INVITED ARTICLE

www.rsc.org/pccp | Physical Chemistry Chemical Physics

A unified view of the theoretical description of magnetic coupling in molecular chemistry and solid state physics Ibe´rio de P. R. Moreira and Francesc Illas Received 7th November 2005, Accepted 20th January 2006 First published as an Advance Article on the web 13th February 2006 DOI: 10.1039/b515732c The magnetic interactions in organic diradicals, dinuclear inorganic complexes and ionic solids are presented from a unified point of view. Effective Hamiltonian theory is revised to show that, for a given system, it permits the definition of a general, unbiased, spin model Hamiltonian. Mapping procedures are described which in most cases permit one to extract the relevant magnetic coupling constants from ab initio calculations of the energies of the pertinent electronic states. Density functional theory calculations within the broken symmetry approach are critically revised showing the contradictions of this procedure when applied to molecules and solids without the guidelines of the appropriate mapping. These concepts are illustrated by describing the application of state-of-the-art methods of electronic structure calculations to a series of representative molecular and solid state systems.

1. Introduction Magnetic properties of matter have attracted the attention of scientists for many years, although the first rigorous descriptions arise from the application of quantum mechanics.1–4 Isolated molecules may exhibit magnetic properties, the paramagnetic character of the oxygen molecule in its electronic ground state or the existence of organic radicals being textbook examples.5 Transition metal complexes of different nuclearity may display magnetic properties and sometimes these are referred to as single-molecule magnets.6,7 In these systems, magnetic properties arise from the interactions between metal paramagnetic centers rather than from a simple sum of the magnetic moments of the individual atoms. The pioneering work of Bleaney and Bowers on the Cu dimer acetate8 already shows that the deviations from a simple paramagnetic system found earlier by Guha in 19519 are due to the interaction between the so-called Cu21 Cramer’s doublets. This work no doubt opened the way for a large number of studies on single-molecule magnets. However, the late eighties and early nineties have witnessed the development of a new exciting field in which spontaneous magnetization has revealed its presence in molecular based materials.10,11 This represented a turning point because of the appearance of a rich variety of magnetic systems with different dimensionalities. The continued research in this field has resulted in novel syntheses exploring various mixed metal assemblies which combine different metal cations and organic ligands12 and may include many spin carriers.13 The building blocks of these materials are organometallic complexes containing transition metal atoms with localized unfilled d orbitals and also p orbital spin bearing organic radicals.14 New materials taking advantage of

Departament de Quı´mica Fı´sica & Centre Especial de Recerca en Quı´mica Teo´rica, Universitat de Barcelona i Parc Cientı´fic de Barcelona, C/ Martı´ i Franque`s 1, E-08028 Barcelona, Spain

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the simultaneous existence of localized magnetic moments and electric conductivity have also been prepared.15 These new classes of molecular materials exhibit interesting properties, which make them good candidates for technological applications such as magnetic storage or optical devices. In particular, molecular based magnets can exhibit phase transitions at a given critical temperature that indicate the existence of long range order as in typical ferro-, ferri- or antiferromagnetic inorganic crystals such as CrO2, Fe3O4, or NiO, respectively. The same kind of phase transition is observed for a broad family of copper16,17 and manganese oxides.18 Upon doping, this particular class of cuprates exhibits high-Tc superconductivity19 while doped manganites exhibit colossal magnetoresistance.20 The fascinating, yet unforeseen, properties of the molecular-based magnets, superconducting cuprates and manganites have raised enormous interest since they are expected to open the way to new technologies. These potential applications triggered a huge amount of research work devoted to understanding the nature of these compounds and the mechanisms governing their remarkable properties. The challenge is enormous and all efforts are welcome. In this sense, the simultaneous use of concepts and methods of molecular chemistry and solid state physics is expected to produce the necessary synergy. From the preceding digression one readily realizes that magnetic interactions are at the heart of the prominent properties of molecular based magnets,6,7,14,21,22 high-Tc superconductors16,17 and manganites18 but they also dominate the chemistry of di- and polynuclear magnetic complexes6,7 and radicals.5–7,21,23 The latter may indeed have important biochemical implications. Magnetic states of metalloproteins, DNA and RNA damaged by radicals, or protein oxidation adducts with radicals are of increasing interest because of their relevance in the biological activity of important carcinogenic and antitumor compounds.24–32 An in depth examination of the specialized literature reveals that the fundamental physics Phys. Chem. Chem. Phys., 2006, 8, 1645–1659 | 1645

governing the magnetic properties of all these systems is essentially the same although often described using different languages for identical physical concepts. For many magnetic systems, the magnetic moments are well localized on a given atom or group of atoms, referred to as magnetic centers or magnetic sites. These centers are characterized by a given effective magnetic moment, Si, which depends on the actual electronic configuration of the magnetic center; for instance, the d8 configuration of Ni21 in NiO or KNiF3 corresponds to an effective magnetic moment of S = 1.6,7,21–23,33,34 The physical description of magnetic coupling, or exchange coupling as it is also often termed, in a broad class of chemical compounds including organic biradicals, inorganic complexes, and ionic solids is based on the use of the well-known phenomenological Heisenberg–Dirac–van Vleck (HDVV) Hamiltonian.6,21–23,33,34 This Hamiltonian describes the isotropic interaction between localized magnetic moments Si and Sj as ^ HDVV ¼  H

X

^j; ^i  S Jij S

ð1Þ

hi;ji

where Jij is a constant giving the magnitude and type of interaction between the Si and Sj localized spin moments (in non-electronically degenerate centers) and the hi,ji symbol indicates that the sum extends to nearest neighbor interactions only. The HDVV Hamiltonian describes the low energy spectrum arising from the interactions of Si and Sj. It is worth pointing out that these magnetic interactions are of quantum mechanical nature and, in general, much stronger than classical interactions between magnetic dipoles. According to the spin Hamiltonian in eqn (1), a positive value of Jij corresponds to a ferromagnetic interaction, thus favoring a situation with parallel spins. The set {Jij} of parameters (their number and magnitude) defining this magnetic Hamiltonian characterizes the magnetic ordering of the system and permits one to describe the lowest part of the excitation spectra of magnetic systems. The sign and magnitude of the relevant (large enough) Jij parameters result from the particular electronic structure that, at the same time, determines the stable crystal structure of the system. Hence, the magnetic order and the crystal structure of the system are consequences of the actual electronic distribution. Nevertheless, one must highlight the fact that in some cases a general spin Hamiltonian containing additional terms may be needed.22,35 The effective Hamiltonian theory described in the next section provides a rigorous way to deduce the spin Hamiltonian of a given system. Usually, the different J parameters of the HDVV Hamiltonian are extracted from experimental measurements such as magnetic susceptibility curves, heat capacity curves or neutron diffraction experiments and by assuming a given form of the spin Hamiltonians. In this way, trial and error procedures are commonly used to fit results and intuition is invoked to choose the relevant interactions. Theoretical studies are clearly needed to understand the physical origin of the magnetic coupling and to provide unbiased predictions about the relative importance of the different terms. Here, it is important to point out that theoretical studies based on first principles or ab initio methods of electronic structure play a very important role although 1646 | Phys. Chem. Chem. Phys., 2006, 8, 1645–1659

