatEbu (4.13)

-added in MP2-3: EtiEaj and EaiEbt (4.14)

-added in MP2: EaiEbj (4.15)

The determinants treated in MP2-2 are precisely the ones that are selected withcondition (4.12). In the MP2-3 calculation all other determinants were addedexcept those connected to a double replacement from the doubly occupied orbitalsto the unoccupied orbitals. The effect of these determinants was checked in thecalculation labelled MP2. By comparing the results of the four differentapproaches a statement can be made about the validity of the selection conditionfor the not exactly degenerate reference space used in this study.

4.3 COMPUTATIONAL INFORMATION

Material model

The smallest cluster suitable for studying the superexchange mechanism inNiO consists of two nickel ions and a bridging oxygen, the Ni2O cluster. In orderto obtain meaningful results from this model the influence of the rest of thecrystal should be modelled. The simplest way to do this is to embed the cluster inpoint charges that account for the Madelung potential in the cluster region. It is,however, by now well known (and it will be illustrated in the next section) that

4.3 Computational information 119

for a reliable estimate of J it is essential to include the first shell of ligandssurrounding the TM - ligand - TM cluster [19-23, 28]. Therefore most of thecalculations for bulk NiO were carried out on the Ni2O11 cluster embedded in aset of optimized point charges that reproduce the Madelung potential in thecluster region. Details are given in Table 4.1, and Figure 4.1 gives a graphicalimpression of the cluster embedded in point charges [38]. For the CIPSI calcula-tions a different set of point charges was used [39] and in almost all cases thebridging oxygen was surrounded by four pseudopotentials as a first improvementof the point charge approximation [40]. These so-called Total Ion Potentials (TIPs)represent the Ni2+ ions by a nickel core potential, with an effective nuclear chargeof +2. The nickel oxygen distance (3.9343 bohr) was taken from experiment [41].

FIGURE 4.1 Ni2O11 cluster embedded in point charges. The thick lines link

cluster atoms; light large spheres represent nickel atoms, dark large

spheres oxygen atoms, while the small spheres represent positive

(light) and negative (dark) point charges.


TABLE 4.1 Electrostatic representation of the environment of the Ni2O11 cluster

representing bulk NiO.

Charge Unique position Number of charges

-2.007499 1/2 1/2 0 4

-2.004800 1 0 0 4

-2.015017 1/2 1/2 1 8

-1.938639 1/2 1 1/2 16

-1.950570 1 0 1 8

-1.164227 1/2 0 3/2 8

2.006299 1/2 1/2 1/2 8

2.000345 1/2 0 0 4

1.997322 1/2 0 1 8

1.977573 1 0 1/2 8

1.894489 1/2 1 0 8

1.384739 0 0 3/2 2

1.705274 1/2 1 1 16

0.795404 1/2 1/2 3/2 8

Standard deviation of the fit is 1.90 x 10-3 au. Exact value of the external Madelung potential at thenickel site is 2.986363 au, the fitted value is 2.986421 au. For the bridging oxygen the exact value is3.255684 au, the fitted value is 3.255831 au. For the two edge oxygens on the Ni -O -Ni axis the exactvalue is 2.672684 au, the fitted value is 2.666179 au, and for the other edge oxygens the exact valueis 2.854412 au, the fitted value is 2.854468 au.

Basis sets

Two different sets of basis functions were used in the calculations: ANObasis sets in the CASSCF/CASPT2 calculations and basis sets with segmentedcontractions for the CIPSI calculations. The ANO basis sets for nickel were builtfrom a primitive set of (21s, 15p, 10d, 6f) functions, for the bridging oxygen aprimitive set of (14s, 9p, 4d) functions and for the edge oxygens a set of (10s, 6p)functions was used. These primitive sets were contracted to (7s, 6p, 4d, 1f) fornickel, [5s, 4p, 1d] for the bridging oxygen and for the edge oxygens to [4s, 3p] as


TABLE 4.2 Overview of segmented basis sets used in the CIPSI calculations.

All-electron

Ni2O Ni2O11 Ref.

N i (14s,11p,6d) / [6s,5p,4d] (14s,11p,6d) / [6s,5p,3d] (a) [45]

bridging O (9s,5p) / [4s,3p] (b) (9s,5p) / [4s,3p] (b) [46]

edge O - (9s,5p) / [3s,2p] (c) [46]

Pseudopotentials

Ni2O Ni2O11 Ref.

N i (3s,3p,6d) / [2s,2p,3d] (d) (3s,3p,6d) / [2s,2p,3d] [21]

bridging O (6s,6p) / [3s,3p] (b) (6s,6p) / [3s,3p] (b) [23]

edge O - (6s,6p) / [2s,2p] [23]

a The contraction scheme was obtained from a calculation on the Ni2+ ions in the field of theother ions using the uncontracted primitive set. (available on request)

b A d-function was added (ζ =1.25) (Ref. [47])c The contraction scheme was obtained from a calculation with one oxygen ion in the field of the

other ions using the uncontracted primitive set. (available on request)d To study the effect of a more flexible basis set in the d-functions, calculations were also done on

the Ni2O cluster with one more d-function left uncontracted.

described in Ref. [42-44]. In some model calculations different contractions wereused, as will be indicated. For all CIPSI calculations segmented basis sets wereused. In Table 4.2 an overview is given of the basis sets used in the all-electroncalculations and the calculations in which the core electrons were replaced bypseudopotentials.

Pseudopotentials

To make calculations on large clusters feasible it can be necessary to usepseudopotentials instead of including the core electrons explicitly. Suchpseudopotentials have shown to be of value in several applications and may wellbe essential in calculations on magnetic properties of 4d- and 5d-transition metalcompounds and in calculations on systems with more than two transition metalions [21, 22, 48]. It has to be established however, whether these potentials can beused for the study of the small energy differences that determine the strength of


magnetic couplings. To investigate the usefulness of pseudopotentials in the caseat hand two sets of calculations were performed, one with the core electronsincluded explicitly and the other with pseudopotentials representing the coreelectrons. For oxygen the 1s electrons and for nickel the 1s, 2s, 2p, 3s, 3p electronsare represented by pseudopotentials [49, 50]. In order to compare the calculationswith and without pseudopotentials it is important that the basis sets are ofcomparable accuracy in both cases. It is impossible to use basis sets of exactly thesame quality because a different number of electrons is to be described. We thinkthat the basis sets explored are adequate for the comparison we want to make. Theuse of pseudopotentials reduces the number of basis functions from 175 to 138 inthe case of a Ni2O11 cluster (see Table 4.2).

