Lecture 3: Modeling molecular magnetism: non-empirical calculation of Heisenberg-VanVleck-Dirac Hamiltonian parameters, modeling spin-dependent (double) exchange Part I : Magnetic properties of polynuclear complexes 1. Borderline between the isolated molecule and the solid state

Fig. 1. — Entities TiCI6, in Cs3TiCl6 (a), Cs3Ti2Cl9 (b) and in TiCl3 (c).

Fig. 5-2.—Variation as a function of temperature of the magnetic susceptibility of TiCl3

2. Interaction between two d1 ions Consider the simple case of two dl ions located respectively in A and in B. Suppose that the ground state 2Γ is orbitally non-degenerate and well separated in energy from the excited states. Moreover, let HA and HB be the electronic hamiltonians associated to each of the two ions, and let φA(1) α(l) and φA(1) β(l) be the eigenfunctions of HA, and φb(1) α(l) and φb(1) β(l) the eigenfunctions of HB. Finally, suppose that A and B are sufficiently close that the two ions can interact. The full hamiltonian H to consider reads then: H = HA + HB + H' (1) where:

H' = !Z1

r1B

+1

r2A

"

# $ %

& ' +

1

r12

+Z2

R (2)

and Z the effective nuclear charge; rlB, r2A and rl2 are defined in figure 3.

Fig. 3.—Signification of the terms in hamiltonian (2)

When R is very large, the system is doubly degenerate. This degeneracy is lifted through the perturbation H' and the orbital eigenfunctions are:

1

S 1± S2( )

!A 1( )!B 2( ) ± !A 2( )!B 1( )[ ]

where S is the overlap integral: S = !A 1( ) !B 1( ) (3)

The symmetrized space function corresponds to the spin singlet and the antisymmetrized space function corresponds to the spin triplet:

!singulet =1

S 1 + S2( )

"A 1( )"B 2( ) + "A 2( )"B 1( )[ ] #2

2$ 1( )% 2( ) & $ 2( )% 1( )[ ]

! triplet=1

S 1" S2( )#A 1( )#B 2( ) " #A 2( )#B 1( )[ ] $

% 1( )% 2( )2

2% 1( )& 2( ) + % 2( )& 1( )[ ]

& 1( )& 2( )

'

( )

* )

+

, )

- )

( 4)

Let’s call lΓg and 3Γu the state associated to the spin singlet and to the spin triplet respectively and E(lΓg) and E(3Γu) the corresponding energies. The indices g et u indicate that Ψsingulet and Ψtriplet are respectively symmetrical et antisymmetrical with respect to an inversion A! B. We easily obtain: E

1!g( ) = 2E0 +

K + C

1 + S2

(5) E

3!u( ) = 2E0 +

K " C

1 " S2

where 2E0 is the eigenvalue of HA + HB and K and C the Coulomb and Exchange integrals respectively:

K = !A 1( )!B 2( ) H' !A 1( )!B 2( )

C = !A 1( )!B 2( ) H' !A 2( )!B 1( ) (6)

The energy difference between the two levels is thus:

E1!g( ) " E 3

!u( ) =2 C "KS2( )1 " S4

(7)

This energy difference is called exchange interaction between A and B. Heisenberg was the first to show that the energy separation between lΓg and 3Γu can be represented by an effective hamiltonian

HEX = !J) S A "

) S B (8)

where:

) S A and

) S B are the effective spin operators for each of the two ions and J is the exchange

constant. This model, called Heisenberg - Dirac - Van-Vleck (HDVV), leads to a great simplification in the interpretation of magnetic properties. E.g. to calculate the eigenvalues of Hex vector coupling of angular momenta applies. Thus, if S is the total spin operator:

) S =

) S A +

) S B (9)

and:

) S A !

) S B =

1

2

) S

2") S

A

2") S

B

2( ) (10)

HEX = !

J

2

) S

2!) S

A

2!) S

B

2( ) (11)

The eigenvalues E(2S+1Γ) of HEX are: E

2S+1!( ) = "

J

2S S +1( ) " SA SA +1( ) " SB SB +1( )[ ] (12)

where S, SA et SB are the quantum numbers associated to the spin operators. In the present case of the coupling between two doublet we get:

E1!g( ) =

3J

4

E

3!u( ) = "

J

4 (13)

J is thus equal to the singlet - triplet separation. When J is negative, the singlet is ground state and the interaction is antiferromagnetic. When J is positive, the triplet is ground state and the interaction is ferromagnetic. The HDVV model can be generalised to complexes with more than two centres. The exchange perturbation reads then: HEX = ! JAB

B"A

#A

# SA $SB (14)

where the summation runs over all ion pairs AB. 3. Magnetic susceptibility Let us come back to the case of an interaction between two dl ions in a binuclear complex an apply an external magnetic field H. There will be no effect on the spin singlet, however the spin triplet will be split into three components lying at -geβH, 0 and geβH with respect to the 3Γu level respectively. This splitting pattern is represented in figure 4 below.

Fig. 4.—Energy diagram for the exchange interaction of 2 d1 ions followed by a Zeeman

perturbation The magnetic susceptibility as a function of temperature is obtained in applying Van Vleck’s formula.

n En(0) En

(1) 1 !

J

4 !ge"H

2 !J

4 0 (15)

3 !J

4 ge!H

4 3J

4 0

Fig. 5a.— Variation of the magnetic susceptibility as a function of temperature in case

of an antiferromagnetic interaction (J < 0) between two ions with 2Γ.

Fig. 5b.—Variation of the magnetic susceptibility as a function of temperature in case

of a ferromagnetic interaction (J < 0) between two ions with 2Γ. All En

(2) vanish; the En(0) and the En

(1) have values noted in (15).

The susceptibility χ is then:

! =N"2

3kT

6ge

3 + exp #J

kT

$ %

& '

(16)

The variation of χ as a function of kT depicted in figure 5 for J < 0 and J > 0. It is seen that in case of antiferromagnetic interaction, χ passes through a maximum at a temperature TM. For the present case TM can be expressed as in terms of K as: TM ! "

J

1.6k (17)

The effective magnetic moment per ion is thus:

µeff = ge3

3 + exp !J

kT

" #

$ %

(18)

When J = 0, we get as expected µeff = 1.73 at any temperature. In figure 6 the variation of µeff as a

function of kT

J is depicted, both for negative J as well as for positive J values.

Copper acetate Cu2(CH3C02)4(H20)2 is probably the most studied exchange coupled system. Its structure is depicted in figure 7. Figure 8 gives the variation of the magnetic susceptibility of Cu2(CH3C02)4(H20)2 as a function of temperature. On the same figure the theoretical curve obtained for J= -295 cm-l is also displayed.

Fig. 6. Variation of the effective magnetic moment in function of

kT

J; case of two 2Γ ions

interacting.

Fig. 5-7.— Geometry of copper acetate Cu2(CH3C02)4(H20)2

Fig. 8. Experimental (dots) theoretical (curve) variations of magnetic susceptibility of

Cu2(CH3C02)4(H20)2 as a function of temperature. 4. Exchange interaction and métal – metal bond The quantitative development of equation (5) is identical to the one proposed by Heitler and London to explain the stability of the H2 molecule (vide infra). Let us describe exchange interaction on the basis of molecular orbital (m.o.) theory, i.e. consider φA and φB two atomic d orbitals centred on atoms A and B respectively. When A and B are close enough we can build 2 m.o.’s σg and σu respectively symmetric et antisymmetric versus inversion: !g =

1

2 1 + S( )"A + "B( )

(19) !u =

1

2 1 " S( )#A " #B( )

The level lΓg stems from the configuration (σg)2 and the level 3Γu from the configuration σgσu. The m.o. model leads to two supplementary level: lΓu originating from σgσu and lΓg originating from (σu)2. In valence bond theory the product φAφB yields an equivalent result as depicted below:

Fig. 9.— Comparison between the valence bonf mo9del and ther m.o. model for the interpretation

of the exchange interaction between two 2Γ Ions. 5. Exchange and superexchange Thus far we implicitly admitted so far that the interaction between two transition metal ions was direct and implied an overlap between orbitals centred on each ions. Such a mechanism leads to an antiferromagnetic interaction when the overlap is large enough. On the contrary, if the overlap is zero or weak, the forces that tend the two electrons to align their spins anti-parallel-wise to make a metal-metal bond is compensated by the forces that tends to place the electron spins parallel-wise in order to minimize the electrostatic repulsion (Hund’s rule). In that case the ground state corresponds to the high spin multiplicity and the coupling is ferromagnetic. In fact, in many case the interaction takes place through diamagnetic bridging group. This interaction is called super-exchange. Many research projects are still devoted to this problem which is far from being solved satisfactorily. To illustrate this situation, let’s consider once more the case studied in paragraph 2, but in placing a diamagnetic atom in O half-way between A and B. We suppose that A and B far enough apart from each other such that there is no direct interaction possible amongst the two magnetic centres.

Suppose that two orbitals centred on A and B are dz2 orbitals and that O possesses a filled pz

orbital. The distances A-O and B-O are such that dz2pz in non-zero. The orbitals φA and φB that

were previously pure atomic orbitals are presently anti-bonding orbitals with respect to dz2 , that

are partially delocalized between A and O and between B and O. These orbitals φA and φB are graphically represented in figure 10 proportional to:

!A = d

zA2 + " pz

!B = #dzB2 + " pz

(20)

where λ is the mixing coefficient.