they also suffer from some limitations. Rigorously speaking, a fully relativistic formalism should be employed. However, the complexity of the n-electron relativistic problem does not permit the required calculations to be carried out in the systems of interest. Still, one can use the non-relativistic Hamiltonian and handle magnetic interactions through a proper introduction of spin coordinates and spin symmetry although the price to be paid is that anisotropy, Dzyalozshinski–Moriya, spin canting and similar effects cannot be taken into account.6,7,35 This review focuses precisely on the theoretical studies of magnetic coupling in radicals, inorganic complexes, molecular magnets and magnetic solids in which the magnetic structure can be explained without invoking these explicit relativistic effects. Therefore, we focus on magnetic interactions in systems with localized magnetic moments where magnetic coupling usually results from electron–electron correlation effects.6,21,34–37 Nevertheless, it is fair to acknowledge the few efforts towards an ab initio description of magnetic coupling including spin–orbit effects.38 The aim of this paper is to provide a unified point of view of the magnetic interactions of the systems described above based on the quantum mechanical, non-relativistic, many-electron theory. To this end, section 2 describes how, for any particular system, a general spin Hamiltonian can be deduced from effective Hamiltonian theory. Next, section 3 couples effective Hamiltonian theory with accurate ab initio calculations enabling one to predict the amplitude of the effective Hamiltonian parameters without any further hypothesis. Section 4 shows that in many cases of interest the effective Hamiltonian parameters can be obtained from a one-to-one mapping of the eigenvalues or energy expectation values of the spin Hamiltonian and those of the exact non-relativistic Hamiltonian, thus justifying values of J obtained from the use of energy differences only. This is an important remark since the mapping procedure can be used either by wave function39–42 or density functional theory (DFT)43–47 based calculations, two commonly used computational methods of electronic structure. The fundamental role of spin restrictions imposed on the exact wave function and the corresponding reduced density matrices will be specially emphasized since neglecting it is a general source of misunderstanding. Section 5 describes the application of the above procedures to a variety of systems chosen as representative examples of the magnetic systems described above. Finally, section 6 presents the conclusions and further general remarks.

2. Spin models from effective Hamiltonian theory Spin models are of interest because they represent a convenient reduction of the problem of magnetic interaction while still preserving the ability to describe the low energy spectrum of magnetic systems and, hence, describe the corresponding macroscopic properties. From a more fundamental point of view, spin models are of interest because of their use in describing dynamic properties. Moreover, in some cases, they can be analytically solved.6,7,48 Spin Hamiltonians can be derived from the exact Hamiltonian through the application of the effective Hamiltonian theory.49–51 The idea behind the effective Hamiltonian theory is simple and elegant. For a This journal is


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system with a given exact Hamiltonian the mathematical structure of quantum mechanics ensures that there is a complete set of eigenfunctions satisfying the time-independent Schro¨dinger equation Hˆ|Cmi = Em|Cmi.

(2)

Usually, one is not interested in the whole spectrum but rather in a small number M of states. The target space S is simply defined by the projector operator on the space spanned by the M states of interest as in eqn (3) X P^target ¼ ð3Þ jCi ihCi j: i¼1;M

According to Bloch’s and des Cloizeaux original theory,52,53 (see also ref. 50 and 51) it is possible to define an isodimensional model space S0, by its corresponding projector Pˆ X ð4Þ P^ ¼ jFi ihCi j i¼1;M

on a given known {|Fii} basis and also an exact effective Hamiltonian acting on the model space and such that the M eigenvalues match exactly those of the exact Hamiltonian and the M eigenfunctions are the projections of the exact wave functions onto the model space

  ^ eff
PC ^ m ¼ Em
PC ^ m ; m ¼ 1; M: H ð5Þ Clearly, Hˆeff only permits the recovery of those states having significant projections onto the model space. The M equations in eqn (5) impose M þ M(M  1) conditions, i.e. they uniquely define the M2 matrix elements of Hˆeff. Schematically, the definition of a given effective Hamiltonian from the exact Hamiltonian can be viewed as that shown in Scheme 1. However, one must notice that the previous definitions do not provide a way to determine Hˆeff because this would imply the knowledge of the exact (or accurate enough) |Cmi states. Nevertheless, if suitable approximations to |Cmi and Em are available, then Hˆeff can be readily obtained from its spectral resolution, as given in eqn (6) X ^ eff ¼ ^ m iEm hPC ^ ?j H jPC ð6Þ m m¼1;M

where |PˆC> m i is the biorthogonal vector associated with |PˆC> m i. In principle, while the different state vectors fulfilling

Scheme 1

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eqn (5) are orthogonal, there is no reason, except for possible symmetry arguments, for the projections of these states onto the model space to be orthogonal.52 Nevertheless, it is always possible to orthogonalize these projections as suggested by des Cloizeaux53 and the corresponding effective Hamiltonian is indeed Hermitian. When the eigenenergies and the eigenvectors of Hˆ are not known, it is still possible to build the effective Hamiltonian from the model space through an order by order expansion, according to the quasi degenerate perturbation theory (QDPT) with a convenient choice of the model space and of the zero-order Hamiltonian Hˆ0.49,51,54–56 The effective Hamiltonian is related to the so-called wave ^ allowing one to obtain the exact wave function operator, O, from its projection on the lower dimensional model space ^ ˆ PˆT = OP

(7)

^ ˆ. Hˆeff = PˆHˆOP

(8)

or

The best choice of the target space is the one minimizing the norm of the wave operator ^ = min 8O8

(9)

and it is spanned by the M eigenstates having the largest linear independent projection onto the model space X   PC ^ i  ¼ max ð10Þ i¼1;M

Of course, the resulting effective Hamiltonian depends on the choice of the M states defining the target space and the effective Hamiltonian is uniquely defined by the choice of the model space and the knowledge of the target space eigenvectors and eigenenergies. However, the identification of the relevant M eigenstates satisfying this condition is not always straightforward. If the model space involves high energy states, they will appear with large coefficients in high energy eigenstates. These are frequently spread over a broad range of energies and the definition of the target space may become impossible. In this case, there is no simple mapping between the states in the model space and those in the target space. Consequently, it is not possible to reduce the physics to a simple effective Hamiltonian defined in that model space. Fortunately, for most of the magnetic systems with localized magnetic moments, a suitable choice of the model space consists of all determinants that can be constructed by maintaining the number of magnetic electrons and total spin per centre. In this way, the model space is simply made of all possible distributions of spin over the magnetic orbitals. Each one of these spin distributions can be represented by a Slater determinant written in a localized basis set of orbitals. The different Slater determinants differ only in the spin distribution in the open shells and share a common orbital space. Moreover, since the space part is made of atomic orbitals, the Slater determinants defined above can be viewed as neutral VB determinants because all of these have the common feature of keeping one electron per site. The set of all neutral VB determinants defines a convenient model space S0 for magnetic problems.55,56 Since all determinants in S0 have the same space part and differ only in spin distribution, the Phys. Chem. Chem. Phys., 2006, 8, 1645–1659 | 1647

corresponding effective Hamiltonian is isomorphic to a spinonly Hamiltonian.

3. Ab initio determination of effective spin Hamiltonians The spectral decomposition in eqn (6) opens the way to an ab initio determination of effective Hamiltonians provided one can obtain suitable approximations to both |Cmi and Em. Nevertheless, before discussing the numerical applications of effective Hamiltonian theory to the prediction of magnetic coupling constants, it is highly illuminating to use the QDPT to show the power of this theory as an analysis tool. Let us for instance assume that, for a given system with one electron per center, the Hubbard model Hamiltonian, defined in eqn (11), provides a suitable description of the low energy spectrum X X ^ Hubbard ¼ ^j þ a^j a^þ ^ia a^þ ^ib ð11Þ H tij ð^ aþ a^þ i a i ÞþU ia a ib a i

hi;ji

^ eff ¼  H

X

^i  S ^ j þ lðS ^i  S ^ j Þ2 g: Jij fS

ð16Þ

hi;ji

Here, tij is the intersite hopping integral, U the on-site effective two-electron repulsion and aˆ i1 and aˆ i the usual creation and annihilation quasiparticle operators. To second-order perturbation theory only two-body interactions appear. Therefore, it is rather easy to show that for the Hubbard model Hamiltonian in eqn (11), the effective Hamiltonian derived from second-order QDPT—using eqn (8) with the wave operator expanded up to second-order—applied to a model space consisting of neutral valence bond determinants (see next section) with different spin ordering has the form of the HDVV spin model Hamiltonian1–23 in eqn (1). The magnetic coupling constant takes the simple form given in eqn (12) Jij ¼ 