Computational methods

In almost all cases discussed below the CASSCF or CASCI wave functionswere constructed from a complete CI expansion in an active space formed by theopen-shell orbitals centred mainly on the Ni2+ ions; 3d(x2 - y2) and 3d(z2). Thismeans that the calculations were carried out at a level which represents theAnderson model. When the active space is extended with other orbitals, it will bementioned explicitly. In all CASPT2 calculations only the (semi-) valenceelectrons were correlated, namely the Ni-3s, 3p and 3d electrons and all O-2s and2p electrons. For a Ni2O11 cluster this implies that 120 electrons were correlated bysecond-order perturbation theory. All CASSCF/CASPT2 calculations wereperformed with the MOLCAS quantum chemistry program package [51]. Thecalculation based on the CIPSI approach were done with the PSHF-CIPSI chain ofprograms [52].

Experimental data

The magnetic ordering in NiO and the other late transition metalmonoxides is determined by the next-nearest neighbour ions in the (100)direction. The nearest neighbour interactions in the (110) direction cancel out inan ideal structure, six ferromagnetically and six antiferromagnetically coupledspin moments for each nickel ion. Experimentally J has been obtained by differenttechniques. In 1972 Hutchings and Samuelsen determined J to be -19.0 meV byneutron scattering [53]. They also found a net value for the nearest neighbour


FIGURE 4.2 Magnetic ordering in rock-salt monoxides as NiO. Only the

transition metal ions are shown. Equal signs on neighbours indicate

ferromagnetic coupling. Opposite signs indicate antiferromagnetic

coupling.

interaction due to small lattice distortions. This interaction was reported to be 1.4meV, indeed very small, and ferromagnetic. Interpretation of experimental databy Shanker and Singh [54] led to a next-nearest neighbour J-value of -17.3 meVand to a small antiferromagnetic nearest neighbour interaction. More recent(1990) are Raman scattering measurements of Massey and co-workers [55].Applying different external pressures, J was found to have a dependence on thelattice parameter a of the following form: J ∝ a-ε, with ε = 9.9 ± 0.5. At the latticeparameter used in the calculations (2 x 3.9343 bohr) J has the value of -19.8 meV.

4.4 BULK NiO: EXPLORING THE COMPUTATIONAL METHODS

Compared to the total energies involved, the energy differences between thedifferent spin states are very small, and at first sight it seems inappropriate tocalculate J as the difference of two numbers that are larger than J by seven ordersof magnitude. However, the major part of the total energy arises from


contributions of core electrons which can safely be assumed to be the same for allspin states. The energy differences between the spin states only stem from thedifference in CI coefficients and as will be shown below, also from the differencein optimal valence orbitals, albeit to a smaller extent. Furthermore, the totalenergies are converged in the self-consistent process up to an accuracy of 10-7 au,i.e., better than 0.003 meV.

Next to be considered is the stability of the magnetic interaction as functionof the applied set of basis functions. It is known that for some molecularproperties, like dipole and higher multipole moments and dissociation energies,the convergence with the size of the basis set is not very rapid (for a more detaileddiscussion see for example Ref. [56]). Although the magnetic coupling is directlyrelated to the total energy, for which the basis set convergence is much morerapid than for the above mentioned molecular properties, it is still warranted tolook at the basis set dependency of J, because of the very small energy differencesinvolved. Casanovas and Illas tested the basis set dependency of J in KNiF3 bycalculating the magnetic interaction with four different segmented basis sets [22]. Jwas found to be fairly insensitive to the size of the basis set. A more extensivestudy is presented for the ANO basis sets in Table 4.3. Values are listed both forCASSCF (upper left corners) and for subsequent CASPT2 calculations (lower rightcorners). In agreement with the findings of Casanovas and Illas, J does not show avery large dependency on the size of the basis set if one applies more thanminimal basis sets on all centres. A further extension of the basis set hardly affectsthe CASSCF value, but has the tendency to slightly increase J obtained withCASPT2, indicating that the smaller basis sets are not converged yet for electroncorrelation. A further increase of the basis set is expected to only have a minorinfluence on the value of J.

One of the fundamental assumptions in describing the magnetic inter-actions between two centres with a net magnetic moment with the HeisenbergHamiltonian is that the spatial part of the wave function is the same for all spinstates. Nesbet already pointed out in 1961 [16] that contributions to the magneticcoupling exist that are not linear in S(S+1) and therefore do not fit into theHeisenberg Hamiltonian. He made an analysis of the effect on the magneticinteraction of different classes of configurations connected to single or doubleexcitations with respect to a zeroth-order wave function as defined in section 4.2,assuming an identical orbital part for all spin states. Following his analysis thefirst non-linear terms arise from configurations connected to single excitationsfrom doubly occupied orbitals either to the open-shell orbitals or to theunoccupied orbitals.

TABLE 4.3 Basis set dependency of J obtained for a Ni2O11 cluster. CASSCF (upper left) and CASPT2 (lower right) values

are given in meV. The ten oxygens surrounding the central Ni - O - Ni cluster are labelled by edge O.

edge O → 2s, 1p 3s, 2p 4s, 3p

O →Ni ↓

2s, 1p 5s, 4p 3s, 2p 4s, 3p 5s, 4p, 1d 6s, 5p, 1d 4s, 3p 5s, 4p, 1d 6s, 5p, 1d

4s, 3p, 2d -10.0 -35.0

-4.3 -14.5

-3.5 -11.1

-3.5 -10.9

-3.5 -11.4

-3.5 -11.2

-3.4 -11.0

-3.5 -11.4

-3.5 -11.3

5s, 4p, 3d -5.8 -17.6

-3.5 -11.3

-3.3 -10.6

-3.3 -10.6

-3.4 -11.0

-3.3 -10.9

-3.2 -10.9

-3.3 -11.3

-3.3 -11.2

6s, 5p, 3d -5.4 -14.9

-3.2 -10.6

-3.3 -10.7

-3.3 -10.6

-3.4 -11.0

-3.3 -10.8

-3.3 -11.0

-3.3 -11.3

-3.3 -11.2

6s, 5p, 3d,1f

-5.4 -17.7

-3.2 -10.8

-3.3 -11.0

-3.3 -11.0

-3.3 -11.4

-3.3 -11.3

-3.2 -11.4

-3.3 -11.6

-3.3 -11.5

7s, 6p, 3d -5.9 -17.2

-4.0 -13.5

-3.3 -10.8

-3.3 -10.7

-3.3 -11.0

-3.3 -10.9

-3.3 -11.2

-3.3 -11.4

-3.3 -11.3

7s, 6p, 4d -5.5 -18.1

-3.9 -13.2

-3.3 -10.9

-3.3 -10.7

-3.3 -11.0

-3.3 -10.9

-3.3 -11.2

-3.3 -11.4

-3.3 -11.4

7s, 6p, 4d,1f

-5.5 -18.8

-3.9 -13.5

-3.3 -11.3

-3.2 -11.0

-3.3 -11.3

-3.3 -11.2

-3.3 -11.5

-3.3 -11.7

-3.3 -11.7


TABLE 4.4 Influence on J (in meV) of the orbital description of the spin states.