Fig. 10.— Orbitals φA and φB in the super-exchange A-O-B interaction

The only difference concerning the nature of the orbitals φA and φB in the treatment of paragraph 2 applies to the super-exchange interaction A-O-B. Hence, it is possible to speak about direct exchange interaction and indirect exchange interaction. There is another mechanism that the one described in paragraph 2 which has also been proposed to describe indirect exchange interaction. This mechanism involves an electron transfer from orbital φA (or φB) to orbital φB (or φA). The configuration φA φB leads to two terms with lower energy lΓg(φA φB) and 3Γu(φA φB) respectively. The corresponding wave functions are given in (eq. 4). The configuration !A

2 (or !B2 )

originating from electron transfer B ! A (or A ! B) to two excited de spin singlets: lΓg(!A2 ) and

lΓu(!A2 ). The wave functions associated to these terms are:

1!g "A

2( ) : 1

2 1+ S2( )"A 1( )"A 2( ) + "B 1( )"B 2( )[ ]

2

2# 1( )$ 2( ) % # 2( )$ 1( )[ ]

(21)

1!u "A

2( ) : 1

2 1 # S2( )"A 1( )"A 2( ) # "B 1( )"B 2( )[ ]

2

2$ 1( )% 2( ) # $ 2( )% 1( )[ ]

Denote ΔE the energy difference between lΓg(φA φB) and lΓg(!A

2 ). These two terms have both the same symmetry and thus couple together to repel each other. lΓg(φA φB) is thus lowered in energy

by !2

"E where α is given as:

! =

1"g #A #B( ) H' 1"g #A

2( ) (22)

α is large if the electron transfer from one orbital to the other easy i.e. if the overlap metal-bridge is large and vice versa. This mechanism stabilizes lΓg(φA φB) versus 3Γu(φA φB) hence leads to an anti-ferromagnetic coupling. These mechanisms considered in this chapter don’t exclude mutually. They are sketched schematically in a diagram in figure 11. Still today it’s not possible to tell which of both mechanisms is dominant in the experimental separation between lΓg(φA φB) and 3Γu(φA φB).

Fig. 11.— Exchange interaction scheme for the two mechanisms considered in this chapter

Finally we list a few examples of binuclear complexes and the corresponding J value (cf. eq.8) in the table below.

Part II : DFT calculations of molecular magnetic properties of coordination compounds Abstract An overview of Density Functional Theory (DFT) based techniques for the calculation of the magnetic properties of molecular and supramolecular assemblies is presented. Three different approaches to compute the exchange coupling constant (Jex) are reviewed, i.e. the Broken Symmetry Technique, the Single Determinant Approach, and the Spin Projection Method. The first one (BS), developed by Noodleman, is undoubtedly most commonly applied, e.g. to clusters containing several paramagnetic metal centres or to paramagnetic organic radical species. The second approach (SD) was originally developed to compute the electronic spectra of transition metal complexes, but was more recently applied to the computation of spin manifold of molecular magnets. The last method, developed by Ovchinnikov and Labanowsky, is mainly an extension of the Hartree-Fock (HF) concept of spin de-contamination to DFT. The performance of the three methods has been evaluated for model systems (HHeH, [Cu2Cl6]2-) and for more complex molecules (Ti(CatNSQ)2 and Sn(CatNSQ)2, Bis-verdazyl diradical (BVD), [Fe2(OH)3(tmtacn)2]2+ and [[Cu3O2L3]3+, L=N-Permethylated (1R,2R)-cyclohexanediamine). A comparison of these results with experimental values and with post-HF results if available is presented as well. In the case of the last two complexes, i.e. mixed valence systems, computation of the vibronic potential energy surfaces is also briefly discussed. Contents Introduction Theory and computational tools The Spin Hamiltonian: a Phenomenological Approach Molecular Magnetism and Computational Chemistry Semi- empirical and qualitative models Quantitative Models

1) The Broken Symmetry Approach 2) The Single Determinant Method 3) The Spin Projection Technique

An illustrative example: the H2 molecule Applications Model Systems: HHeH and [Cu2Cl6]2- Real Systems Ti(CatNSQ)2 and Sn(CatNSQ)2 BVD [Fe2(OH)3(tmtacn)2]2+ [[Cu3O2L3]3+ Concluding Remarks

References

1. Introduction Although the magnetic behaviour of some materials, e.g. lodestone, has been well known since antiquity, a deeper understanding of the microscopic processes related to magnetic interaction is still an interesting topic in modern theoretical chemistry and a difficult task in computational chemistry [1]. The basis of the theory of magnetic interactions between paramagnetic sites was developed in the first half of this century, but even now the computational effort needed is still not generally affordable. From a theoretical point of view, scientists interested in explaining the magnetic behaviour of insulators were the first to study magnetic interactions. Two names must certainly be mentioned in this field i.e. those of J. H. Van Vleck and of P. W. Anderson who can be considered as the founders of modern magnetism. In this contribution we borrow the definition of the late Olivier Kahn1: "Molecular Magnetism deals with magnetic properties of isolated molecules and/or assemblies of molecules" to give a more pragmatic picture of the topic. The main reason to deal with a molecular magnet instead of with common inorganic magnets e.g. of transition metals oxides is the possibility to add various other properties to the compounds such as, e.g. optical properties. The possibility to increase the complexity of the 3D architecture and exchange pathway using paramagnetic organic ligands [2] and paramagnetic metal centres [3] was an idea for several preparative approaches. On the other hand, the study of biological processes involving active sites containing metal ions is related to the interaction between paramagnetic metal centres, which is not so different from that which happens in a magnetic material [4]. This chapter will be organised as follows: after a short description of the phenomenological Spin Hamiltonian approach used to describe magnetic interactions, the unavoidable link between the interpretation of experimental data and theory, we will present in their original formulation three formalisms that have been used in the framework of DFT in order to study exchange interaction in organic and organometallic systems. Finally, an overview of the the performance of the methods in the case of model and chemical systems will be given. 2. Theory and computational tools 2.1 The Spin Hamiltonian: A Phenomenological Approach The main idea of Spin Hamiltonians is to represent the physics of the interactions between magnetic centres through spin operators which act only on local effective spin eigenfunctions. The system is therefore described through its spin eigen-states The spin eigenfunction reads: S1S2S3...Si..SM{ }

where S is the total spin state due to the coupling of all the Si's (i.e. the local spin of the i-th site). This idea is justified in considering that e.g. for insulators the spin and the magnetic moments whose alignments lead to magnetic effects are certainly localisable, so that from a phenomenological point of view at least, their interactions can be described by spin operators acting on local spin states. The spin Hamiltonian is not able to give electronic information on the interactions between the metal centres. But it can be used, and commonly is used, to interpret the variations of the different total spin state energies as a function of empirical parameters derived from different kind of experiments such as EPR, NMR, Mössbauer, optical spectroscopy and/or magnetic susceptibility measurements at different temperatures.

1 Kahn, O., in "Molecular Magnetism" pg. VI, VCH publishers, 1993.

The most-commonly used Spin-Hamiltonian is the so-called HDVV [5] (Heisenberg-Dirac-Van-Vleck), which implies an isotropic interaction between the paramagnetic centres described through an exchange coupling constant Jex. It is a pair interaction and amongst the different formulation found in literature2, we will use the following notation:

!

HHDVV = Jij

ex ˆ S i "ˆ S j

i> j

# (1)

Where i and j denote the paramagnetic centres. In case of a dimer, the Hamiltonian can be expressed in a simpler form as:

!

HHDVV

= JAB

ex ˆ S A" ˆ S

B=

JAB

ex

2

ˆ S 2# ˆ S

A

2# ˆ S

B

2( ) (2)

If an external field is absent, each of the total spin states are (2S+1)-fold degenerate. Sometimes the HDVV spin Hamiltonian is not sufficient to describe the magnetic interaction in materials, therefore either single centres or interaction terms are added. The ZFS (Zero Field Splitting) and the biquadratic HDVV terms are respectively an example of both corrections. The ZFS justify for instance the lack of degeneracy of the different Ms values for a given S total spin state in absence of external field due to the spin-orbit interaction, while the biquadratic HDVV is sometimes used to describe non-linear interaction between the magnetic centres. A deeper description and the list of Spin Hamiltonians commonly used to interpret magnetic data can be found, for instance, in reference [6]-[8]. These non-Heisenberg terms are often considered as a first order perturbation to the energy obtained using a HDVV, spin-only, Hamiltonian. As mentioned previously the use of a HDVV spin Hamiltonian presupposes that spins are localised on each centre. This phenomenon is usually called valence trapping and this description is inadequate for systems where the valence electrons can be delocalised. It is the case for mixed valence systems, where metal centres with different oxidation states are present. Anderson and Hasegawa [9] developed a spin Hamiltonian that describes these phenomena of electron delocalisation at the beginning of the 1950s and introduced the concept of double-exchange. This term may appear rather obscure to many chemists and we propose to use the term: spin dependent electron delocalisation [10] because just so-called Hund-states, i.e. states with the same spin multiplicity, will be consider to contribute to the delocalisation phenomena, as depicted in Figure 1 below.

(a) (b) (c) Figure 1. Hund (a and b, without spin flip) and non-Hund (c, with spin-flip) mechanism of delocalisation.

2 Other notations commonly used are:

!

HHDVV = " Jij

ex ˆ S i #ˆ S j

i> j

$ ;

!

HHDVV = "2 Jij

ex ˆ S i #ˆ S j

i> j

$

Later on, more complicated interactions schemes were developed, e.g. the exchange-transfer [11] model, where the delocalisation involves three centres and which can take place via non-Hund states (cf. Fig. 2).