4t2ij : U

ð12Þ

However, one must point out that this is an approximate model since higher-order terms in the QDPT expansion can be relevant in some cases and, a priori, cannot be discarded. In fact, at the fourth-order expansion four-body operators appear by permuting all spins in a four-member ring,57,58 for instance a square or rectangular plaquette. These four-body operators can be formally written as Kijkl[(Sˆi  Sˆj)(Sˆk  Sˆl) þ (Sˆi  Sˆl)(Sˆj  Sˆk)  (Sˆi  Sˆk)(Sˆj  Sˆj)] (13) with Kijkl ¼ 

80t4ij ; U3

ð14Þ

a quantity which, in many circumstances, is not negligible.54,55 Hence, the general spin Hamiltonian reduces to X X ^j  ^i  S ^ j ÞðS ^k  S ^lÞ ^i  S ^ ¼ H Jij S Kijkl ½ðS i;j

In systems where each magnetic site bears two unpaired electrons in well defined and localized singly occupied (or magnetic) orbitals, as for instance in the case of the Ni21 cations in NiO or K2NiF4, similar four-body terms can appear, as shown recently in one of the few applications of the effective Hamiltonian theory.59 In this case, the crystal field fixes the two unpaired electrons in well defined atomic-like orbitals of eg symmetry. In this case, one has four electrons in four orbitals and while the usual Heisenberg Hamiltonian assumes that the two electrons gives rise to an effective particle with total S = 1 per magnetic sites, this assumes that the two unpaired electrons per magnetic site are ferromagnetically coupled in an atomic triplet state. This oversimplified picture neglects biquadratic terms which appear in the ab initio effective spin Hamiltonian as in eqn 16

i;j;k;l

ð15Þ

^ l ÞðS ^j  S ^ k Þ  ðS ^i  S ^ k ÞðS ^j  S ^ l Þ þ ::: ^i  S þ ðS Indeed, the application of effective Hamiltonian theory together with the embedded cluster approach has allowed Kijkl to be obtained for La2CuO4.57,58 Malrieu and Maynau have also pointed out that other higher-order terms can be important in other topologies.54–56 1648 | Phys. Chem. Chem. Phys., 2006, 8, 1645–1659

This proves that the resulting spin Hamiltonian does not necessarily reduce to the simple form of eqn (1) in agreement with the detailed analysis of Herring.60 Nevertheless, the numerical determination of the effective Hamiltonian for K2NiF4 shows that in this system the amplitude of the biquadratic terms is small. This provides ab initio support to the pioneering work of Nesbet61–63 who suggested that two magnetic centers with an unpaired number of electrons leading to a maximum total spin S per center should interact according to the Heisenberg Hamiltonian precisely with an effective spin S per center. The analysis of the final effective Hamiltonian also permits the checking of the consistency of the model. For instance, in the case when the model space is built up of unphysical magnetic orbitals the resulting effective Hamiltonian is also unphysical. Hence, the effective Hamiltonian theory also permits one to reach an unbiased definition of the magnetic orbitals. In spite of the predictive power of the effective Hamiltonian theory, this has seldom been used. The applications have been mainly carried out by Malrieu et al. and cover molecular magnets,64–66 cluster models of cuprates,58,67 nickel potassium fluoride59 and mixed valence compounds.68 The same group has employed the effective Hamiltonian theory to develop spin-only Hamiltonians for conjugated p systems.54–56,59,69 In part, the limited use of this elegant formalism comes from the need to find suitable approximations to all |Cmi in the target space. This implies that one must find a series of low energy roots of large configuration interaction (CI) matrices and this is not always straightforward. In the case of the effective Hamiltonian of K2NiF4, a new diagonalization technique had to be implemented to efficiently search for the desired roots of the CI problem.70 Fortunately, the dominant magnetic coupling constants can often be deduced from a simpler approach which only makes use of the lowest roots of the CI matrix or even just energy expectations values. The J values are then obtained from total energy differences. The justification for this energy difference approach is discussed at length in the next section. This journal is


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4. Effective parameters from ab initio calculations: mapping approaches 4A.

Mapping approaches within spin adapted wave functions

To restrict this section to a minimum extent while including the material necessary for following the forthcoming discussion and to understand its relationship with previous periodic calculations let us consider two particular cases. On one hand, systems with two magnetic centres with localized spin moments of S = 1/2 in each centre and, on the other hand, a more general case involving two centres with effective S = 1 spin moments. For these systems involving two magnetic centres only, the HDVV Hamiltonian takes the simple form, as in eqn (17) HˆHDVV = J Sˆ1  Sˆ2

(17)

and making use of the well known ladder operators Sˆ1 and Sˆ eqn (17) may be rewritten as   ^þ  S ^ þ S ^  S ^þÞ þ S ^ z1  S ^ z2 ^ ¼ J 1 ðS H ð18Þ 2 where Sˆ1 = Sˆ11 þ Sˆ12, Sˆ1 þ Sˆ2, and Sˆ11, (Sˆ1), are the spin-up (spin-down) operator for particle 1 and Sˆz1, Sˆz2 the operators for the z-component of spin for particles 1 and 2, respectively. Inorganic Cu dinuclear complexes and organic diradicals are representative examples of systems with two magnetic centres with S = 1/2. These systems have just one unpaired electron per centre and the only possible spin eigenstates are one singlet and one triplet with total spin S of 0 and 1, respectively. Since the spin states are eigenfunctions of both Sˆ2 and Sˆz the spin states are then denoted as |SMsi. Thus, we denote these eigenstates with two quantum numbers S and Ms. Hence, one has |00i and {|11i, |10i, |1i} for the singlet and three components of the triplet, respectively. In this very simple example the spin model space will consist of two neutral VB determinants |abi and |bai where each spin is localized on one center. One can nevertheless use the two combinations giving rise to |00i and |10i which span the same vector space, now in a delocalized description. Note that the spin model space constructed in such a way has a defined value of Sz. The next step consists of defining an isomorphic model space also constructed from two VB determinants but now including all electrons in the system. These two VB determinants will have the two open shell |abi and |bai configurations and with the open shells localized on each magnetic center. Alternatively, one can use any delocalized description of the open shell orbitals obtained by a suitable unitary transformation. In any case, there is a one-to-one correspondence between the spin eigenfunctions in the spin model space and the spin eigenfunctions built up from the VB N-electron determinants. This is because of the isomorphism between the Nelectron model space and the spin model space. Using eqn (18), it is straightforward to prove that |00i and |10i are eigenfunctions of the HDVV Hamiltonian in eqn (1) with eigenvalues of 3/4J and 1/4J, respectively.71–73 Mapping the N-electron E|Si and E|Ti singlet and triplet state energies onto corresponding eigenvalues in the spin model space leads This journal is