Es - Et (Es - Eq)/3 Average

CASCI-q -2.8 -2.8 -2.8

CASPT2 -9.7 -8.9 -9.3

CASCI-s -3.6 -3.6 -3.6

CASPT2 -9.8 -9.1 -9.5

CASSCF -3.4 -3.2 -3.3

CASPT2 -12.3 -11.1 -11.7

In CASCI-q the orbitals were optimized for the quintet, in CASCI-s for the singlet state.

These configurations were estimated by second-order perturbation theory to havea small but not negligible antiferromagnetic contribution to the interaction. Thisindicates that the optimization of the orbitals for the different spin states can yieldsmall contributions to the value of the superexchange constant that cause adeviation from the ‘exact’ Heisenberg splitting of the relative energies of the spinstates. Table 4.4 compares a CASCI calculation in which a common set of orbitalsis used with CASSCF calculations in which the orbitals were optimized for theseparate spin states. The listed values for J were obtained from the singlet -quintet and the singlet - triplet energy differences: Es - Eq = 3J, Es - Et = J. The one-to-one average of these values is also given.

As expected, the choice of taking the quintet orbitals as the common setsomewhat underestimates the magnetic interaction, while using the singletorbitals causes the opposite, an overestimation compared to the case where theorbitals for all spin states were optimized separately. The CASPT2 results obtainedafter CASCI do not depend very much on the choice of orbitals, indicating thatsome part of the orbital relaxation for the different spin states is incorporated inthe second-order perturbation treatment.

For the other methods to improve on the Anderson model mentioned inthe previous section it has to be checked whether the criterion for selectingdeterminants for the CI (equation 4.12) is still valid with a reference space inwhich not all determinants are degenerate. This was done by studying the effectof the determinants not included in the CI with second-order perturbationtheory. To reduce the computational cost of the CI calculation the Ni2O clustermodel was used. In Table 4.5 an overview of the results is given, using theacronyms as explained in section 4.2. The numbers show that the differences

4.4 Bulk NiO: exploring the computational methods 127

TABLE 4.5 Influence of different classes of determinants on J (in meV) inves-

tigated by second-order perturbation theory, compared to DDCI-2.

CASCI MP2-2 MP2-3 MP2 DDCI-2

Es - Et -1.5 -3.5 -3.2 -3.3 -3.5

(Es - Eq)/3 -1.5 -3.3 -3.3 -3.3 -3.5

TABLE 4.6 Comparison of pseudopotentials and all-electron results for J (in

meV) obtained for the Ni2O cluster.

Pseudopotentials All-electron

CASCI MP2 CASCI MP2

Ni-3d + O-2s, 2p + Ni-3s, 3p

Es - Et -1.0 -2.2 -1.5 -2.9 -3.1 -3.3

(Es - Eq)/3 -0.9 -2.2 -1.5 -2.9 -3.1 -3.3

between the specific method of improving on the Anderson model are verysmall. In a second-order perturbation treatment only the determinants ofequation (4.13) contribute to the energy differences between the three spin states.Because the values of all methods are essentially the same, it is concluded thatthe selection criterion, although not valid strictly, applies very well.

Next, we studied the validity of the use of pseudopotentials for determiningthe small energy differences involved in calculating J. Due to the fact that in thecalculations with pseudopotentials different numbers of electrons are availablefor correlation, some extra all-electron calculations were performed restricting thenumber of electrons being correlated. The comparison of the all-electroncalculations versus the calculations with pseudopotentials is presented in Table4.6 for the Ni2O cluster and in Table 4.7 for the Ni2O11 cluster.

The comparison of the pseudopotentials versus all-electron results can bemade at two points, namely CASCI and MP2 with the same set of electronscorrelated: the Ni-3d and O-2s, 2p (see the one but last column in Table 4.6 forNi 2O, and in Table 4.7 for Ni2O 11). The present implementation of thepseudopotentials indeed influences the value of J. For Ni2O they decrease J by40% for CASCI (-1.5 versus -1.0) and by 30% for MP2 (-3.1 versus -2.2) and forNi2O11 a similar behaviour is found, a decrease of about 20% is observed both forthe CASCI and MP2 values. Probably, the number of electrons presently included


TABLE 4.7 Comparison of pseudopotentials and all-electron results for J (in

meV) obtained for the Ni2O11 cluster.

Pseudopotentials All-electron

CASCI MP2 CASCI MP2

Ni-3d + O-2s, 2p + Ni-3s, 3p

Es - Et -3.0 -7.8 -3.8 -6.2 -10.0 -10.4

(Es - Eq)/3 -3.0 -7.8 -3.8 -6.2 -10.1 -10.6

TABLE 4.8 The magnetic interaction (in meV) for two different mulitreference

perturbation theory implementations: CASPT2 and MP2 (CIPSI).

Method Es - Et (Es - Eq)/3 Average

CASCI -2.6 -2.6 -2.6

MP2 (CIPSI) -7.6 -7.7 -7.6

CASPT2 -8.4 -7.8 -8.1

in the pseudopotential for nickel is too large and a smaller dependency mightoccur when the Ni-3s and 3p electrons are treated explicitly.

Before discussing calculations aiming at a more physical investigation of themagnetic interaction one last test calculation is presented in Table 4.8. It concernsthe comparison of the two different multireference second-order perturbationtheory implementations applied in this study of the magnetic interaction. Table4.8 presents two calculations in which all calculational details are the same, exceptfor the second-order treatment of the external dynamical correlation effects. Thecalculation was performed for a Ni2O11 cluster model embedded in only pointcharges. All electrons were explicitly treated and the appropriate segmented basissets from Table 4.2 were applied. The CASCI wave function describes theAnderson model and the MOs were optimized for the quintet state. The valuesfor the CASCI and MP2 (CIPSI) are somewhat different from those in Table 4.7because in the present calculation no TIPs were used to represent the four Ni2+

ions around the bridging oxygen not included in the cluster.The Heisenberg behaviour (Es - Et = (Es - Eq)/3 = J) of the CASCI energies for

the different spin states is practically maintained by the MP2 (CIPSI), while asmall deviation appears in the CASPT2 approach. Nevertheless, the (average)values obtained with both methods are quite similar, indicating that neither the

4.4 Bulk NiO: exploring the computational methods 129

precise details of the construction of the first-order wave function (contracted vs.uncontracted) nor the two different choices of the zeroth-order Hamiltonianinfluence the estimate for J obtained with these second-order perturbation theorymethods.