2

1 3

Figure 2. Exchange transfer model. When electron delocalisation can occur, the whole set of spin states representing the system will be: {|S1...Si...SnSM>}1⊗.......{|S1...Si...SnSM>}i........⊗{|S1...Si...SnSM>}n. where the subscript i indicates the centre where spin excess is localised. To illustrate this, let us consider for the sake of clarity a dimer A-B where SA = 1/2 = SB with a spin excess se=1/2. Two different sets of spin states will arise, corresponding to the two different localised situations depicted in Figure 3. | SAse(SA)* SB SM>A | SA (seSB) SB*SM>B

A B A B

I II

Figure 3. Graphical representation of localised spin states for a AB dimer. Next we define a transfer operator as: HAB = βTAB (3) where TAB is the operator which transfer an electron from the site A to B, i.e.: TAB|SA...SM>A=KAB|SA...SM>B (4) The resulting Hamiltonian matrix is then :

| SA ...SM>A | SA ...SM>B

!

Jex ' ˆ S

Aˆ s

e+ J

ex( ˆ S

A+ ˆ s

e) ˆ S

B" #1

" #1 Jex' ˆ S

Bˆ s

e+ J

ex ˆ S A( ˆ S

B+ ˆ s

e)

$

% & &

'

( ) ) (5)

where Jex’ is the intra-atomic exchange coupling constant and Jex is the inter-atomic one. If we ignore non-Hund states we obtain:

!

EA+ J

ex ˆ S A

* ˆ S B

" #1

" #1 EB+ J

ex ˆ S A

ˆ S B

*

$

% &

'

( ) (6)

where EA and EB are the energies of the two isolated centres when the electrons' excess is localised onto them. For a binuclear system i.e. one where each of the two centres has n electrons occupying the d-orbitals and a 'delocalised' electron oscillating between them (se=1/2), Anderson and Hasegawa [9] derived the following analytical formula to express the energies of each spin state as a function of what they call the double-exchange parameter β:

E(S) = ±!S+1 2

2 S0+ 1 (7)

Due to the double exchange interaction each spin states is split into two levels corresponding to an in-phase (gerade) and an out of phase (ungerade) combination. Since the stabilisation is proportional to S+1/2 (in case of a dimer), it is straightforward to see that the double exchange process will favour the stabilisation of the spin state with highest multiplicity S=Smax (cf. Fig. 4).

2 4 6 8 10

-1,0

-0,5

0,0

0,5

1,0

E

2S+1

Figure 4. Energy of the spin states (in units of β) for a d5-d6. Finally, in the case of a mixed valence dimer, when taking into account both exchange and double exchange interactions, the energy expression of each spin state as a function of exchange (Jex) and double exchange (B) parameters thus reads:

E(S) =J

2S S +1( ) ± B S +1 2( ) (7a)

2.2 Molecular Magnetism and Computational Chemistry It is well known that magnetic interactions arise from weak bonding while a strong interaction between paramagnetic centres leads to a covalent bond formation and therefore to diamagnetic molecules. It is important to find computational techniques, which are able to correctly describe these weak interactions. A semi-quantitative evaluation of magnetic interactions and the corresponding parameters is possible using Orbital Interaction Models where a simplified Hamiltonian and limited basis-sets are used. A quantitative estimation will need the use of extensive post-HF or DFT calculations in order to fully include the dynamical and the quasi-degeneracy correlation of the electrons. 2.2.1 Semi -empirical and qualitative models Several orbital interaction models have been developed in the seventies to describe the magnetic interaction. Amongst these, let us mention both, the model of Hoffman et al. [12] and the model of Kahn et al.[13]. Both of them have been widely used. We will illustrate shortly the two methods next. Consider a full electronic Hamiltonian: H = h(i)

i

! +1

rijj> i

! + ˆ h SO

The following approximations are usually made: • Active electron approximation: Just the unpaired electrons are taken into account. All the inner

electrons are just considered as belonging to non-interacting cores. • The spin-orbit interaction is usually neglected. • A minimal basis set is used • Usually, semi-empirical methods are used to evaluate the integrals e.g. Extended Hückel (EH). The first approximation is valid only if the magnetic orbitals, i.e. those containing the unpaired electrons, are well separated in energy, both, from the doubly occupied and from the unoccupied or virtual ones. This is not always the case. For the sake of simplicity, only the model of Hoffman et al. [12], which is based on m. o. theory (Kahn and Briat’s model is based on valence bond theory) and is usually applied within the EH formalism, will be described in some details here. Suppose we have a dimer A-B where each of the moieties contains an unpaired electrons occupying a d-orbital. We will denote by dA and dB the two magnetic orbitals on site A and B respectively. Hence, φ1 and φ2 will be the molecular orbitals arising from their interaction i.e.

!1 =1

2(dA + dB)

!2 =1

2(dA " dB)

(8)

In the absence of a strong direct interaction ( dA dB ! 0), these two orbitals will not be much perturbed. Within the active electron approximation described here, just the configuration arising from different occupation of magnetic orbitals will be considered. Thus, we get the following spin-states: 1 triplet and 3 singlets. They are depicted below in Figure 5:

Figure 5. Different configuration arising from the magnetic orbitals. Where: T = !1"!2" ; S1 = !1" !1# ; S2 = !2"!2# ; S3 =

1

2(!1" !2# - !1#!2" ) .

Within this manifold, the triplet state is unique and does therefore not interact if spin-orbit coupling is neglected. The singlet states however do and can be expressed as !S = "1!S1 + " 2!S2 + " 3!S3 (9) Usually λ3≈0 since φ1 and φ2 often belong to two different, or almost different, irreducible representations. Thus, ψS3 has a different symmetry than ψS2 and ψS1 and does therefore not interact. The triplet state is correctly described by the single determinant ψT. In the case of non-interacting orbitals (φ1 and φ2 are degenerate): λ1 = λ2. While for an interacting system, we have |λ1|>>|λ2|. At this level of approximation, the energies of the states can be expressed in terms of Coulomb (J12), Exchange (K12) and core matrix elements (hi). Thus,

!

ETripl. " ESing. = J12 "K12 "1

2(J11

+ J22) +1

2(2h

1+ J

11" 2h

2+ J

22)2 + 4K

12

2[ ]1/ 2 (10)

where: hi= !i (1) h(1)!i (1) ;

J12= !1(1)!2 (2)1

r12!1(1)!2(2) ;

K12= !1(1)!2 (2)1

r12!1(2)!2(1) .

This can be rewritten in terms of orthogonal and localised orbitals φA and φB mainly centred on site A and B i.e.

!A =1

2(!1 + !2 )

!B =1

2(!1 " !2 )

(11)

Thus, the singlet to triplet energy difference can be evaluated as:

ETripl. ! ESing. = !2KAB +("1 ! "2 )

2

JAA ! JAB (12)

where: ε1 = h1 + J12 - K12 and ε2 = h2 + J12 - K12. Moreover, since ε1-ε2 = h1-h2 = !1 + !2 h !1 " !2 = 2hAB we get the final expression:

!

ETripl. " ESing. = "2KAB +(2hAB )

2

JAA " JAB= JAB

ex

(13)

From these expressions we can derive some qualitative conclusions regarding the stabilisation of a ferromagnetic (triplet) versus an anti-ferromagnetic (singlet) ground state. It is evident that two antagonistic contributions to the exchange-coupling constant (

!

JAB

ex ) do exist: (i) a ferromagnetic one: JF and (ii) an anti-ferromagnetic one: JAF. Considering that the exchange integral KAB is always positive, the contribution: JF = -2KAB will favour the ferromagnetic interaction (the triplet is the ground state). This term is referred to as “potential exchange” in Anderson's theory [14]. The second contribution:

JAF =(2hAB)

2

JAA ! JAB

will on the other hand favour the anti-ferromagnetic interaction. The term JAB is usually referred to as "kinetic exchange"[14]. This model can be extended to any binuclear system containing m electrons per site. The general expression is: Jex=JF+JAF (14)

where JF = !

2

m2Kij

j"B

#i"A

# (15)

JAF =1

m2(!2i " !2i"1)

2

JAi,Ai " JAi,Bii =1

m

# . (16)

Before passing to non-empirical techniques, it is worthwhile to mention a fully empirical approach developed in the fifties which is based on the overlap between the two magnetic orbital and which

does remarkably well predict the ferro- or anti-ferromagnetic nature of their interaction. We summarize below briefly the so-called Goodenough-Kanamori [15] rules: • when the interacting centres have the lobes of the magnetic orbitals directly pointing towards

each others and when the overlap between them is substantial, then the interaction (direct exchange) is strongly anti-ferromagnetic.

• If a magnetic orbital of one centre overlaps with an empty orbital of the other centres the direct interaction is ferromagnetic.

• when the magnetic orbitals are orthogonal and hence the overlap is zero, then the direct interaction is ferromagnetic.