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to E|Si = þ3/4J and E|Ti = 1/4J and hence E|Si – E|Ti = (3/4J)  (1/4J) = J. Note that here E|Si and E|Ti are the total energies of the singlet and triplet states of the corresponding N-electron dinuclear complex or diradical. This is indeed the basis for the mapping approach widely used to predict magnetic coupling constants if the singlet and triplet energies can be computed with sufficient accuracy. The difference dedicated configuration interaction (DDCI)74,75 and multiconfigurational second-order perturbation theory (CASPT2)76,77 approaches are among the techniques which can provide the desired degree of accuracy36,37,78–83 whilst working with spin eigenfunctions. Density functional theory (DFT) based approaches can also provide useful information although the computed values exhibit a very large dependence on the choice of the exchange–correlation potential.84,85 Moreover, DFT approaches do not usually maintain both S and Ms spin quantum numbers.71,72,86,87 This is a source of problems and misunderstanding as will be discussed in the next section. Next, let us consider the slightly more complex case of two particles with total spin S = 1; NiO and the KNiF3 or K2NiF4 perovskites constitute representative examples of this kind of system. The electronic configuration of the Ni21 cations can be effectively described as a near 3F atomic state arising from the d8 open shell configuration. Here, coupling the spin angular momentum of the two magnetic centres gives rise to the following spin eigenfunctions manifolds: one singlet |Si {|0,0i}, one triplet |Ti {|1,1i, |1,0i, |1,1i} and one quintet state |Qi {2,2i, |2,1i, |2,0i, |2,1i and |2,2i}. Again, using eqn (18) it is easy to prove that ES = 2J, ET = J and EQ = –J, respectively.88–90 It is easy to define an isomorphic model space now constructed from six VB determinants including all electrons in the system but with the four open shells in |ababi |abbai |baabi|babai|aabbi and |bbaai configurations and with the open shells localized on the magnetic centers. In this way, there is a one-to-one correspondence between the |Si = |00i, |Ti = |10i and |Qi = |20i spin functions and the lowest energy spin eigenfunctions built up by a suitable combination of the N-electron neutral VB forms obtained from suitable electronic structure calculations. Using eqn (18), it is straightforward to prove that |Si = |00i, |Ti = |10i and |Qi = |20i are eigenfunctions of the HDVV Hamiltonian in eqn (1) with the eigenvalues 2J, J and J, respectively. It is straightforward to show that, independently of the value of the Ms component, one has E|Si  E|Ti = J

(19)

E|Ti  E|Qi = 2J.

(20)

and

In this case, accurate calculations of the three spin states provide an internal test of the consistency of the results and whether or not the system behaves according to the HDVV Hamiltonian. For cases with two particles with total spin Si, one can easily prove that the energy difference between the different 2S þ 1 multiplet states (S = 2Si) of the dimer obeys the Lande´ rule given in eqn (21) E(S  1)  E(S) = JS.

(21)

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Deviations from the HDVV may indicate an inadequate computational approach or, if the latter can be trusted, that many-body effects cannot be neglected. In such a situation a careful analysis using effective Hamiltonian theory is mandatory. Here, let us point out that deviations from the Lande´ rule can occur when using the CASSCF method.91 However, this does not mean that the system does not behave according to the HDVV Hamiltonian as claimed by some authors.92 This is because the set of orbitals used to construct the wave function for each multiplet is effectively built on a different Hilbert subspace. Using the language of effective Hamiltonian theory outlined in section 2, each state is built upon a different model space thus making it impossible to define a suitable spin Hamiltonian. We end this section by mentioning that similar mappings can be derived for cases with more than two magnetic centers.88,89,93 4B. Mapping approaches within spin unrestricted formalisms In the spin polarized formalism one uses one single Slater determinant to approach the N-electron wave function as in unrestricted Hartree Fock (UHF) methods or to describe the density of the N non-interacting electrons as in the spin polarized version of the Kohn–Sham implementation of DFT. In either of these two approaches, the total square spin operator is not properly defined and a space and spin broken symmetry solution is required to describe the open shell low spin state. Therefore, the mapping procedures described above are not valid and other relationships have to be searched for. This is an important limitation of periodic Hartree–Fock or DFT based studies where all of the computational methods implemented to date have used a one Slater determinant description of the electron density. From a strictly fundamental point of view, one can argue that since DFT is an effective one-electron theory one has no right to ask about the value of the total square spin, which is a two-electron operator. The consistent follow up to this line of reasoning is to give up using DFT to predict magnetic coupling constants since one cannot assign a defined multiplicity to the single determinant description of the low spin state. Still, one could argue that for a system with a total Sz = 0 the variational theorem ensures that even in the case of using a single Kohn–Sham determinant the self consistent procedure will converge to the lowest singlet state which has precisely Sz = 0. This has indeed been claimed by a group of authors to justify the good agreement with experiment of magnetic coupling constants of various compounds computed by means of the B3LYP functional.94,95 While this can be a convenient numerical recipe to unravel useful magnetostructural relationships,96 there is no fundamental justification for such a claim. On the contrary, there are many strong arguments against it. In the case of UHF, which can be considered as a particular case of DFT, one can analytically show that the single determinant solution for the low spin state lies approximately half way between the singlet and triplet.71 For the case of two particles with S = 1 there are two single determinant broken symmetry solutions |aabbi and |bbaai with two parallel electrons per site and with total Sz = 0, they are degenerate with respect to the exact non-relativistic Hamiltonian and 1650 | Phys. Chem. Chem. Phys., 2006, 8, 1645–1659

hence can be combined without changing the energy expectation value. It is easily proven that one of the combinations is indeed a spin eigenfunction with total spin square of 1 and hence represents a triplet state.72 It is hard to maintain that this broken symmetry solution provides an approximation to the singlet state. Even worse, for periodic systems such a hypothesis will lead to the unphysical conclusion that in an antiferrromagnetic solid the broken symmetry antiferromagnetic solution represents a singlet state. Finally, it has been shown very recently that B3LYP and other functionals lead to very poor results when spin restricted DFT calculations for the open shell singlet are carried out by means of the restricted ensemble Kohn–Sham (REKS) formalism.86 Therefore, the good agreement found for some compounds between experiment and B3LYP values, calculated assuming that the broken symmetry solution energy approaches that of the singlet state, are the result of error cancellation. The agreement is fortuitous and disappears when, within the same exchange–correlation potential, the singlet energy is evaluated using a method that properly defines the square of the total spin. The obvious message from the previous discussion is that the use of DFT to compute and predict magnetic coupling constants needs to be used with appropriate caveats. In fact, there are many interesting problems where DFT is the only possible way to investigate electronic structures within a minimum ‘‘first principles’’ basis. This includes periodic approaches in solid state physics which are all single-determinant by construction. A possible way to tackle the spin problem with the spin polarized formalism is again to make use of the mapping approaches described in the previous section, but conveniently modified. A possible way consists of further simplifying the spin Hamiltonian. In particular, the Ising model Hamiltonian, which results from neglecting the ladder operator terms in eqn (18), provides a suitable choice. For a two-particle system the Ising Hamiltonian is simply given by HˆIsing = J SˆzlSˆz2.

(22)

A mapping between electronic states representing approximate eigenfunctions of the exact non-relativistic Hamiltonian and the eigenfunctions of the Ising model Hamiltonian can be derived in quite a straightforward way. Note, however, that the eigenfunctions of the Heisenberg Hamiltonian in eqn (17) and (18) are not, in general, eigenfunctions of the Ising Hamiltonian given by eqn (22). Moreover, different spin eigenfunctions that are degenerate either for the exact or Heisenberg Hamiltonian exhibit different energy expectation values for the Ising Hamiltonian. For instance, for two interacting S = 1/2 particles one has h10|HˆIsing|10i = h00|HˆIsing|00i = J/4 whereas for the exact Heisenberg Hamiltonian those functions are non-degenerate. For two interacting S = 1 particles, h20|HˆIsing|20i = J/3 but h22|HˆIsing|22i = J, whereas the exact and Heisenberg Hamiltonians |20i and |22i are, obviously, exactly degenerate. After some simple algebra it can be proven that some particular eigenfunctions of the Heisenberg Hamiltonian are also eigenfunctions of the Ising model Hamiltonian. For example, it can be shown that ^ Ising j1  1i ¼  J j1  1i for two interacting S = 1/2 partiH 4 cles and HˆIsing|22i = J|22i and HˆIsing|10i = þJ|10i for two interacting S = 1 particles. This journal is


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Let us now consider again the case of two particles with S = 1/2 and the |aai and |abi or |bai single Slater determinants, which in practice are computed as approximations to the ferro- (|Fi) and antiferromagnetic (|AFi) coupling of the two magnetic centers. It is easy to show that within the Ising Hamiltonian E|Fi = 1/4 J and E|AFi = þ1/4 J and hence E|AFi  E|Fi = J/2.