From these investigations we conclude that a reasonable estimate of themagnetic interactions in transition metal materials can be obtained applyingstandard quantum chemical techniques, provided the following points are keptin mind:• The cluster should consist of the two magnetic centres plus the complete first

shell of neighbours as the edge ligands strongly affect the magnetic interaction.• The basis set should be of at least moderate size for all centres considered in

the cluster, including the edge ligands.• The correct sign for J is obtained at the level of the Anderson model, but the

absolute value is strongly increased by including configurations not present inthis model. This improvement on the Anderson model can be retrievedeither with second-order perturbation theory or with methods based on CI.The precise details of the methods are not essential.

• The optimization of the orbitals for all spin states individually introducesterms that cause deviations in the ‘exact’ Heisenberg splitting of the spinstates. The choice of the orbitals of the quintet state as a common set of orbitalsfor all spin states leads to a slight underestimation of J, while the use of thesinglet orbitals leads to an overestimation.

• The present implementation of the core potentials causes a decrease of J by asubstantial amount compared to the all-electron values. It is, therefore, notadvised to use these pseudopotentials in the calculation of the magneticcoupling.

4.5 BULK NiO: MECHANISMS DETERMINING J

At this point, we turn to the discussion of the physical mechanismsdetermining the strength of the magnetic interaction in transition metalcompounds. As was observed above, a cluster model containing two magneticcentres and the bridging oxygen shows the correct magnetic behaviour ofantiferromagnetic coupling, but gives far too small an interaction. This is


drastically improved by adding the first layer of oxygens to the cluster (cf. Table 4.6and 4.7). This might seem somewhat surprising because in the cluster modelapproach these edge oxygens do not act as bridging ligands between two magneticcentres; they do not directly participate in a superexchange mechanism. The roleof the edge oxygens was studied by a series of computational experiments,designed to isolate the different mechanisms that could influence J. Furthermore,a Constrained Space Orbital Variation (CSOV) analysis [22, 57, 58] of the wavefunction was made for the Ni2O and Ni2O11 cluster models.

The first cluster model in the series of experiments was a Ni2O10 clustermodel which is obtained by replacing the bridging oxygen by a point charge q = -2(Model I). This model excludes the role of the bridging oxygen ion. Because thetwo ions are separated by more than 4 Å it is not expected that the exchangeinteraction is of any significance.

NiO

O

O

O

O

ONi2-

O

O

O

O (Model I)

In the second cluster model the Ni2+ ions are connected by a frozen chargedistribution representing an O2- ion in a NiO crystal (Model II). Although themagnetic centres are not really isolated, no delocalization effects as described byAnderson and Nesbet [13, 15] can occur, because the open-shell orbitals on theNi2+ ions are orthogonalized to the orbitals of the bridging ligand. This modelpermits to investigate the role of the bridging oxygen in the superexchangemechanism.

NiO

O

O

O

O

ONi

O

O

O

O

O

(Model II)

4.5 Bulk NiO: mechanisms determining J 131

TABLE 4.9 Magnetic interaction (in meV) for various cluster models (see text)

CASSCF CASPT2

Ni2O -1.3 -3.1

Model I 0.0 -0.3

Model II -0.8 -2.2

Model III -3.4 -9.3

Ni2O11 -3.3 -11.2

The third model consists of a Ni2O unit which is embedded in the frozen chargedistribution of the ten edge oxygens (Model III). No covalency between the Ni2Ofragment and the edge oxygens is allowed in the wave function in this model,neither in the CI expansion nor in the composition of the orbitals. The influenceof the ten oxygens is purely static in this model.

NiO

O

O

O

O

ONi

O

O

O

O

O

(Model III)

CASSCF/CASPT2 calculations were performed as described in thecomputational information, applying the following contracted basis sets: for Ni[7s, 6p, 3d] and for all oxygens [4s, 3p]. The frozen charge distributions in Model IIand III were obtained in a calculation on the respective fragment with the rest ofthe cluster replaced by point charges with the formal ionic value. The total clusterwave function was constructed by orthogonalizing the orbitals of the cluster partthat will be relaxed onto those of the frozen fragment. The virtual orbitalsassociated with the ions that are represented by frozen charge distributions werediscarded from the calculations. Table 4.9 gives the results of the models I, II andIII extended with the results for Ni2O and Ni2O11.

As expected the exchange between two isolated Ni2+ ions does not give riseto splittings of the spin states of any importance; for Model I both the CASSCFand the CASPT2 value for J are essentially zero. In Model II, in which the point


charge between the two nickels is replaced by a frozen O2- charge distribution, thesituation is not substantially changed. The CASSCF value is very small and alsothe CASPT2 value can by no means be compared to the value for the Ni2O11

cluster with full orbital relaxation. It shows that the magnetic interaction indeedoriginates from the covalent interference of the magnetic orbitals with theorbitals on the bridging ligand as described by Anderson and Nesbet [13, 15].Model III elucidates the role of the edge oxygens in the cluster model approach tothe magnetic coupling mechanism. It shows that this role is purely static and thatwithout allowing for covalency between the Ni2O unit and the edge oxygens anestimate for the magnetic interaction is obtained which almost coincides with thevalue obtained from a Ni2O11 cluster model.

The second method used to analyze the mechanisms that lead to a magneticinteraction is the CSOV method, introduced by Bagus et al. [57, 58]. Starting withtwo totally ionic fragments, the wave function is fully relaxed in a series ofconstrained variations in which the variational space is modified in eachsuccessive step. Depending on the choice of the two starting fragments, physicalmechanisms can be isolated that influence the magnetic interaction. Thedelocalization effect of Anderson can be studied by dividing the Ni2O cluster intoa Ni2

4+ fragment (fragment A) and an O2- fragment (fragment B). The role of theedge oxygens can be investigated by splitting the Ni2O11 cluster into Ni2O2 +

(fragment A) and O1020- (fragment B). The wave functions for these ionic fragments

were obtained by ROHF calculations on the fragments embedded in the pointcharges of Ref. [39]. The missing centres, which in the standard calculations arepart of the cluster, were replaced by a point charge with the formal ionic value. Ateach step of the CSOV J was calculated at the CASCI level only. This is because themethods used to improve on the Anderson model account at each step of theCSOV for at least part of the orbital relaxation, in this way circumventing theconstraints given by the CSOV. The all-electron segmented basis sets as indicatedin Table 4.2 are used in these calculations.