The interaction can occur also through diamagnetic ligands, which have doubly occupied or empty orbitals. In this case the interaction is referred to as super-exchange. A classic example of failure of the semi-empirical models mentioned before is given by the study of the magnetic behaviour of one of the first molecular magnetic material synthesised, i.e. the diaquo-tetra-µ-acetato-dicopper(II). In this case, the two copper(II) are antiferromagnetically coupled and the separation between the lowest singlet and the lowest triplet has been estimated from magnetic susceptibility data as equal to 296 cm-1 [16]. In Figure 6 we reported the singlet-triplet energies separation computed [17] at different levels of theory and compared to the experimental one. Figure 6. Computed Singlet-Triplet separation (in cm-1) for diaquatetra-µ-acetato-dicopper(II). Where the ferromagnetic contribution (eq. 15) and the antiferromagnetic one (eq. 16) are denoted 1st and 2nd order respectively. It is easy to notice that we need a post Hartree-Fock (HF) treatment in order to obtain a quantitative agreement with the experimental J-value. Moreover, the simple Anderson approach is completely wrong! Why? In fact, the reason is simple in this case. The main approximation (active electron) is not correct and several configurations are to be taken into account in order to reproduce the data correctly. Generally speaking, we need a computational tool, which is able to include correlation and to reproduce the intrinsic multi-determinantal nature of the spin manifolds. Post-HF techniques are certainly good candidates. Unfortunately, they can only be used for relatively small systems. As soon as the dimension of the system increases, the consideration of extended configuration interactions to account for correlation and of huge basis sets needed to represent correctly the electron cusp, render these approaches frequently unfeasible (cf. Calzado et al.: J. Chem. Phys. 116, 2002, 3985).

I II

1st order 2nd order higher orders exp

-244 -286

I : JF = 2Kab

II : JAF = -(εg- εu)2 / (Jaa - Jab)

Therefore, beginning in the eighties, several new techniques based mainly on Density Functional Theory (DFT) were developed and applied to solve the problem of calculating multiplet energies and thus to the first principle determination of magnetic spin Hamiltonian parameters. In this work, we will describe three of them, that is: (i) the Broken Symmetry approach, (ii) the Single Determinant method and (iii) the Spin Projection technique. All these methods have been applied to real and to model systems (cf. next section: Application). In case of smaller model systems a comparison to post-HF results are also included. 2.2.2 Quantitative Model 2.2.2.1 The Broken Symmetry Approach The Broken Symmetry (BS) approach, developed by Noodleman [18-19] and co-workers, relates the energy of a broken symmetry single determinant, which is not an eigenstate of the S2, to the energy of a pure spin state. The most important feature of this technique is that only the calculations of the energy of single determinants are needed to approximate the energies of all spin manyfolds. On the other hand, the main limitation is that the method is based on a pure HDVV spin Hamiltonian, thus all non-Heisenberg system will be described incorrectly for this reason. First, we have to remember that the spin state of highest multiplicity can always be correctly described by a single determinant. Thus, the energy of the other component of the same multiplicity needed for the calculation of J can easily be obtained. The Broken symmetry state is artificially constructed in order to obtain an electronic symmetry which is lower than the actual nuclear symmetry. This is done in order to allow the two magnetic orbitals, which usually belong to different irreducible representations, (e.g. g and u or a and b etc.) to interact by using different orbitals for different spins α and β. The two different orbitals which accommodate electrons with different spins (spin polarised calculation) are usually (vide infra for an extended discussion of this) located on each centre in order to have the maximum local mS and are coupled together in order to have the minimum resulting MStot. This corresponds to a globally antiferromagnetic state where the spin α and β density is localised on different paramagnetic centres (Fig. 7).

Figure 7. Broken symmetry α and β component of the HOMO of for diaquatetra-µ-acetato-dicopper(II). Noodleman showed that the Broken Symmetry state (|ψB>) can be approximated by a weighted average of the different spin state of the system:

!B = A(S) "S

S

# (17)

Consider two interacting centres A and B and the related, spatially orthogonal, magnetic orbitals ai and bi on A and B respectively. A set of two non-orthogonal orbital a i = ai + cbi and b i = bi + ca i can be constructed and the overlap between them is therefore S

a i b i= 2c . If each of the centres has

just one unpaired electron occupying the magnetic orbitals a1 & b1, the Broken Symmetry determinant reads: |ψB>=(N!)!1/ 2M!1 / 2

det (a1 + cb1)"a2".......an"(b1 + ca1)#b2#......bn#[ ]=

(N!)!1/ 2

M!1 / 2

det a1"a2".......an"b1#b2#......bn#[ ] + c(det b1"a2".......an"b1#b2#......bn#[ ] +

det a1"a2".......an"a1#b2#......bn#[ ]) + c2det b1"a2".......an"a1#b2#......bn#[ ]

$

% &

' &

(

) &

* & (18)

where M is the normalisation factor. This determinant can be expressed as the sum of four different contributions: !B = M

"1/ 2(#1 + c(#2 + #3 ) + c

2#4) (19)

Dropping out φ4, which depends on c2, the other contributions are respectively, φ1: the purely covalent (A-B)-state, φ2 and φ3: the charge transfer states corresponding to A+-B- and A--B+. The orthonormality condition of ψB leads to: !B !B =

1

M"1 "1 + c

2"2 "2 + "3 "3( ) + 2c "1 "2 + "1 "3( )[ ] = 1

M1 + 2c

2( )=1 (20)

therefore: M=1+2c2. Using the Lowdin and Nesbet spin projection technique, any normalised state φν can be projected by the MS

SO operator onto its component |S MS>. Therefore, each state can be expressed as:

!µ = MS

SO !"

MS

#S

# (21)

Suppose φν is a Slater determinant constituted by orthonormal orbitals, we then get:

MS

SO !" = AS(S,Ms ) !" + #"µ (S,MS)!"µ

µ

$%

& ' '

(

) * *

(22)

where φµν are all the Slater determinants, which can be obtained by a permutation of the spatial part of φν keeping the spin part constant. Using the orthogonality theorem for irreducible representations, we obtain:

A!"1MS

SO #! A!

"1MS

SO #! = A!

"1 #! #! + $!µ#!µµ=1

n'

% = A!"1

MS

SO #µ H MS'

S'O #! = #µ H MS'

S'O #! &SS'&MM'

(23)

and:

MS

SO !µ H MS'

S'O !" = A" !µ H !" + #"µ!"µ

µ=1

n'

$ %SS'%MM' (24)

The energy of the Broken Symmetry state is therefore

!B H !B =1

M

"1 H 0

SO"1 + c

2 "2 H 0

SO"2 + c

2 "3 H 0

SO"3

+2c "1 H 0

SO"2 + 2c "1 H 0

SO"3

#

$

% % %

&

'

( ( ( S

) (25)

where all the element depending on c2

have been neglected. Finally, we have:

!B H !B =1

M

A1(S) "1H"1 + #1µ"1µµ$ + c2A2 (S) "2H"2 + #2µ"2µ

µ$

+c2A3(S) "3H "3 + #3µ"3µ

µ

$

+2cA1(S) "2H"1 + #1µ"1µµ$ + 2cA1(S) "3H"1 + #1µ"1µ

µ$

%

&

' ' ' ' ' ' ' ' ' '

(

)

* * * * * * * * * *

S

$ (26)

where the Ai(S) are the vector coupling coefficients of the φi states which can be expressed in terms of S, n and Smax. Let n be the total number of magnetic electrons on the two centres. The resulting state φ1 will be:

|SASB> = n

2 !n

2 .

And

A1(S) =

!

cn

2 n

2 S;n

2 "n

2 0

#

$ %

&

' (

)

* +

,

- .

2

where c(SA SB S; Ma Mb M) is a Clebsh-Gordan coupling coefficient. Substituting 1

M=

1

1+ 2c2= 1! 2c

2

and

c2!" H !" + #µ"

µ

$ !µ" % E"

and expressing with E1 = <φ1|H|φ1>; E2 = <φ2|H|φ2> = E3 = <φ3|H|φ3> , the energy of the Broken symmetry state becomes:

!B H !B = "1 H "1 + A1(S)n #S(S +1)

n2

$

% & '

( ) S* aibj

1

r12

aibj +i, j*

A1(S)n(n +1) # S(S +1)

n2

$

% &

'

( )

S* 2S

a b a1 h b1 + a1b1

1

r12

a1a1 +S

2a b

2(E2 # E1)

$

% & &

'

( ) )

+ , -

. -

/ 0 -

1 -

(27)

The energy of a pure spin state ψS is given as

!S =M

M'0

SO!B

where the normalisation constant M' is: M’ = <φ1|0

SOφ1> + c2<φ2|0

SOφ2> + <φ3|0

SOφ3> = A1(S) + 2c2A2(S). (28)

Thus, the energy of a pure spin state is therefore: !S H !S = "1 H "1 +

n # S(S +1)

n2

$

% & '

( ) aibj

1

r12

aib ji, j

* +

n(n +1) # S(S +1)

n2

$

% &

'

( ) 2S

a b a1 h b1 + a1b1

1

r12

a1a1 +1

2S

a b

2 (E2 # E1)$

% & &

'

( ) )

+ , -

. -

/ 0 -

1 -

(29)

From equation (27) and (29) we can see that the energy of the broken symmetry state is a weighted average of the energies of pure spin states and it is also a function of

!

Sa b

, the overlap integral

between the magnetic orbitals. Variational minimisation of the energy i.e. imposing that !E

!S= 0

permits one to determineSa b

and the energies of the two states. The value of

!

Sa b

, variationally determined from the Broken Symmetry state is equal to the one determined from a pure spin state S, i.e.:

Sa b

= 2c =2 !2 H !1

E1 " E2

=

2 a1 h b1 + a1b1

1

r12

a1a1

#

$ % %

&

' ( (

E1 " E2

(30)

This means that we will obtain the same result when starting from a BS wavefunction and projecting on a pure spin state S or vice versa. From a computational point of view the use of BS wavefunctions is particularly convenient because its energy can be obtained through a simple SCF procedure while the energy of any intermediate spin state would need a multiconfigurational procedure. This feature makes the BS Method very popular for systems with large number of unpaired electrons. To correlate the energy of the BS state with the HDVV spin Hamiltonian parameters we have just to remember that the energy of the BS state is a weighted average of the energy of the pure spin states. Therefore for a binuclear system AB where SA=SB and Smax= SA+SB, we have:

!