(23)

Now, note that the |Fi and |AFi states are isomorphous to the high spin and broken symmetry low spin solutions that can be obtained by spin polarized calculations (either UHF or UDFT). Therefore, eqn (23) allows one to obtain J from energy differences as in the case discussed in the previous section. For the case of two particles with S = 1 one can readily prove that E|AFi  E|Fi = 2J

(24)

where |AFi = |aabbi or |bbaai and |Fi = |aaaai or |bbbbi. The mapping procedure can be easily extended to two particles with a general S or to cases where one particle has spin S1 and the other one has spin S2.97 In this case one has 2 EjAFi  EjFi ¼ Smax

J 2

ð25Þ

where Smax is just S1 þ S2. A general development of these ideas for two-center spin models with S up to 5/2 particles has been reported recently by Whangbo and Dai.97 The mapping procedure just described is the basis of a method suggested earlier by Noodleman et al.,98–100 which was also initially in the framework of Xa. Later, Yamaguchi et al.101–103 used the same idea to predict magnetic coupling constants through the UHF method. The broken symmetry approach has since been used by many authors and has been the subject of many research papers, although it has not always been used carefully.71,72,86,104,105 The mapping procedure between low spectrum energy states of the exact non-relativistic Hamiltonian and those of the Ising model Hamiltonian provides a rather rigorous way to extract magnetic coupling constants using spin unrestricted formalisms. Still, the approach has a clear weak point. It assumes that the magnetic coupling constant of the Heisenberg and Ising models is the same. In the following, we will present a new line of reasoning which circumvents this inconvenience. The idea behind the mapping between eigenstates of the Heisenberg and exact Hamiltonians is the one-to-one correspondence described above. Instead of finding mappings between the energies of the eigenstates of the exact and model (Heisenberg or Ising) Hamiltonians one can think of finding mappings between expectation energy values for given trial functions. This is precisely what is done in practice to find a variational approximation to the energy of the |Fi and |AFi trial functions. Hence, one can derive the expectation value of these two trial functions for the Heisenberg Hamiltonian. For the case of two particles with spin S = 1/2 one finds ^ HDVV jaai ¼  J hEjFi i ¼ haajH 4

ð26Þ

J : 4

ð27Þ

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Subtracting these two equations one finds hEjAFi i  hEjFi i ¼

J : 2

ð28Þ

Note that eqn (28) is equivalent to eqn (23), although in the former one compares the expectation value of the exact and Heisenberg Hamiltonians whereas in the latter one uses the expectation values of the exact Hamiltonian for the |Fi and |AFi states and map the difference to the exact spectrum of the Ising Hamiltonian. We have just proven that in the case of two particles of spin S = 1/2 the two approaches are equivalent. This adds further strong support to the claim that for the case of two particles with spin S = 1/2, the correct way to extract J from spin polarized calculations is to use either eqn (23) or eqn (28). There is no evidence to support the use of E|AFi = E|Si in spin polarized calculations. It is straightforward to prove that for the situation where there are two particles with S = 1 one has hE|Fii = haaaa|HˆHDVV|aaaai = J

(29)

hE|AFii = haabb|HˆHDVV|aabbi = J

(30)

and

and subtracting these two equations one finds hE|AFii  hE|Fii = 2J

(31)

which is equivalent to eqn (23). The algebra involved in the Ising Hamiltonian is by far simpler than that necessary for the Heisenberg Hamiltonian. Hence, from a practical point of view the latter is recommended. 4C. Mapping approaches for periodic systems Let us now return to the Ising model, but now assuming that instead of two interacting particles with spin S one needs to deal with a periodic array of such particles. If we assume additivity of the two-body interactions between nearest neighbours only and that each magnetic centre interacts with z neighbours, eqn (23) and (28) reduce to E|AFi  E|Fi = 2zS2J

(32) 106,107

in which is precisely the equation used by Dovesi et al. their periodic Hartree–Fock study of magnetic coupling in KNiF3 and K2NiF4. More recently, this approach has been used by many other authors, as described in section 5C. We must point out that eqn (31) is consistent with the form of the Heisenberg Hamiltonian in eqn (17) and (18). To close this section, we just remark that using the periodic approach it is possible to obtain the relevant coupling constants. Hence, one does not have to restrict the search to nearest neighbour interactions. Again, the use of expectation values for the |Fi and various |AFi broken symmetry solutions is the clue to the extraction of the various parameters of a general spin Hamiltonian with many effective two-body interactions. To this end one needs to find consistent field (SCF, using either Hartree–Fock or DFT) solutions representing |FMi = mmm. . .mi and various |AFM(i)i = |mk. . .kmki phases with different spin alignments along the relevant dimensions.108 Phys. Chem. Chem. Phys., 2006, 8, 1645–1659 | 1651

5. A critical survey of representative examples For any finite system such as organic diradicals or single molecule magnets, the most accurate approach consists of carrying out large configuration interaction calculations such as DDCI74,75 for the spin states of interest and using the mapping procedures outlined in the previous sections. Alternatively, one can still make use of second-order perturbation theory applied to multireference wave functions for the spin states of interest as shown by de Graaf et al.78,79 However, the very large computational resources needed to carry out this type of calculation as well as the necessary skills to properly handle these calculations resulted in a limited number of studies. On the other hand, the computationally efficient ‘‘quick and dirty’’ procedure inherent to the broken symmetry approach triggered a much larger number of studies although these were not always carried out with the necessary care. In fact, note that for the very simple H–He–H system, a paradigm of superexchange interaction for which the exact solution with a given basis set is known,109,110 none of the standard available exchange–correlation potentials was able to reproduce the exact solution.71,86,87,104 Nevertheless, the broken symmetry approach still provides the only possible way to compute magnetic coupling constant in periodic systems using band structure calculations on periodic models. The recommendation to use CI based calculations does not only come from its theoretically well-grounded correctness but also from the need to unravel the physical mechanisms behind a given magnetic coupling. One does not only need to reproduce or anticipate an experimental result, understanding is also a necessary and fundamental scientific task that cannot be disregarded. CI wave functions offer a better way of analysis as evidenced in a couple of elegant papers by Calzado et al.65,66 extending the pioneering work of de Loth et al.64 On the other hand, broken symmetry DFT calculations may provide useful information and reliable magnetostructural correlations but the mapping approach also provides the theoretical ground for a proper comparison to experiment. In the following section, we illustrate the above ideas with examples from the recent literature on organic and inorganic magnetic molecules. For pedagogical reasons we will limit the discussion to cases where results are available simultaneously from CI and DFT calculations. A number of examples taken from solid state physics will also be discussed. This will permit us to illustrate that in the three families of compounds the underlying physics is essentially the same, to describe state-ofthe-art calculations for these kinds of systems, and to critically revise the strength and limitations of each approach. We hope that this will be useful to experimentalists trying to gain a deeper understanding of the theoretical methods used to extract the magnetic coupling parameters and to provide a critical reading of the theoretical calculations reported in the literature. We also hope to help computational chemists and solid state physicists choose appropriate computational frameworks or to at least highlight the limitations of different computational methods. 5A.