The results listed in Table 4.10 of the CSOV analysis on the Ni2O clusterclearly confirm the superexchange mechanism of Anderson; again J is completelydetermined by the mixing of the open-shell orbitals on the Ni2+ ions with theorbitals of the bridging oxygen. Furthermore, the role of the edge oxygens foundwith the model studies described above is again recognized to be completely static.In the first step of the CSOV on the Ni2O11 cluster, in which the two purely ionicfragments are superimposed and no covalency between the two fragments isallowed, the magnetic interaction already has the same magnitude as it has forthe fully relaxed cluster model.


TABLE 4.10 CSOV analysis of J (in meV).

Step in CSOV variational space Ni2O Ni2O11

1. Frozen ions _ -0.2 -3.5

2. Polarisation fragment A doubly occ. A +virt. A

-0.0 -4.5

3. Charge transfer A → B doubly occ. A +virt. A + virt. B

-0.1 -4.1

4. Polarisation of B doubly occ. B +virt. B

-0.1 -4.0

5. Charge transfer B → A doubly occ. B +virt. A + virt. B

-0.1 -3.9

6. Mixing open- & closed- shells

doubly occ. A & B+ open A

-1.5 -3.7

SCF full space -1.5 -3.8

From the results presented so far some conclusions may be drawn on thephysical mechanisms that lead to the antiferromagnetic ordering as observed inNiO. The mechanism proposed by Anderson of the magnetic mechanism [13]indeed leads to antiferromagnetic coupling between two Ni ions, butunderestimates the strength of the interaction. With a Ni2O cluster model only avery small antiferromagnetic interaction is obtained. This is improved by almosta factor of three by including the first shell of oxygens into the cluster modeldescription. The role of these oxygens was found to be purely static. Furthermore,the inclusion of remaining electron correlation effects by means of second-orderperturbation theory or a CI approach drastically increases J (see Tables 4.4-4.9).

The reasoning that a more realistic description of the direct surroundings ofthe nickel centres increases the magnetic interaction also holds for the bridgingoxygen. A substantial contribution to J can be expected when the point chargedescription is improved for the four other nickels surrounding the bridgingoxygen. To mimic these Ni2+ ions different approaches were used. First of all, asalready mentioned in the computational information, in all CIPSI calculationsTIPs were used. Table 4.11 indicates the effect on J of these pseudopotentials.


TABLE 4.11 Effect on J (in meV) of various representations of the four Ni2+ ions

around the bridging oxygen that are not included in the Ni2O11

cluster.

CIPSI CASCI MP2

point charges -2.6 -7.6

TIPs -3.8 -10.6

MOLCAS CASSCF CASPT2

point charges -3.3 -11.7

Frozen Mg2+ ions (a) -4.1 -13.9

Mg2+ AIEMPs -4.6 -15.8

Mg2+ AIEMPs + [1s, 1p] -4.1 -14.2

Ni2+ AIEMPs + [1s, 1p] -4.2 -14.3

Ni2+ AIEMPs + [1s, 1p, 1d] -4.1 -14.0

a The frozen charge distribution of the four Mg2+ ions was obtained by an SCF calculation of thefour centres embedded in the point charges of the Ni2O11 cluster, the atoms of the cluster werereplaced by formal point charges. A minimal basis set was used to describe the Mg2+ ions.

Another method of improving on the point charge approximation is to replacethem by frozen charge distributions. In the present case this would mean thatopen-shell orbitals have to be kept frozen, a feature which is not implemented inour programs. Instead, calculations were done with frozen charge distributions ofMg2+ ions. The divalent magnesium ion has an ionic radius that does not differmuch from Ni2+: 0.69 Å for Ni2+ and 0.66 Å for Mg2+ [59]. The third method toimprove on the point charge approximation was to represent the four nickelsaround the bridging oxygen by Ab Initio Embedding Model Potentials (AIEMP)[60, 61]. Two types of AIEMPs were used. Firstly, Mg2+ potentials optimized forMgO [62] were applied in order to compare with the frozen ion description. Tomaintain the strong orthogonality between the cluster and the Mg4-fragment,extra basis functions were added to the potential. Secondly, spherically averagedNi2+ potentials optimized for NiO were used [63]. The results of these calculationsare added to Table 4.11.

Irrespective of the details of the change in the point charge approximation—TIPs, frozen ions, AIEMPs or AIEMPs with basis functions— we see that themagnetic interaction has increased considerably both for the CASSCF and the


CASPT2. An explanation is that by accounting for the short-range repulsion thecharge on the bridging oxygen becomes more localized in the region between theNi ions, and can therefore participate more efficiently in the superexchangemechanism. A closer look teaches us that the AIEMPs without basis functionsoverestimate this localization effect. The J value calculated in the material modelin which the point charges are represented by Mg2+ AIEMPs without basisfunctions that can maintain the strong orthogonality relation (1.6) are higherthan the value calculated in a material model in which the same ions arereplaced by frozen charge densities. The addition of an s- and a p-function to theAIEMP largely repairs this overestimation. A small effect is observed by adding ad-function and it is therefore believed that the results for the Ni2+ AIEMPs with[1s, 1p] functions reproduce the results that would be obtained if it were possibleto represent open-shell ions with a frozen charge distribution.

It is interesting to find out to what extent the increment in J caused by thetreatment of the electron correlation effects not incorporated in the CASSCFwave function can be ascribed to charge-transfer effects from the bridging oxygen.From the CASSCF/CASPT2 results discussed so far this is not very easy toinvestigate, as these charge transfer contributions are hidden in the large numberof small contributions to the first-order wave function. Furthermore, asexplained in the first chapter, an important part of the effect cannot be recoveredby perturbation theory when a zeroth-order wave function which is obtainedfrom an active space in which only the Ni-3d orbitals are included. In otherwords, to study the effect on the magnetic interaction of charge transferconfigurations connected to excitations from the bridging oxygen to the transitionmetal it is inevitable to enlarge the active space in the CASSCF part of thecalculation. Following the conclusions derived from the NOCI calculations ofvan Oosten et al. [26, 27], a proper inclusion of ligand to metal charge transferconfigurations into the CASSCF wave function should lead to a considerableincrease of the superexchange interaction. With an active space that is largeenough to account for the complete relaxation of the charge transfer configura-tions the value of J obtained with CASSCF should in principle coincide with theNOCI value and, also, the contribution of the perturbation theory shoulddecrease considerably.