HHDVV

= Jex ˆ S

A" ˆ S

B

the difference in energy between the highest spin states and the BS states is given by:

E Smax( ) ! EB = Smax(Smax +1) ! A1(S)S(S +1)S=0

Smax

"#

$ % %

&

' ( ( J

2 (31)

Using

A1(S)S(S +1) = SmaxS

Smax

!

we get: E(Smax) - EB = Smax

2 J

2 (32)

For binuclear systems where SA≠SB the equation (31) becomes:

E Smax( ) ! EB = Smax(Smax +1) ! A'1 (S)S(S+1)S=n1

Smax

"#

$ % %

&

' ( ( J

2 (33)

Where: |SA-SB| = ni, |SA-SB| ≤ S ≤ SA+SB and A’S = [c(SA SB S; mA mB -ni)]2. Therefore the following expression can be derived: E(HS)-EB=

!

2JexSASB

(34) All this has been derived by supposing that the magnetic orbitals are orthogonal. Normally, a correction for the overlap between magnetic orbital has to be applied since we are using unrestricted non-orthogonal magnetic orbitals. In the case of a dimmer, the following, overlap corrected, expression for the J value can be derived:

!

Jex

=E(HS) " E(BS)

2SASB(1+ s

AB

2)

(34a)

where sAB is the overlap between the magnetic orbitals. This expression yields J-values that can differ by 100% considering the two limits sab

2=0 (strong localised limit) or sab

2=1 (strongly delocalised orbital). Calculations using both regimes can be found in literature. Some authors, claiming a tendency of DFT to give more delocalised magnetic orbitals with respect to UHF, are commonly assuming the strongly delocalised limit [20]. Malrieu et al. [21] have deeply analysed the effect of the non-orthogonality of magnetic orbitals in the broken symmetry approach in the case of reference (HHeH and [Cu2Cl6]2-) and more complex ([Cu2O11]18- embedded cluster) systems by explicitly computing Sab

2. Their conclusion is that there is no evidence for assuming a strong delocalised limit, also when using hybrid functionals such as B3LYP, even if in that way nice agreement with experiment is found. The strong localised limit or, better, the direct evaluation of Sab

2 should be preferred especially in the case where direct interaction can’t be neglected.

In the section on applications (vide infra) the strong localised limit was assumed and in one of the examples (BVD radical) a direct evaluation of Sab

2 was performed by monitoring the expectation value of S2 (see the Spin Projection Technique). It must be stressed that, beside the fact that almost all the calculation using BS have been done using DFT, there are no theoretical limitations to the use of Unrestricted-Hartree-Fock wave function as well for the same purpose. Finally, it’s worth citing a work of Mouesca [22] where an harmonization of the results obtained for exchange coupling evaluation when using Broken Symmetry and qualitative methods is given. 2.2.2.2 The Single Determinant Method The Single Determinant Method [23] (SD) makes use of the full multi-determinant nature of an open-shell state expressed as a linear combination of Slater Determinants (Single Determinant). This is done by using vector coupling (symmetry adaptation of the N-electrons wave function) and yields first order expressions for the multiplet energies consisting of linear combinations of SD energies. In order to calculate these quantities, both spatial and spin symmetry are fully exploited. This method was first developed for the calculation of optical transitions [24], but has also been successfully applied to the study of magnetic properties. The wave function, ψi, of a multiplet whose spatial component belongs to the irreducible representation Γ, component mΓ, and whose spin part is described through S and mS i.e. ψI = |αΓ mΓ S mS> can be expressed as: ! i = Aiµ"µ

µ

# (35)

Where the φµ's are Slater determinants expressed in terms of the spin orbitals χi as φµ = |χ1χ2χ3........| and where Aiµ is a square matrix of the expansion coefficients. The energy of each mono-determinantal state E(φµ) can directly be obtained from a non-empirical quantum chemical calculation. It is usually convenient to optimise the orbital corresponding to an Average Of Configuration (AOC) in a SCF calculation and subsequently to use these frozen orbitals to calculate the energies of the corresponding SD. Neglecting the contribution common to this AOC, the energy of each mono-determinantal state can be expressed in terms of electrostatic two-electrons integrals, as: E !µ( ) = "p

*(1)"q

*(2)G12"p (1)"q (2)dV1dV2 #$$ "p

*(1)"q

*(2)G12"p (2)"q(1)dV1dV2$$[ ]

p<q

%

(36) where G12 =

1

r12

.

The two-electron operator G12 can be developed in terms of one-electron operators gk,f! (1) and gk,f! (2) which are equivalent to the solid harmonics rk1Y1

lm(θ1ϕ1) and rk2Y2lm(θ2ϕ2)

used in electronic structure theory of atoms. The resulting expansion reads: 1

r12= gk,f!

k

"f!

" (1)gk,f! (2) (37)

where

!

gk, f"(i) are mono-electronic operator of symmetry f with component ϕ. The sum k accounts for multiple occurrence of each irreducible representation f.

Hence, the expression for each bi-electronic integral becomes: <aα(1)bβ(2)|G12|cγ(1)dδ(2)>=

k,f!" <aα(1)|

!

gk, f"(1)|cγ(1)><bβ(2)|

!

gk, f"(2)|dδ(2)>=

Va

!

c

"

f

#

$

% &

'

( )

f ,#* V

b

+

d

,

f

#

$

% &

'

( ) a gkk

* c b gk d (38)

V.

.

.

.

.

.

!

" #

$

% & are the V-coefficients of Fano and Racah, related to the 3j as:

Vj1

m1

j2

m2

j

!m

"

# $ $

%

& ' ' = (!1)

j1 + j2 + j j1

m1

j2

m2

j

!m

"

# $ $

%

& ' ' (39)

We define reduced bielectronic matrix element R(ab,cd,f) the sum over k of the monoelectronic reduced matrix elements: R(ab,cd,f) = a gk

k

! c b gk d (40)

For symmetry reason the matrix elements <m||gl||n> are different from zero if and only if m⊗n∈l. Moreover, if m=n, l is restricted to the symmetrized square: [m⊗n]sym The energy of a single determinant E(φµ)is therefore:

E !µ( ) = Bµ""=1

N

# R" (41)

where Bµρ is a rectangular m×n matrix where m is the number of microstates which belongs to this configuration, and n is the number of the electrostatic reduced matrix elements which can be defined in the space of the spinorbitals χ1χ2.... χn which contributes to the µ-th microstate (SD). B can be reduced to a full rank matrix U(µi,j) which relates m microstates φµ to a subset of r (r<m) non-redundant φµi . In terms of non-redundant microstates we get:

E !µ( ) = CµjE(!µj )j=1

r

" µ=1,.....,m (42)

R! = D!jE("µj )j=1

r

# ρ=1,...,n (43)

Finally the energy of each multiplet ψi can be expressed as

E ! i( ) = ! i G! i = Fijj=1

r

" E(#µj ) (44)

The energy of a multi-determinantal spin state (ψI) can be finally expressed as a linear combination of energies of single determinants. The higher is the symmetry the greater is the performance of the method because the number of non-redundant microstates decreases significantly. Some approximations and limitations in the practical implementation of the method are:

a) The method does not consider the two-electron second order terms, which are neither exchange nor coulomb, and which cannot, in general, be expressed in terms of energies of single determinants but involve off-diagonal elements !µ G !" .

b) The method is usually applied within the active electrons approximation. In this case it is possible to include second order electrostatic interaction exactly (vide infra). Hence, frequently, only the microstates arising from different occupations of the active orbitals are taken into account.

In order to illustrate the method, let us consider two metal centres A and B with one unpaired electron on each, interacting through the doubly occupied orbitals of a diamagnetic bridging ligand X (super-exchange). The following interaction diagram can be drawn:

|! A> | !B>

| b >

| a >

| ! 0>

| !X>

Figure 8. Interaction diagram for an A-X-B molecule. Considering just the Semi-Occupied Molecular Orbitals (SOMO) as active orbitals, four multiplets will occur, i.e.: | a2 1A> ; | b2 1A> ; | ab 1B> ; | ab 3B>. Applying the energy decomposition as explained in the text we obtain the following scheme:

Figure 9. Multiplet energies in term of single determinants It is easy to notice that the energy of each state is expressed just in terms of the energy of a single determinant and that the configuration interaction of state with the same symmetry (1A) is also taken into account as a second order correction. 2.2.2.3 The Spin Projection Technique The Spin Projection technique presented here was developed by Ovchinnikov and Labanowski [25]. These authors present the problem of Spin Contamination, already known for HF calculation, in a general formalism, which can also be applied to DFT calculations. It is important to underline that this topic is presently the subject of much debate. We can easily assume that the spin-state of highest multiplicity is almost a pure one and that it can be represented by a single determinant. Whereas the lower spin states are usually contaminated by the higher spin state components of the same MS-value. Let us first consider the easiest case i.e.: a binuclear system with one unpaired electron per site. Two different total spin states can arise i.e.: a singlet (S) and a triplet (T). Any unrestricted wavefunction ψunr,S with Ms = 0 can be expressed as a linear combination of pure Singlet and Triplet spin states as:

! unr,S = a!S + b!T (45) with: a2 + b2 = 1 Thus, if we apply the total spin operator S2

to both sides of equation 45 we generate the following expression for b2: b2 =

1

2!unr,S

S2 !

unr,S (46)

|b2 1A>

|a2 1A>

|ab 1B>

|ab 3B>

E(b2) - Kab/[E(a2)-E(b2)]

E(b2)

E(a2) + Kab/[E(a2)-E(b2)]

E(a2)

2E(a↑b↓)-E(a↑b↑)

E(a↑b↑)

where Kab is the exchange integral between the orbitals a and b

Kab =

E(1 B) ! E( 3B)

2= E a " b#( ) ! E a " b"( )

Applying also the Hamilton operator to both sides again we get: Eunr ,S = ! unr,S H!unr ,S = a

2!S H!S + b

2!T H!T (47)

Hence, the energy of the pure singlet state can be expressed as:

ES=Eunr,S

! b2ET

1 ! b2 (48)

and the difference in energy between the singlet and the triplet is:

!ST = ES " ET =Eunr,S " ET

1 " b2 (49)

This approach can be generalised for any spin system. Considering a system where the maximum spin state is Smax, any unrestricted wavefunction with MS=Si, where Si is an intermediate spin state, will be described as: ! unr,Si

= aSi (S)!S + aSi (S +1)!S+1 + ... + aSi (Smax )!Smax (50) Applying the operator: BL = (S

!)L(S

+)L

to the left hand side of the equation above we get:

!unr ,Si BL !unr ,Si = L! det V(j, j' )[ ]j, j'=1

L

" (51)

where the V matrix is defined as: V(j, j' ) = ! j, j' " sjk

#$s j' k#$nk$

k

% (52)

For the right hand side of the same equation we can use the properties of the spin ladder operator to obtain the following expression:

!(S,M)BL !(S,M) = S S +1( ) " M +1( ) M + i +1( )[ ]i =0

L"1

# (53)

Defining:

TK(P,S) = S S +1( ) ! K + i( ) K + i + 1( )[ ]i=0

P!1

" (54)

yielding the following set of equation:

ASi (S) +ASi (S+ 1) +ASi (S + 2) + ... + ASi (S + L) =1

TSi (1,S +1)ASi (S +1) + TSi (1,S+ 2)ASi (S+ 2) + ... + TSi (1,S + L)ASi (S+ L) = B1

...

...

TSi (L,S + L)ASi (S + L) = BL

(55) where: ASi S( ) = aSi S( )

2 The solution of this linear system of equations yields the ASi(S) coefficient and therefore the mixing of the pure spin state into the unrestricted wavefunction. The energy of each intermediate pure spin state is thus:

Ecorr (S) =Euncor (S) !ASi (S +1)Ecorr (S +1) !ASi (S+ 2)Ecorr (S+ 2) ! ...

1!ASi (S +1) !ASi (S + 2) !ASi (S + 3) ! ... (56)

A final remark concerning the formula derived for the singlet-triplet energy difference: it is easy to prove that the Broken Symmetry (formula 34a) and the Spin projection formalism (formula 49) are exactly identical for this case.

d (a.u.)

An illustrative example: the H2 molecule The H2 molecule, at long inter atomic distance, presents two unpaired electron, one on each centre, and the interaction between these two electrons can be considered as an example of magnetic interaction. We can therefore analyse the behaviour of the different spin states, i.e. the triplet with S=1 and the singlet with S=0 as a function of the inter-atomic distance. This example has all the ingredients of a molecular magnet and all the computational concepts introduced so far will be illustrated and become clear. Figure 10 represents the dissociation curve for the H2 molecule as computed in a minimal basis set, full-CI, treatment

-0.50

0.00

0.50

1.00

1.50

2.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Figure 10. Dissociation Energy curves for the triplet and lowest energy singlet of H2 molecule. As expected, the triplet-state is always higher in energy with respect to the lowest singlet and their respective energies become identical at long distance, where the interaction between the centres is negligible. This situation is in full agreement with the qualitative semi-empirical rules outlined in section 2.2.1 . If we consider just the singlet ground state, then the full CI result can be compared to a Restricted HF (RHF) and a Restricted DFT/LDA (RDFT/LDA) calculation (cf. Fig. 11). By restricted we mean that two electrons with different spins α and β do occupy the same spatial orbital. The horizontal mirror symmetry (D∞h) of H2 requires that the two magnetic orbitals belong to two different irreducible representations: σg

+ and σu+. By Unrestricted, on the contrary, we mean that

the spatial shape of orbitals which do accommodate α− and β-electrons are free to relax i.e. the

Singlet

Triplet

E (Ry)

horizontal mirror plane of H2 is artificially removed (Broken symmetry: D∞hC∞v) and the orbitals which now do interact, are individually optimised according to the variational principle.

0 1 2 3 4 5 6-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 11. Full-CI (dot-dashed line) vs. RHF (full line) and RDFT-LDA (dashed line) energies for the lowest energy singlet state of H2 as a function of the intermolecular distance. Of course the dissociation behaviour is not correctly reproduced because of the restricted single determinant approximation. However the singlet-triplet separation, for which the correct behaviour is to vanish when the inter-atomic distance increases, is much closer to reality for the RDFT-curve than for the RHF-curve. This behaviour is of course related to near degeneracy correlation, which is much better reproduced in DFT than in HF calculations. We can also comment on the well known over-binding feature observed for the LDA calculation, i.e. the minimum RDFT is too low. In order to properly describe the long-range behaviour of the dissociation curve, an unrestricted formalism must be used. The role of correlation energy must also be considered. The so-called dynamical correlation i.e. the instantaneous correlated electrons motion is already included if one uses DFT, whereas the so-called near degeneracy correlation is not. This is the reason why in Figure 12 the corrected asymptotic behaviour is observed for both, the BS and the SD method where near degeneracy correlation is partially taken into account (in fact completely at infinite separation). It is also interesting to notice that in this case the BS behaviour is the closest to the full-CI one.

d (a.u)

E (Ry)

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.00 2.00 4.00 6.00 8.00

d(H-H)

E (

Ry)

Full-CI

RDFT

Single Det

UDFT (BS)

Figure 12. Full-CI vs. RDF, UDFT and SD energies for the lowest energy singlet state of H2 as a function of the intermolecular distance. Moreover, the different behaviour of the RDFT vs. the UDFT at long range can be related to the extra degree of freedom allowing for localisation of the spin α and β on different centres in the Unrestricted calculation. This can easily be shown by monitoring <S2> as a function of distance (cf. Fig13). The calculation of <S2> is analogous to the spin de-contamination described in section 2.2.2.3 . It can easily be seen that localisation takes places for <S2> > 0, as soon the RDFT and UDFT curves start to differ. This terminates our methodological part. We also hope that the experimentalists are now confident about the computational tools involved in the coming section.

-0.35-0.30-0.25-0.20-0.15-0.10-0.050.000.050.100.150.20

0.00 2.00 4.00 6.00 8.00d(H-H)

0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00d(H-H)

RDFTUDFT<S2>(UDFT)

Figure 13. a) RDF and UDFT for the lowest energy singlet state of H2 as a function of the intermolecular distance. b) <S2> for the UDFT calculation as a function of the intermolecular distance 3. Applications Some typical applications of DFT to model systems and to real paramagnetic systems are reported. As soon as the dimensions of the systems increase, less demanding techniques have been applied and instead of a comparison to sophisticated post-HF calculations, direct comparison with the experimental values is given. Concerning the model systems, two kind of examples were considered: the HHeH and the [Cu2Cl6]2-. The first one (HHeH) is a perfect candidate to study the exchange interaction between two paramagnetic sites (the H atoms) through a diamagnetic bridging ligand (the He atom), in the simplest possible way. The [Cu2Cl6]2- system has been used in experimental and theoretical work to understand the magnetic interactions between two metallic paramagnetic sites and the effect of structural deformations (magneto-structural correlations). In the case of real systems: i.e. Ti(CatNSQ)2, Sn(CatNSQ)2 or BVD (vide infra), due to the dimension of the systems, extensive post-HF treatments were not possible and a structural modeling was performed. The calculated values are directly compared to the experimental ones. Finally, an example of the possible extension to mixed valence systems is given in the case of the well characterized [Fe2(OH)3(tmtacn)2]2+ and [Cu3O2L3]3+ complexes. Technical details: For the sake of clarity the following notation is used throughout in order to specify the different approaches: CAS(n,m) = complete active space multiconfigurational SCF post HF calculations involving a set of m orbitals with n electrons;

b

a

QCISD(T) = quadratic CI including single, double and triple perturbative substitutions; BS = Broken Symmetry assuming the strongly localised limit Sab

2=0(vide infra); SD = Single Determinant (vide infra); RU-: using ROHF wave function; PU-: using Projected UHF triplet energies. The effect of the different exchange-correlation functionals used for the computed exchange coupling constant was also studied. The following notation is used: Xα = Slater's modification of Dirac's local exchange (α= 0.7) [26]; VWN = Dirac's local density approximation for exchange and Vosko, Wilk and Nusair's expression for correlation energy of a uniform electron gas [27]; VWN+Stoll = same as VWN plus the Stoll's local correction to correlation [28]; BP = gradient corrected functional combination of Becke's gradient corrected exchange functional [29] and Perdew's correlation functional [30]; PW91 = gradient corrected exchange and correlation functional proposed by Perdew and Wang [31]; BLYP = gradient corrected functional combination of Becke's gradient corrected exchange functional [30] and Lee, Yang and Parr's correlation functional [32]; B3LYP = hybrid functional including exact HF exchange in the ratio proposed by Becke [33]. If not differently specified HF and post-HF calculations or calculation involving hybrid exchange correlation functionals, were performed using Gaussian 94 [34]. All other DFT calculations were performed using the ADF program package [35] (Version 2.01 to ADF2000.02). The basis sets used are specified in the text or in the corresponding references. 3.1 Model Systems3: HHeH and [Cu2Cl6]2- Amongst the possible model systems, which can be designed in order to mimic the main interaction involved in a super-exchange pathway, the linear H-He-H molecule is certainly the simplest. The different spin states arise from the distribution of three electrons in four different orbitals, as schematically depicted in Figure 14. Figure 14. One electron energy level diagram for H-He-H. Due to its size, this molecule can be fully characterized at an ab-initio post-HF level and therefore it has been chosen as benchmark for the DFT techniques. 3 For a more detailed discussion of the model systems' results address to the original paper [36].