Organic magnetic systems

The 1,1 0 ,5,5 0 -tetramethyl-6,6 0 -dioxo-3,3 0 -biverdazyl biradical (Fig. 1) is one of the few organic systems which has been 1652 | Phys. Chem. Chem. Phys., 2006, 8, 1645–1659

Fig. 1 Schematic representation of the geometry of the 1,1 0 ,5,5 0 tetramethyl-6,6 0 -dioxo-3,3 0 -biverdazyl biradical.

studied by a complete set of theoretical methods. The experimentally measured singlet–triplet difference is in the [95,110] meV range depending on whether the measure is carried out in solution or in the crystal.111 The molecular structure is also known from X-ray measurements and corresponds to a geometry where the two aromatic rings are coplanar. Note, however, that even if the biverdazyl radical is planar in the solid state, vibrational averaging by torsional motion remains important and only a proper consideration of this effect allows solid state and solution results to be reconciled; otherwise, assuming that the radical is planar in the former case and significantly distorted in the latter could not be reconciled with the similar magnetic properties measured in both situations.112 The size of the biverdazyl radical is large enough to prevent DDCI calculations from being carried out. Nevertheless, CASPT2 calculations using a moderately large basis set and a minimal active space consisting of 2 electrons in 2 orbitals predict an antiferromagnetic coupling of 151 meV. This too large value can be attributed to a deficiency of the zero-order wave function. In fact, a more accurate selection procedure of the active orbitals and electrons leads to a complete active space involving 14 electrons in 12 orbitals. The corresponding CASPT2 singlet–triplet gap calculated with the enlarged CAS now becomes 114 meV,78, which is very close to the experimental result. This is the typical behavior of ab initio wave function based calculations. Improving the calculation in a systematic way leads to a systematic improvement of the calculated property. Now, let us briefly discuss the results reported by Barone et al.113 from broken symmetry based calculations. These authors have used the B3LYP exchange–correlation functional and have been able to compute the magnetic coupling in vacuo and in Cl3CH solution. Using the same mapping approach discussed above they found magnetic coupling values of 70 and 93 meV, respectively, but used a rather different geometry which deviated considerably from planarity. Averaging over all values corresponding to temperature accessible energy variations of the torsional angle led to values of 89 and 104 meV at 298 K for the ‘‘in vacuo’’ and in CCl3H solution, respectively. Thus, Barone et al. reproduce the experimental magnetic coupling constant and the difference between the measurements in solution and in the crystal. Nevertheless, a caveat is necessary since the calculations in the absence of solution are carried out for the molecule ‘‘in vacuo’’ and not for the crystal structure. Hence, the This journal is


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CASTP2 values commented on above and the broken symmetry results of Barone et al.113 are not directly comparable. A proper comparison requires the use of the same molecular geometry and when this is done the results are quite surprising. The B3LYP broken symmetry calculated magnetic coupling constant is of the order of 150 meV, almost twice the experimental value. The same value is reported by Barone et al.113 for the coplanar geometry. The conclusion from this comparison is that the broken symmetry B3LYP seems to overestimate the magnetic coupling constant between centers with localized S = 1/2 magnetic moments unless one incorrectly assumes that the B3LYP broken symmetry solution provides the singlet energy. To definitively prove that this is not the case the triplet and open shell singlet energies have been calculated using the spin adapted restricted open shell Kohn–Sham (ROKS)114 and REKS115 methods, respectively, and for the B3LYP functional. For the planar structure, the calculated magnetic coupling constant is 59 meV; which is far enough from the experimental result to cast serious doubt on the accuracy of the B3LYP calculated magnetic coupling constants. In the next sections we will provide further evidence for such a conclusion. The discussion above merits a further, important comment. The good agreement between calculated B3LYP values and experiment is only achieved when forcing the adoption of a non-planar geometry, which does not correspond to the crystal structure. The experimental measurement of the magnetic coupling corresponds precisely to the crystal structure. In section 5C we will provide compelling evidence that the magnetic coupling constant is a local property and, hence, it can be properly reproduced by a local model (in this case, the isolated molecule at the crystal geometry). The poor agreement between properly calculated B3LYP magnetic coupling constant values and experiment is perhaps not so surprising. In fact, B3LYP is a hybrid DF method that mixes Fock and the gradient-corrected Becke exchange116 and the Lee–Yang–Parr correlation functional.117 However, the mixing has been parameterized to reproduce the thermochemistry of a large set of organic molecules116 and there is no reason why this parameterization should also work for the energy differences involving magnetic states. Still, there is increasing evidence that time dependent DFT based on the B3LYP functional provides a good estimate of the optical spectra of molecules118–120 and even of point defects in solids.121–123 For these reasons, the B3LYP method can be considered as a semiempirical approach, as it works well for the properties where it has been parameterized and for some extra properties not included in the parameterization, but a general correct behaviour cannot be guaranteed. Here, let us just add that Barone and Adamo realized that the key to the success of the B3LYP functional is the use of an amount of Fock exchange close to 25%. This led them to define the PBE0 functional with only one empirical parameter, an ad hoc 25% admixture of Fock exchange with no further fitting to experiment.124 This new functional produces results which are similar to those of the B3LYP although it can be considered as less empirical since it depends on just one parameter. We will come back to this point later on when discussing magnetic coupling on inorganic solids. This journal is


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The difficulties of DFT to describe magnetic coupling in organic radicals are not restricted to the B3LYP potential, as shown by the work of Barone et al.125 on the 1,4-bis(4 0 ,4 0 dimethyloxazolidin-N-oxyl)cyclohexane biradical, a typical nitroxide biradical. These authors also show the extreme dependence of the calculated result on the exchange–correlation potential chosen to carry out the DFT calculations. This is in line with the findings of Martin and Illas for inorganic solids.84,85 The problems encountered by DFT to describe the magnetic coupling in radicals are likely to be related to the multireference character of the wave function for the open shell singlet.126 This has been shown to be the case for didehydrogenated pyrene127 and other radicals.128,129 From the preceding discussion, one can suggest that for very large systems where DDCI or CASPT2 calculations cannot be carried out, the B3LYP broken symmetry approach, always within the appropriate mapping procedure, provides at least a cheap but often unreliable alternative. Nevertheless, one must add that even if the B3LYP calculated absolute values of magnetic coupling constants significantly differ from experiment, the trends along a given series are usually well reproduced and for this purpose the B3LYP method can be a good choice. The results above illustrate how delicate it is to compute the magnetic coupling constants, in particular in the DFT framework, and indicate that further theoretical development is necessary. In this sense, we quote the initiatives aimed at introducing dynamic correlation through DFT applied to ad hoc representable ‘‘spin adapted’’ densities arising from CASSCF wave functions of defined multiplicity. The resulting methods are generally termed CAS-DFT130–132 and provide a promising alternative. Nevertheless, Cremer and Filatov have already pointed out the problems of finding a suitable correlation potential and have shown that the widely used LYP correlation potential is not adequate to describe magnetic properties.133

Fig. 2 Schematic representation of the geometry of the diaquo-tetram-acetato-dicopper(II) complex.