Table 4.12 gives an overview of the calculations with different active spacesin the CASSCF. The table indicates which group of orbitals is active and inparentheses the number of active orbitals and electrons is given. The 2p orbitalon the bridging oxygen pointing towards the nickel ions is the only O-2p orbital


that has an overlap with an open-shell orbital on the Ni ions. Therefore, thisorbital is responsible for the occurrence of the delocalization effect described byAnderson. It is therefore expected that the charge transfer configurations arisingfrom excitations out of this orbital give the largest contribution to the magneticcoupling of the Ni ions. Adding only this oxygen orbital, denoted by O-2pσ, to theactive space cannot affect the total energy for the quintet state (5A 1g). Forsymmetry reasons only single excitations can be added to the referenceconfiguration. In case of a single-determinantal wave function single excitationsfrom a doubly occupied orbital to a singly occupied orbital only cause a unitarytransformation of the wave function; the wave function can always be rewrittenin its one-determinantal form:

Φ = a |. . . . pdd| + b |. . . . ppd| = |. . . . p'p'd'| (4.15)with: p' = ad + bp

d' = bd - ap

For the singlet (1A1g) and triplet (3A2u) states a very small contribution wasfound. As observed for the d-d excitations (see Chapter 2) the O-2pσ orbital stayspractically doubly occupied and the CASSCF value of J changes only by ~0.2 meV.

The next extension of the active space that was considered was adding fourvirtual orbitals with the same symmetry character as the open-shell orbitals to theactive space. From a Mulliken population analysis these virtuals were, as in thecase of the calculations on the d-d excitations, identified as the Ni-3d' orbitals.This extension of the active space hardly affects the magnitude of the interaction;only a tiny increase in J is observed, although it opens up the possibility for somerelaxation of the charge transfer configuration.

Adding an occupied orbital of the proper symmetry character, a2u, to theactive space indeed leads to an increase in the superexchange interaction.However, the character of this orbital, determined by Mulliken populationanalysis (MPA) (Table 4.13), was found to be delocalized over all oxygen centresand cannot really be considered as 'the' O-2pσ orbital. This extra orbital localizeswhen another virtual orbital is added to the active space. This virtual acts as acorrelating orbital for the O-2pσ and will be named O-2pσ'. This extension of theactive space shifts part of the treatment of the dynamical correlation effect fromthe CASPT2 to the CASSCF in the same way the Ni-3d' orbitals do. The amountof dynamical electron correlation, mainly an atomic effect, obtained within theCASSCF is maximized when the pair of occupied and correlating orbitals for theO-2p electrons is as localized as possible on the bridging oxygen. The analysis of


TABLE 4.12 J (in meV) as a function of the size of the active space.

CASSCF CASPT2

Active: Ni-3d (4 orb. / 4 el.)

Es - Et -3.4 -12.3

(Es - Eq)/3 -3.2 -11.1

Average -3.3 -11.7

Active: Ni-3d + 3d' (8 orb. / 4 el.)

Es - Et -3.5 -12.9

(Es - Eq)/3 -3.4 -11.2

Average -3.5 -12.1

Active: Ni-3d, 3d' + O-2pσ (9 orb. / 6 el.)

Es - Et -5.2 -15.6

(Es - Eq)/3 -4.8 -13.0

Average -5.0 -14.3

Active: Ni-3d, 3d' + O-2pσ + O-2pσ' (10 orb. / 6 el.)

Es - Et -4.2 -14.3

(Es - Eq)/3 -4.0 -12.1

Average -4.1 -13.2

the character of the orbitals in Table 4.13 shows that the active space now indeedconsists of the Ni-3d and 3d' orbitals, and the O-2pσ orbital plus its correlatingcounterpart, the O-2pσ'. The superexchange interaction shows an increase withrespect to the active space in which no charge transfer effects were includedimplicitly; at the CASSCF level J is enlarged by almost 30% and at the CASPT2level by 12%. Nevertheless, the absolute difference between the two approachesdoes not change with the larger active space.

The results obtained with the largest active space compare reasonably well tothe experimental number (J ~ 19.5 meV) for the magnetic interaction; about 70%is recovered. The remaining difference can partly be explained by the inadequate


TABLE 4.13 Mulliken gross population of the active orbitals for some active

spaces considered.

OrbitalOccu-pancy Ni-s Ni-p Ni-d

bridgeO-p

edgeO-s

edgeO-p

Active: Ni-3d, 3d' + O-2pσ

17 a2u 1.987 -0.01 0.00 0.17 0.43 0.01 0.39

18 b1u 1.000 0.00 0.00 0.97 0.00 0.00 0.03

19 a2u 1.000 0.00 0.00 0.85 0.07 0.00 0.07

20 b1u 0.012 -0.02 -0.01 1.03 0.00 -0.02 0.01

21 a2u 0.000 0.00 0.00 1.01 0.00 -0.02 0.01

Active: Ni-3d, 3d' + O-2pσ + O-2pσ'

17 a2u 1.987 -0.03 0.00 0.08 0.94 0.00 -0.01

18 b1u 0.999 0.00 0.00 0.97 0.00 0.00 0.03

19 a2u 1.000 0.00 0.00 0.90 0.08 0.00 0.02

20 a2u 0.013 -0.09 0.02 0.00 1.10 0.00 -0.03

21 b1u 0.001 -0.03 -0.01 1.02 0.03 -0.02 0.00

22 a2u 0.001 0.00 0.00 1.01 0.00 -0.02 0.01

representation of the direct environment of the bridging oxygen. It was foundthat the point charge approximation for the four nickel neighbours not includedin the cluster leads to an underestimation of the magnitude of the magneticinteraction. A representation by frozen charge distributions or pseudopotentials isexpected to bring the calculated value for J closer to the experimental value (seeTable 4.11).

Although it is sometimes important to calculate numbers that are in verygood agreement with experiment, it is often more appropriate to look at thetrends that can be reproduced, explained or even predicted with ab initio

computational methods. To illustrate the ability of reproducing trends accurately,we calculated the pressure dependence of the magnetic interaction by performinga series of calculations on the Ni2O11 cluster model in which the lattice parametera is varied. Experimentally, the dependence of J on a was determined to followthe power law: J ∝ a-ε, with ε = 9.9 ± 0.5 [55]. Table 4.14 lists the results of the


TABLE 4.14 Pressure dependence of J (in meV).