2σg

1σg

1σu He

The 1Σg+ is predicted to be the ground state and the lowest triplet state is the 3Σu

+ In Table 1 the computed energy differences between the triplet and the singlet states are reported for three different H-He distances. Ab initio calculations were performed with the programs GAMESS [34], Gaussian 94 [34] and HONDO-CIPSI [38]. Table 1. Singlet-Triplet Energy differences (in cm-1) computed for H-He-H at different H-He distances. H-He distance (Å) Method 1.25 1.625 2.00 Ab Initioa

CAS(2,2) 4204 476 48 OVB-MP2(2,2) 4358 420 26 CAS(4,3) 4294 484 46 OVB-MP2(4,3) 4530 492 48 QCISD(T) 4928 790 202 RU-BS 4298 526 60 PU-BS 4580 554 60 Full CI 4860 544 50 DFTb

Xα BS 6004 646 58 LDA-BS 12432 760 158 BP-BS 10529 1268 134 Xα-SD 5631 745 64 LDA-SD 9050 1553 183 BP-SD 7799 1170 123 a 6-311G** basis set.b Standard ADF triple-ζ Slater type basis set. All the techniques are able to reproduce the correct singlet ground state and the variation of the singlet-triplet energy gap as a function of the distance, which can be considered as a very simple example of "magneto-structural correlation". Assuming the full-CI results as the exact value (they are in agreement with previous calculation by Hart et al. [39]), it easy to notice that the CAS(2,2) approach is in reasonably good agreement with the exact results, showing that a fairly large part of correlation is taken in to account already at this level. On the other hand, due probably to the spin contamination of the reference wavefunction, the QCISD(T) approach does not lead to results of the same quality. The BS approach, especially when using HF wavefunction, gives results in good agreement with the full-CI calculations. Regarding DFT based calculations, an overestimation of the singlet-triplet energy gap is found especially when going from Slater Xα exchange only functional to local or gradient corrected exchange and correlation functionals using both the SD and BS approaches. In any case the correct order of magnitude of the splitting is reached although we can not say that the results are quantitatively correct. In the case of BS calculations, Malrieu et al. [21] have shown that when including a correction for overlap between the magnetic orbital better agreement with post-HF data can be found, the effect being larger the shorter the H-He distance, as expected. Several magneto structural correlation studies have been experimentally established for [Cu2Cl6]2- [40-41] therefore this molecule is an optimal candidate for testing the performance of the different computational models. A schematic sketch of the molecule is depicted in Figure 15.

Cu Cl

Cl Cu

Cl

Cl

Cl

Cl

x

yz

!

Figure 15. Schematic structure of [Cu2Cl6]2-

The results in best agreement with the experiment are those obtained by variational CI calculations of the singlet-triplet energy gap through a judiciously chosen post-HF procedure [42]. These results are collected in Table 2 as LOC and DELOC, corresponding to the use of localized or delocalized molecular orbitals respectively. Table 2. Singlet-Triplet Energy differences (in cm-1) computed for [Cu2Cl6]2- 1. ϕ (deg)

Method 0 45 70 Ab initioa

Localized2 36 -58 873

Delocalized2 6 -47 753

ROHF-BS 1620 1331 1901 DFTb

Xα SW-BS4 198 -210 113 Xα BS 122 -309 72 LDA-BS 362 -246 191 BP-BS 256 -200 109 Xα-SD -41 -437 -106 LDA-SD 160 -374 45 BP-SD -35 -402 -145 Exp. 0 to 405 -80 to -906 -- 1 For geometrical parameters see reference [36]. 2 From reference [42]. 3 Computed for ϕ = 90°.4 From reference [43]. 5 From reference [40]. 6 From reference [41]. a Basis Set: Hay-Wadt basis set was used for Cu and terminal Cl atoms. The core orbitals were treated using the appropriate pseudopotential. The bridging Cl atoms were described using a 6-311G all-electron basis set. A p polarisation function was applied to all Cl atoms. b Standard ADF double-ζ basis set was used for Cu and Cl atoms. Orbitals up to 2p for Cu and Cl atoms were kept frozen. The BS approach reproduces correctly the order of the singlet and of the triplet states, however not the magnitude of the splitting. The SD on the other hand shows the poorest agreement. This is probably due to a lack of description of correlation for the singlet state in the case of the latter approach. 3.2 Real Systems As a matter of fact, it is important to stress the importance of geometrical parameters in the determination of the magnetic properties of all the systems. Therefore, modeling and neglecting the environmental influences e.g. solvent-effects or crystal packing can strongly affect the final results [44]. This is the reason why, when possible, magneto-structural correlations are often investigated

and why it is important to reduce to a minimum the modelisation when direct agreement between the experimental and theoretical data is sought. An example of a possible model of a complex molecular magnetic system, within the QM/MM framework, and the effect of modelling on the J values are reported here in the case of the [Cu3O2L3]3+ complex. 3.2.1 Ti(CatNSQ)2 and Sn(CatNSQ)2

4

These molecules can be considered to be typical examples of interaction between two paramagnetic radical ligands through a formally diamagnetic d0 Ti4+

or Sn4+ metal ion. The structure and the magnetic properties of both systems have been fully characterized experimentally [45]. Surprisingly enough, the interaction is ferromagnetic as measured by magnetic susceptibility as a function of temperature. From recent calculations [46], it has been shown that the empty metal orbitals provide an efficient super-exchange pathway between the two radical ligands whose unpaired electron is mainly localized on the donor N atoms. Since the magnetic orbitals are practically orthogonal, an overall ferromagnetic interaction is determined. A significant amount of metal density is, in any case, found in the magnetic orbitals (up to 4%). These results are in agreement with recent neutron diffraction data [47]. The energy of the lowest spin states was computed for a model system (Figure 16) were the tert-butyl groups were substituted by H (see reference [46] for a detailed description) within the SD formalism.

N

OTi

O

N

O O

1

3

2

4 1.89

2.12

1 2105°

150°

75°

N

OO

N

O O

1

3

2

4

1 2105°

150°

75°

Sn

2.10

2.07

z

y

x

Figure 16. Schematic structure and main geoemtrical parameters of Ti(CatNSQ)2 and Sn(CatNSQ)2 model systems. Using this approach, the energy of the 3A2, of the 1A2 and of the 2A1 states arising from different occupations of the magnetic orbitals (of symmetry b1 and b2) can be expressed as: E(3A2) =E|b1

+b2+|

E(1A2) = 2E|b1+b2

-| -E|b1+b2

+| And the energies of 2 1A1 come from the diagonalisation of the following 2 x 2 matrix:

!+++!+

++!+!+

!

!=

222121

212111

1

1 )(bbEbbEbbE

bbEbbEbbEAE

The results are shown in Table 3.

4CatNSQ=(3,5-di-tert-butyl-1,2-semiquinonato 1(2-hydroxy-3,5-tert-butyl-phenyl)immine).

Table 3. Computed energy splitting (in cm-1) for Ti(CatNSQ)2 and Sn(CatNSQ)2 a. The experimental values are reported in square brackets. State Ti Sn 3A2 0.0 0.0 1A2 57. [56(1)] 58. [23(1)] 1A1 9510 11934 1A1 9571 11991 a Standard ADF triple-ζ basis set was used for Ti and Sn atoms. A standard ADF double-ζ basis set was used for N, C, O and H atoms. Electrons up to 3p for Ti and 4p for Sn and up to 1s for C and N atoms were kept as frozen core. For a complete description of the model used see reference [43]. An excellent, but probably fortuitous, agreement with the experimental data was obtained for the Ti complex while the Sn complex shows the usual reasonable, but semi-quantitative, agreement with the magnetic data. Nevertheless, it is interesting to see how relatively cheap calculations can give a clear, although not fully quantitative, insight on the mechanism of magnetic interaction on real complex systems. 3.2.2 BiVerDazyl (BVD) biradical Very few examples of stable ferromagnetic free radicals, e.g. galvinoxyl [48], nitroxides [49], nitronyl-nitroxides [50], and bi-nitroxides [51], have been studied so far experimentally and characterized theoretically [52]. The interest for stable organic radicals that can exhibit ferromagnetic ground states upon chemical functionalization is therefore increasing. BVD is a biradical whose constituting radical moieties (verdazyl [53]) are antiferromagnetically coupled to give a singlet ground state. A schematic representation of the model used in the calculation is given in Figure 17. The Singlet-Triplet energy gap has been experimentally determined by ESR measurements [54]. The Triplet lies 887 cm-1 higher in the crystal and 760 cm-1 higher in solution.