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5B. Inorganic magnetic systems The first molecular magnetic material synthesized, the Cu dimer acetate complex (diaquo-tetra-m-acetato-dicopper(II); Fig. 2), is a dinuclear Cu(II) antiferromagnetic complex which has become a classic example of the subtle mechanisms underlying the magnetic interaction only accessible by including nearly all dynamic correlation effects. This compound has been well characterized experimentally and the reported magnetic coupling constant is antiferromagnetic with J = 37 meV.8,134 Moreover, this is a theoretically interesting system for testing the models and methods proposed to describe magnetic coupling because of the strong competition between direct exchange, superexchange and higher-order mechanisms.64 A series of values have been obtained using the ab initio wave function based methods of decreasing accuracy DDCI, CASPT2, CASSCF and of broken symmetry DF calculations using different exchange–correlation potentials.135 The DDCI method applied to this compound at the experimental geometry predicts J = 28 meV, which is rather close to the experimental value but clearly below it. The underestimation of J by the DDCI method can be attributed to the limited basis set and one can ensure an improved prediction by improving the basis set. The CASPT2 method predicts J = 14 meV, with the difference between this and the DDCI result due to the use of a minimal complete active space with two electrons and two orbitals only. The secondorder correction applied to a contracted CASSCF wave function is not enough to recover all important dynamic correlation effects. A solution to this problem is to enlarge the CAS, but this has a concomitant loss of interpretative power.78 The extremely important contribution of dynamic correlation effects is precisely evidenced by the CASSCF value for the magnetic coupling constant, J = 3 meV. Here it is important to point out that the UHF method with the broken symmetry approach and the proper mapping (see section 4B) predicts J = 7 meV. The spin polarization introduces configuration mixing and hence non-dynamical as well as a small part of dynamic correlation effects. This statement can easily be proven by inspection of the UHF natural orbitals for the broken symmetry solution.136 The need for intensive inclusion of dynamic correlation and, hence, the ability to go beyond the CASSCF approach was already pointed out in the pioneering study of de Loth et al.64 Later on, this fact has also been shown in many other binuclear complexes such as Cu2–oxalato,137 binuclear copper(II) compounds bridged by a ferrocenecarboxylato(1) and a hydroxo or methoxo ligand,138 Ni2–oxalato,139 Na6Fe2S6,140,141 oxamido-bridged Mn(II)Cu(II) heteronuclear complexes,142 linear oxo-bridged heterobinuclear complexes of titanium(III), vanadium(III), and chromium(III)143 and sulfur-bridged binuclear Ni(II) complexes144 among others. The need to include dynamic correlation effects triggered also many studies (see, for instance, ref. 94–96, 104, 105, 109, 135, 145 and 146) aimed at taking advantage of the promise of an accurate value provided by the Hohenberg and Kohn theorem and the practical Kohn–Sham implementations of DFT. Therefore, let us go back to the Cu2–acetate complex and direct our attention to the results predicted by broken symmetry DF calculations. 1654 | Phys. Chem. Chem. Phys., 2006, 8, 1645–1659

The predicted J values range from 262 meV for the local density approach (LDA) to the exchange–correlation functional to 74 meV for the B3LYP,135 which are all quite far from the experimental or DDCI values. Still, one may argue again that the B3LYP broken symmetry solution provides the energy of the singlet state and, in this case, the predicted J value is J = 74/2 = 37 meV, in excellent agreement with experiment. This goes against all of the theoretical arguments provided in section 4B based on the mapping procedure applied between either the high spin and broken symmetry solution and the corresponding eigenfunctions of the Ising Hamiltonian or between the former and the expectation values of the Heisenberg Hamiltonian. The forthcoming discussion will definitively prove that this line of reasoning is not sustained by theoretical arguments. At most one can accept that it is a numerically convenient approach permitting one to deduce important magnetostructural correlations and, hence, providing very useful information to experimentalists.94,95 The [Cu2Cl6]2 dinuclear complex (Fig. 3) has also been a prototype for inorganic molecular complexes and has been studied by a variety of methods.71,72,74,75,86,87,109 However, depending on the counterion, the experimental values of the magnetic coupling constants lie over a broad range. Nevertheless, this system is convenient because it has also been studied by the spin restricted ROKS/REKS implementations of DFT. This is indeed a difficult case because the dominant interaction is governed by dynamic correlation effects to the point that CASCI calculations, carried out with the molecular orbitals of the triplet state, or CASSCF calculations, with orbitals optimized for each spin state, predict a qualitatively incorrect ferromagnetic coupling constant. The DDCI method correctly predicts the compound to be antiferromagnetic and J E 1 meV, which is well within the 0 to 5 meV experimental range.86 The DFT results are again strongly dependent on the exchange–correlation potential chosen. However, for the B3LYP result, the calculated value using the broken symmetry approach and the appropriate mapping is antiferromagnetic with J = 11 meV while a ferromagnetic coupling constant of J = þ14 meV is predicted by this functional when both singlets and triplets are properly calculated through the ROKS/REKS method.86 This result unambiguously proves that the broken symmetry solution does not provide the energy of the singlet state. Moreover, the final results are quite deceiving because when DFT calculations are carried out properly the result is qualitatively incorrect while a qualitatively correct result is found when using a qualitatively incorrect approach which does not respect symmetry. The failure of the ROKS/REKS method is clearly due to the use

Fig. 3 Schematic representation of the geometry of [Cu2Cl6]2.

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of exchange–correlation potentials which were constructed without taking into account spin symmetry. Here, it is worth pointing out that the problems of DFT outlined above are not restricted to the domain of magnetic coupling constants. Similar problems and contradictions appear when dealing with the interaction of NO with NiO(100) or a Ni-doped MgO(100) surface, which is due to a problem with an open shell electronic configuration. In this problem, there is experimental evidence that the unpaired electron of NO is transferred to the unfilled d shell of Ni21; this produces a change in the on-site electron–electron repulsion that is not correctly described by DFT. Unrestricted DFT calculations incorrectly predict the existence of three open shells whereas restricted DFT calculations incorrectly predict the bonding geometry and bonding energy of NO to these surfaces.147–149 The exceedingly high computational cost of CI based calculations when applied to systems of chemical interest and the intrinsic problems of DFT on describing magnetic coupling constants triggered new approaches. We already commented on the CAS-DFT based methods.130–132 Here we add a new idea which uses the Kohn–Sham orbitals to construct multireference spin adapted CASCI wave functions and then obtain the dynamic correlation by second-order perturbation theory.150 5C.

Extended systems: solids and surfaces

The electronic structure of periodic solids is usually treated by the standard band structure methods which permit one to properly handle the periodic symmetry of these systems. There is a broad family of band structure methods which go from the simplest empirical tight-binding approaches to sophisticated ab initio Hartree–Fock or DFT implementations. The latter are usually preferred in solid state physics applications because electronic correlation is included in at least an approximate way. For bulk semiconductors and metals the LDA method generally provides a correct description overall of the crystal geometry, band structure and even of macroscopic properties obtained by molecular dynamics approaches.47 However, for magnetic insulators, LDA is unable to provide even a qualitatively correct description. This is clear when analyzing LDA results for a prototypal system such as NiO: LDA describes it as a metal151,152 where experimentally is an insulator with a band gap of ca 4 eV.153 This incorrect description is not limited to NiO but also occurs for almost all magnetic oxides and halides. The reason for this failure is the presence of localized d open shells on the metal cations. The electrons in these open shells exhibit strong electronic correlation effects which are not at all similar to the uniform electron gas upon which LDA is parameterized. For this reason, these compounds are usually referred to as strongly correlated systems. This deficiency of LDA is not repaired by generalized gradient methods (GGA), and the GGA band gap of NiO is still too small.154–156 It has been suggested that the difficulty of LDA (and GGA) to properly describe NiO and other narrow-band insulators is related to the self-interaction error inherent in the local exchange functional. Introducing a self-interaction correction (SIC) to LDA results in a qualitatively correct (B3 eV) gap in the spectrum, and improves the magnitude of the This journal is