Ni-3d Ni-3d, 3d' + O-2pσ + O-2pσ'

a [Å] CASSCF CASPT2 CASSCF CASPT2

4.18 -3.16 -11.24 -3.94 -12.69

4.1639 (a) -3.31 -11.68 -4.13 -13.18

4.16 -3.34 -11.79 -4.16 -13.29

4.14 -3.53 -12.36 -4.40 -13.93

4.12 -3.73 -12.95 -4.64 -14.60

4.10 -3.94 -13.58 -4.91 -15.30

4.08 -4.16 -14.24 -5.18 -16.05

4.06 -4.40 -14.94 -5.48 -16.83

4.04 -4.65 -15.67 -5.79 -17.66

4.02 -4.91 -16.45 -6.12 -18.54

ε (b) -11.27 -9.74 -11.24 -9.71

a Lattice parameter used in all other calculations.b The standard deviation of ε is less than 0.06 at all levels of approximation.

calculations and it shows that already at the simplest level of approximation, theAnderson model, the power law is described rather satisfactorily and onincluding the electron correlation the agreement is even better. No significantchanges are observed when the description of the N-electron wave function isimproved by extending the active space. Therefore we conclude that independentof the question of how close the absolute calculated values compare toexperiment, trends in the magnetic interaction can not only be reproduced butalso predicted, as will be shown in the next section.


4.6 COMPARISON OF THE MAGNETIC INTERACTION IN BULK NIOAND AT THE (100) SURFACE

One general conclusion from the discussion of the magnetic couplingmechanisms in bulk NiO in the previous two sections is that with quantumchemical schemes a very reasonable estimate to the magnetic coupling can beobtained. Another conclusion is that quantum chemistry provides the tools toaccurately calculate various trends. This has been demonstrated in recent studieson this subject. For example for the systems studied in Ref. [48] it is observed thatJ is systematically underestimated, but going from compound to compound thetrend in J as observed in experiments is reproduced. Furthermore, the geometrydependence of J can be reproduced rather accurately. In various molecularsystems J was calculated as a function of the metal - ligand - metal angle [19, 24,25]. For some ionic solids the pressure dependence of J was reproducedsatisfactorily [23]. These observations allow us to extend the study of the magneticinteraction to new systems for which no experimental data exist yet.

One of these new systems is the (100) surface of NiO, for which, moreover, atheoretical prediction based on a model Hamiltonian was recently published byPothuizen et al. [64]. Following these authors J can be written as a function of theHubbard parameters V, U and ∆:

J ≅ -2V4

∆2

1∆

+ 1U

(4.16)

This expression predicts an increase for the magnetic interaction caused by thedecrease of the Madelung potential at the surface, which leads to a lower chargetransfer energy ∆. The increase of J going from bulk to surface was estimated byPothuizen et al. to be roughly a factor of 1.5.

In the present study the comparison of the magnetic interaction in NiO bulkand the (100) surface has been approached from a different point of view. Thematerial is represented by a cluster Ni2Ox embedded in the Madelung potential ofthe remainder of the crystal. Contrary to the model Hamiltonian study byPothuizen et al. [64] the exact (non-relativistic) N-electron Hamiltonian of thecluster is considered. The Hamiltonian is not replaced by a simpler, parametrizedexpression, but instead approximations to the eigenfunctions of the relevantstates are calculated in steps of increasing accuracy. We will discuss the effect onthe magnetic interaction of extending the active space. The material model,

4.6 Comparison of the magnetic interaction .... 141

graphically represented in Figure 4.3, chosen to represent the NiO (100) surfacecontained two nickels and the nine surrounding oxygens, Ni2O9. The rest of thecrystal was modelled by point charges obtained by the Evjen method.

Table 4.15 compares the magnetic interaction in bulk and surface obtained atdifferent levels of approximation to the exact N-electron wave function. First theAnderson model, represented by the CASSCF calculation with only the Ni-3dopen-shell orbitals in the active space. Within this approximation the magneticinteraction between two Ni2+ ions at the surface is only 80% of the interaction inthe bulk, -2.7 meV versus -3.3 meV. This decrease remains when dynamicalelectron correlation is included and also when the charge transfer effects from thebridging oxygen are more properly included in the wave function. At the mostsophisticated level of approximating the N-electron wave function a decrease ofthe magnetic interaction of 15% is observed going from bulk to surface. This incontradiction to the increase of 50% predicted based on the Hubbard model.

One of the important advantages of computational chemistry is the ability toperform numerical experiments and thus to have access to information that is

FIGURE 4.3 Ni2O9 cluster embedded in point charges. The thick lines link cluster

atoms; light large spheres represent nickel atoms, dark large spheres

oxygen atoms, while the small spheres represent positive (light) and

negative (dark) point charges.


TABLE 4.15 Comparison of J (in meV) for bulk NiO and NiO (100).

bulk NiO NiO(100)

CASSCF CASPT2 CASSCF CASPT2

Active: Ni-3d (4 orb. / 4 el.)

Es - Et -3.4 -12.3 -2.8 -10.4

(Es - Eq)/3 -3.2 -11.1 -2.6 -9.2

Average -3.3 -11.7 -2.7 -9.8

Active: Ni-3d, 3d' + O-2pσ + O-2pσ' (10 orb. / 6 el.)

Es - Et -4.2 -14.3 -3.7 -12.3

(Es - Eq)/3 -4.0 -12.1 -3.4 -10.2

Average -4.1 -13.2 -3.5 -11.2

impossible, or very difficult to obtain from experiments in the laboratory. Toanalyze the results listed in Table 4.15 and to find out what makes the magneticinteraction at the surface smaller than in the bulk two different analyses weremade. First, we investigated the influence of the Madelung potential on themagnetic interaction, which is large within the Hubbard model. Secondly, ananalysis was made of the effect of the number of oxygen neighbours of the nickelcentres in the crystal. An indication of a large dependency on the number ofneighbours is already given by the increase of the magnetic coupling with a factorof three going from the Ni2O to the Ni2O11 cluster model.

To isolate the effect of the Madelung potential on the magnetic interaction, Jwas calculated for a Ni - O - Ni cluster within the Anderson model with reducedexternal Madelung potentials. These calculations resulted in a J value of -1.4 meVfor the cluster in a bulk Madelung potential. The interaction is slightly increasedto J = -1.5 meV when the potential in the cluster region is reduced by 25% and to J= -1.6 meV for a reduction to 50% of the normal bulk Madelung potential. Thisshows that a reduction of the external Madelung potential in the cluster regionindeed leads to an increase in the absolute value of J, as the predictions based onthe Hubbard model indicate. Note, however, that the reduction of the Madelungpotential going from bulk to surface is only minor, 4% at the surface latticepositions, and is much smaller than the reductions tested here. Therefore theeffect of the Madelung potential on the magnetic interaction is expected to berather small.

4.6 Comparison of the magnetic interaction .... 143

FIGURE 4.4 Ni2O9 cluster embedded in point charges reproducing the Madelung

field of the bulk. The thick lines link cluster atoms; light large

spheres represent nickel atoms, dark large spheres oxygen atoms,

while the small spheres represent pos (light) and neg (dark) PCs.

In order to quantify how the oxygens directly surrounding the nickel ionscontribute to the difference in J for bulk and surface, a Ni2O 9 cluster wasembedded in a bulk Madelung field (see Figure 4.4) and J was calculated withinthe Anderson model. Going from Ni2O11 in a bulk Madelung potential to Ni2O9

in a bulk potential the nickel coordination is reduced from six to five, withoutchanging the Madelung field. Going from Ni2O9 in a bulk potential to Ni2O9

embedded in the surface potential, the Madelung field is reduced withoutchanging the nickel coordination. In this series the magnetic interaction is -3.3,-2.7 and -2.7 meV for the three different cluster models respectively. This provesthat the reduction of the Madelung potential only has a minor effect and that thelowering of the nickel coordination plays a key role in the difference of themagnetic interaction of bulk NiO and the NiO (100) surface.


One effect not yet considered in the calculations is the influence of surfacereconstruction on the magnetic coupling. Although the surface reconstruction isnot expected to be large (if present at all) even a small reconstruction mightchange a small quantity like the magnetic interaction significantly. Within thecluster model approach a hypothetical reconstruction is considered in which theNi - O - Ni angle is distorted by 1 and by 5 degrees, moving the O-ions outwards,and the Ni-ions inwards parallel to the surface normal. The magnetic interaction,calculated within the Anderson model, is found to decrease by a very smallamount, from -2.69 meV for the unreconstructed surface to -2.66 meV for adistortion by 1 degree, and to -2.20 meV for a 5 degrees distortion. These resultsindicate that a surface reconstruction would slightly decrease the magneticinteraction at the surface.

The conclusions of this section can be summarized in a concise way: twomechanisms were found that lead to a change of J going from bulk NiO to theNiO (100) surface. The first mechanism, the lowering of the Madelung potential,leads to a lower relative energy for the excitations from the bridging oxygen to thenickel centres, and from this point of view an increase of J is expected for the NiO(100) surface comparing it to the bulk value. Nevertheless in the present studythe magnetic interaction at the surface is found to be smaller. This is attributed toa second mechanism, the change in the nickel coordination by oxygens. Adecrease is found going from six oxygen neighbours in the bulk to five at thesurface. The net effect of the two opposing mechanisms is a decrease of J of about15% going from bulk NiO to the NiO (100) surface.

4.7 FINAL REMARKS AND SUMMARY

In the introduction of this chapter a connection was suggested between thetrue spin-adapted antiferromagnetic ground state and the experimentalinterpretation in terms of a Néel state. Within the unrestricted formalism of theHartree-Fock (UHF) theory the magnetic interaction is calculated as the differenceof a completely ordered ferromagnetic state and a broken spin symmetry state.This can be done either for molecules or complexes [65, 66], or for solid-statecompounds [29, 67-71]. Of particular interest are the results of the UHF bandcalculations [67-71] performed with the CRYSTAL program [72, 73]. This approachexplicitly calculates the energy of one of the Néel states and compares it to theenergy of the state with all spins aligned parallel. Unfortunately, there is no RHF

4.7 Concluding remarks and summary 145

implementation of the method. Nevertheless, the comparison of UHF bandcalculations with cluster calculations —both restricted and unrestricted— cangive a numerical justification of approximating the true ground state by the spinsymmetry broken Néel state. Recently, the magnetic coupling parameter J of theTM materials KNiF3 and K2NiF4 has been calculated within these three differentapproximations [29, 70, 71]. The analysis in Ref. [29] of the results clearly indicatesthat within certain small errors the magnetic coupling parameter J is identical forthe three models.

Another matter of concern is the generalization of the exact description ofthe magnetic interaction of a two-electron system (equation (4.3)) to a manyelectron, many centre system (equation (4.4)). Besides the non-linear termsdiscussed by Nesbet [16], also the more than two-centre interactions and the next-nearest (and further) interactions are neglected in this generalization. Althoughthe effect of both approximations is expected to be rather small, it can explicitly betested by extending the cluster model with more magnetic centres. In recentcluster model studies [48] we showed that irrespective of the number of magneticcentres the value of J was almost unchanged from clusters with only twomagnetic centres. For example, the difference between J calculated from a Ni2O11

cluster and a Ni3O16 cluster (both representing bulk NiO) is only 0.3 meV. In thelatter cluster with three magnetic centres both next-nearest neighbour and morethan two-centre interactions are in principle accounted for, but from the results itcan be concluded that neither of them is very important. For other ionic TMmaterials the same trend was observed in this study.

In this chapter we have presented a quantum chemical approach for theinvestigation of antiferromagnetic interactions in TM materials. From a clustermodel to represent the material the magnetic interactions can be studied bymapping spin adapted wave functions with different spin-spin couplings on theHeisenberg Hamiltonian. It has been shown how these spin adapted wavefunctions are connected to the more approximate Néel state and other spinsymmetry broken states used to extract information about the magnetic couplingfrom spin unrestricted methods. The properties and numerical accuracy of thequantum chemical approach were tested thoroughly by applying the method tothe magnetic coupling in bulk NiO. We investigated the basis set dependency; theinfluence of optimizing the orbitals for the different spin states; the validity ofrepresenting the core electrons by pseudopotentials; and the difference betweensome quantum chemical approaches to account for the electron correlation


effects. From this exploration of the method a number of conclusions could bederived, which are presented at the end of section 4.4.

There are several different mechanisms that determine the size of J. First ofall the superexchange mechanism described by Anderson and Nesbet [13, 15] wasrecognized. The influence of the edge oxygens is rather large, but in contrast tothe superexchange interaction mediated by the bridging oxygen, it is completelystatic in nature. An additional contribution to the magnetic coupling was foundwhen the point charges surrounding the bridging oxygen were replaced by frozenions or model potentials. The magnitude of J is strongly increased when electroncorrelation is accounted for. By performing a series of calculations with differentactive spaces in the way described in Chapter 1 we attempted to estimate theimportance of charge transfer configurations. Although an increase of J wasfound when the CT configurations were accounted for (including a certain degreeof relaxation), we did not observe the drastic increase of J that was found in theNOCI calculations of van Oosten et al. [26, 27]. The fundamentally differentconcepts of the two quantum chemical schemes makes it rather complex tocompare the results with respect to the importance of CT configurations.

After determining the pressure dependence of J for bulk NiO, which wasreproduced in good agreement with experiment, we estimated the size of J for theNiO (100) surface. The magnetic interaction was calculated to be about 15%smaller than in the bulk, contrary to the theoretical prediction of Pothuizen et al.[64] on the basis of a model Hamiltonian. Our analysis shows that the effect of thechange in Madelung potential going from bulk to surface is very small, while thedecrease in the number of oxygen ions directly surrounding the Ni ions hasmuch more influence.

4.8 References 147

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