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magnetic moments and the value of the lattice constant in NiO.157,158 Nevertheless, the computation of the SIC is very costly and in practice is not simple. An alternative approach supplementing the LDA with an effective, semiempirical, onsite repulsion U has also become popular.159,160 This approximation also improves the gap and lattice constant.158 Martin and Illas have suggested that hybrid DF methods can lead to an appropriate description of these systems by comparing the magnetic coupling constant predicted by different cluster models.84,85 More recently, the B3LYP implementation in the CRYSTAL code161 cleared the way for hybrid DF calculations on periodic solids using localized Gaussian type basis sets.156 This is also becoming possible in the Gaussian-03 program162 where periodic boundary conditions are applied instead of making use of translational symmetry.163 This included the calculation of magnetic coupling constants at the B3LYP level,156,164–168 thus improving the previous UHF estimates.106–108 Very recently, an implementation of hybrid DF methods has been presented for calculations within plane waves, a computationally convenient choice of band structure calculations.169 Hybrid DF calculations with the popular B3LYP functional carried out for a large number of systems showed that the introduction of part of a non-local Fock exchange solves many of the problems of LDA and GGA and, in particular, leads to a more accurate prediction of band gaps in this type of solids. The use of hybrid functionals also improves the prediction of magnetic coupling constants. However, obtaining J from the energies of the ferro- and various antiferromagnetic broken symmetry solutions and the appropriate mapping as described in section 4C leads to values which are too large compared to experimental values.166 A careful analysis of the electronic structure and magnetic properties of NiO led Moreira et al.166 to conclude that for this type of system, the inclusion of a 35% Fock exchange leads to an overall correct description of the geometry, band gap, form factors and magnetic coupling. This has been confirmed by the recent work of Feng and Harrison in a representative list of systems.170 The use of hybrid functionals leads to a substantial improvement over the LDA and GGA predictions of the electronic structure of strongly correlated systems. Still, the fact that the results depend so much on the amount of Fock exchange and that this Fock exchange mixing can be system dependent171 gives a semiempirical flavor to this approach and hence one still needs to look for alternative procedures which do not exhibit such an empirical character. An obvious choice consists of cutting a finite part of the crystal and treating it using accurate methods of quantum chemistry such as DDCI or CASPT2. The problem is, of course, how to take into account the effect of the rest of the crystal in the cluster model. In the last few years, it has been shown that a cluster model containing the magnetic centers, their nearest neighbor ligands, a set of total ionic potentials surrounding low coordinated ligands and a set of point charges to reproduce the Madelung potential is adequate to extract the magnetic coupling constants and other local properties.36,90,91,166,172–174 The adequacy of the embedded cluster model for the description of magnetic coupling in these systems is clear from the results reported in Table 1, which correspond to values obtained using the same Phys. Chem. Chem. Phys., 2006, 8, 1645–1659 | 1655

Table 1 Magnetic coupling constants for a series of magnetic insulators predicted by the UHF method applied to embedded cluster and fully periodic models System

Cluster model/meV

Periodic model/meV

KNiF3 K2NiF4 KCuF3(D): Jc Jab NiO: (J1) (J2) K2CuF4 Nd2CuO4 La2CuO4 Sr2CuO2F2 Ca2CuO2Cl2 Sr2CuO2Cl2

2.7 2.8 7.2 þ0.3 þ0.4 4.4 þ0.6 33.4 36.1 33.7 31.7 30.0

2.6 2.8 7.9 þ0.2 þ1 5.4 þ0.5 31.8 38.3 34.9 32.5 29.1

computational scheme and either cluster or periodic models. The values reported in Table 1 correspond to UHF calculations carried out with basis sets of similar quality. UHF was chosen simply because it does not make use of numerical procedures which are common in DF calculations. Nevertheless, calculations carried out with the B3LYP or Fock-35 functionals lead to the same conclusion. Of course, the magnetic coupling constants reported in Table 1 are far from the experimental values due to the lack of dynamic correlation effects in these calculations. However, once it is clear that the embedded cluster model provides an adequate representation of the system one can make use of the DDCI or CASPT2 method and the corresponding mapping procedure. This approach leads to excellent agreement with experiment36,37,78,79,166,172–174 and to the important conclusion that, even in solids, the magnetic coupling constant is a local property and can be accurately predicted by appropriate electronic structure calculations (Table 2). The accuracy and predictive power of the computational models and methods described in this subsection are clearly illustrated in the problem of the Li2CuO2 spin chain compound. Here, ab initio parameters were used to define an effective spin Hamiltonian that enabled one to perform Monte Carlo simulations of the magnetic system to simulate macroscopic properties as a function of the temperature and to calculate the Neel temperature. The simulated specific heat and magnetization curves predicted a Neel temperature in good agreement with experiment.175 Table 2 Magnetic coupling constants for a series of magnetic insulators predicted by the DDCI method applied to embedded cluster models compared to experiment. Values are taken from refs. 36, 37, 78, 79 and 166 System

DDCI/meV

Experimental values/meV

KNiF3 K2NiF4 KCuF3(D): Jc Jab NiO: (J1) (J2) K2CuF4 Nd2CuO4 La2CuO4 Sr2CuO2Cl2

7.6 7.7 28.8 þ0.5 þ1.9 16.4 þ2.1 126 145 130

7.4 [7.9; 9.1] [31.6; 33.8] þ0.5 þ1.4 [17.0, 19.8] [þ1.4, þ1.8] 125  6 135  6 125

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We end this section by commenting that magnetic interactions at surfaces are also of interest although so far the number of applications using these interactions are quite limited. Geleijns et al.176 have studied the magnetic interaction at the NiO(001) surface and predicted a considerable decrease of the magnetic coupling constant, this has been later confirmed by experiment177,178 thus providing further evidence of the predictive capability of the computational methods described in this paper. In a similar way, Staemmler and Fink studied the CoO(001) surface including spin orbit effects.179 Nevertheless, subsequent calculations showed that the average of the two anisotropic J values calculated with the inclusion of spin–orbit interactions give exactly the CASSCF values obtained without including these effects,180 again indicating the usefulness of non-relativistic ab initio calculations to predict and analyze this important magnetic property. A more extended and general discussion about magnetic interactions at surfaces can be found in a recent paper by Sousa et al.180

6. Concluding remarks We have shown that the magnetic interactions occurring in organic diradicals, inorganic complexes or ionic solids has a common physical background. This is because in all these systems the magnetic interactions occur between localized spins. The magnetic interactions between these localized spins govern the low energy part of the energy spectrum and are well described by spin model Hamiltonians. These model Hamiltonians contain different parameters which are commonly fitted to reproduce experimental values. In some cases, however, more than one set of parameters can reproduce the experimental property thus evidencing the need for an unbiased prediction from first principle calculations. The clue to this prediction lies in the mapping approach between calculated total energies for a series of spin states and the corresponding energy values for the spin Hamiltonian chosen to represent the low energy physics of the system. This is quite straightforward when the energies involved in the mapping procedure are obtained from ab initio calculations on pure spin trial wave functions. A more delicate question arises when the energies have to be calculated through DFT based methods. There is no doubt that these methods have been able to provide accurate magnetostructural correlations and have been and will continue to be very useful when, for a given system, no other approach is possible. Nevertheless, these methods need to be used with special care because on the one hand the energy differences are strongly dependent on the exchange–correlation functional and, on the other hand, there is no clear way to compare the DFT energies from broken symmetry solutions and the analytical results for the spin model Hamiltonian except if one relies on the proper use of the mapping procedure. This reasoning offers a simple yet elegant way out of this problem. Indeed, this is precisely the way to extract magnetic coupling constants in solids from ab initio UHF or DFT periodic calculations. Still, there are cases where energy differences are not enough. Here, effective Hamiltonian theory provides an adequate framework for unravelling the relevant interactions, and in addition offering a powerful interpretation tool This journal is


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which permits one to define the spin Hamiltonian without a priori constraints.

Acknowledgements Financial support from the Spanish Ministerio de Ciencia y Tecnologia (projects UNBA05-33-001 and BQU2002-04029C02-01 and the Ramon y Cajal program (I de P. R. M.)) and in part from the COST-D26-08-02 and Generalitat de Catalunya (projects 2001SGR-00043 and Distincio´ de la Generalitat de Catalunya (FI)) is fully acknowledged.

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