C

NN

C

N N

C

NN

C

N N

H

H

O

H

H

O

Figure 17. Schematic structure of BVD model system. The structural, spectroscopic and magnetic properties of this oxo-verdazyl bi-radical have been studied using a model complex were the methyl groups have been substituted by H [55]. DF calculations were performed using Gaussian 94 [34] within the framework of the BS approach explicitly taking into account the correction for the overlap of the magnetic orbitals. The effect of the environment on the tuning of magnetic and structural characteristics of biverdazyls was analyzed by applying the Polarizable Continuum Model (PCM) [56]. The Free Rotor Model was used through out. Therefore for each inter-ring torsion angle (α) a full geometry optimization of the triplet state was performed. On each optimized geometry a SCF calculation of the BS state was done and the singlet-triplet energy gap determined. The Jex computed value (in cm-1) for different model chemistry is shown in Figure 18 as a function of the α angle.

Figure 18. Computed depence of Jex on the inter-ring torsion angle, α. PCM/B3LYP (•); vacuo/BLYP (■); vacuo/B3LYP (▲). Experimental value in frozen CHCl3 solution (···) and in solid state (− · −) are reported for comparison. Because the difference in energy between the conformation with α smaller than the equilibrium value is small, at room temperature all these conformations can be populated. Therefore a vibrational average over large amplitude motions was performed in order to obtain a correct Jex value using the following expression:

!

Jex

T= J

min

ex +

j "Jex jj

# eE0$E j( ) / kT[ ]

eE0$E j( ) / kT[ ]

j

#

The final results including both solvent and vibrational average are presented in Table 4. Table 4. Equilibrium angle (αmin) and corresponding Jex value (

!

Jmin

ex ), and vibrationally avaraged Jex values at 0K and 298K for BVD (units: cm-1 and degrees)a. Vacuob CHCl3 αmin 34.6 26.8

!

Jmin

ex 566 (606) 757 <

!

Jmin

ex >0 K 585 (601) 774 <

!

Jmin

ex >298 K 721 (729) 844 Crystal Solution Exp.1 887 760 a Basis set: 6-31G*.b The values in parentheses have been obtained using the geometries computed in vacuo and J computed in CHCl3. 1 From reference [54].

Although in this case direct solvent effects are negligible the effect of solvation on the structure is not. The results obtained for the computed singlet triplet splitting are in quantitative agreement with the experimental data. Furthermore the increase of J value with increasing polarity of the environment (i.e. going from vacuo to CHCl3) is in agreement with the experimental observation (i.e. going from solution to crystal). A detailed description of the results and a characterization of the monomer are given in the original paper [55]. 3.2.3 [Fe2(OH)3(tmtacn)2]2+ 5 This complex is an example of a mixed valence system where the two iron centres (namely a Fe(III) d5 and an Fe(II) d6) are completely equivalent from an experimental point of view both on the long time scale (X-ray) and on short time scale (Mössbauer) [57-58]. This complex is often referred to as a Class III complex in Robin-Day classification [59]. Because valence electron delocalisation between the metal centre is possible, a spin Hamiltonian taking into account not only exchange but also the double-exchange terms have to be considered. From a magnetic point of view it has S=9/2 high spin as ground state and the difference in energy between the ground and the first S=7/2 excited state was evaluated as 720 cm-1. The experimental Jex value was determined as J<235cm-1 and the B parameter was determined, from UV-Vis spectra, as B=1350cm-1 [58]. The BS approach was applied in the modification needed to taken into account the double exchange interaction [60]. A model complex, schematically depicted in Figure 19, where each tmtacn macrocyclic ligand was substituted with 3NH3 molecules was used in all the calculation [61].

Fe

OH

HO

HO

Fe

H3N

H3N

H3N

NH3

NH3

NH3

Figure 19. Schematic structure of the model system [Fe2(OH)3(NH3)6]2+ The computed Jex and B values are 137 cm-1 and 1369 cm-1 respectively which nicely compare with the experimental data. From the constructed spin ladder (shown in Figure 20) the splitting between the ground and first excited spin state can be evaluated as 752 cm-1 in good agreement with the experimental value (720cm-1).

5 Tmtacn = trimetyltriazacyclononane.

13690

6570

8144

9856

11704

+

+

+

+

-

+

-

-

-

-

3832

2668

1642752

9/2

7/2

3/2

5/2

1/2

HDVV+ Double ExchangeHDVV Figure 20. Computed spin ladder for [Fe2(OH)3(NH3)6]2+ Furthermore to fully characterise the system an analysis of the PES along a simulated normal coordinate corresponding to the anti-symmetric breathing motion was performed (see ref. [61] for detailed description). This motion is usually referred to as the active normal mode for electron delocalisation [62]. Therefore the presence of a single minimum in this surface corresponding to the symmetric structure of the metallic sites is consistent with a fully delocalised picture of the complexes (Class III). From the analysis of the computed PES (see ref. [61]) an estimate of the corresponding harmonic frequency was given. This frequency is in good agreement with the experimental one (ωcomp=330-360cm-1 ; ωexp= 306cm-1). 3.2.4 [Cu3O2L3]3+ 6 The [Cu3O2L3]3+ was recently the subject of serious experimental [63]-[64] and theoretical investigation [64]-[66]. This system, shown in Figure 21 is indeed extremely interesting since it is the first 3:1 metal:O2 complex which shows oxygen scission (d(O-O)=2.37Å).

6 L= N-Permethylated (1R,2R)-cyclohexanediamine

9/2 7/2

5/2 3/2 1/2

Figure 21: Schematic structure of [Cu3O2L3]3+, L= N-Permethylated (1R,2R)-cyclohexanediamine. In fact, contrary to what happen in nature, where each of the copper atom in the Cu active sites supplies only one electron to reduce O2, here one of the copper atoms supplies 2 electrons and the other two 1 electron each thus leading to a formal Cu(II)-Cu(II)-Cu(III) -O2

2- complex. All experimental data (NMR, SQUID, MCD [63]-[64]) are consistent with a description of a polynuclear complex built up from two paramagnetic S = 1/2 Cu(II) centres, ferromagnetically coupled to give an S = 1 ground state and one diamagnetic (d8) Cu(III). The system is experimentally characterised as a completely localised mixed valence system where a hole is localised on the d8 Cu(III) center. Previous studies [64] based on a model system where each of the ligands (L = N-Permethylated (1R,2R)-cyclohexanediamine) was substituted by two ammonia molecules, showed that the localisation is driven by the Jahn-Teller effect [67]. In fact, the computed electronic structure for the S = 1 state in the highest symmetry possible for the model compound (i.e. D3h) shows that the SOMOs are of a2” and e1” symmetry. The resulting degenerate 3E’ ground state is therefore subject to Jahn-Teller instability and it will distort to remove the degeneracy. The distortion mode (QJT) can be derived by a symmetry argument and it leads to a descent in symmetry from D3h to C2v. By performing a set of structural optimisations on the model systems both in D3h and in C2v we determined the minimum energy localised structure and the values of the Jahn Teller parameters (� , RJT and EJT) reported in Table 5 Table 5: Jahn-Teller parameters computeda for the model system [Cu3O2(NH3)6]3+. EJT in eV, Δ in cm-1 and RJT in Å [68]. Rel = Pauli relativistic. LDA BP PW91 LDArel BPrel PW91rel EJT 0.192 0.185 0.183 0.203 0.188 0.190 � 9.54 -8.01 3.84 11.00 -3.94 5.16 RJT 1.93 0.13 0.15 0.11 0.30 0.12 a Calculations performed using the ADF2000.02 QM/MM implementation [70]. Orbitals up to 3p for Cu, and 1s for N and O were kept frozen; valence shell of H, N, O were described by a double-ζ STO set plus one polarisation function. In the case of Cu atoms a valence triple-ζ plus one polarisation function STO basis was used. Using the methodology of Daul et al. [69] the full potential energy surface corresponding to the E-e Jahn-Teller coupling was also computed. Concerning the calculation of the isotropic coupling constant (Jex), Root et. al. [64] computed the J value using the BS approach obtaining a value of 878 cm-1 which greatly overestimates the experimental value (14 cm-1 [64]). They attribute the large difference between the experimental and the theoretical value to the fact that in the Broken Symmetry approach the contribution of the close lying LUMO is neglected. We applied the Single Determinant method and computed the full spin manifold arising from the occupation of the SOMO (12b2 and 9a2) and of the LUMO (13b2): a 2 electron in 3 orbital problem. Thus we obtain one Single Determinant (SD) for (12b2)2, one SD for (9a2)2, one SD for (13b2)2, four SD for (9a2)1(12b2)1, four SD for (9a2)1(13b2)1 and four SD for

(12b2)1(13b2)1, i.e. a total of fifteen microstates. Out of this set, the energy of only nine of them (the so-called non redundant single determinants) is needed. They are represented in fig. 22 below.

Figure 22: Schematic representation of the nine Non-Redundant Single Determinants (NRSD). By computing the energy E(i) of the nine NRSD depicted in Fig. 22 above, we can get the energies of the 9 spin states (four 1A1, two 1B1, two 3B1 and one 3A1) via the following relationships:

!!!!

"

#

$$$$

%

&

+'

+'+'

+'+'

+'+'

=

)3(2)8(000

0)2()3()8()6()5(

0)3()8()1()7()0(

0)6()5()7()0()4(

1

1

EE

EEEEE

EEEEE

EEEEE

A

!"

#$%

&

+'

+'=

)6(2)5(0

0)7()0(1

1

EE

EEB

!

3B1 =

E(0) 0

0 E(5)

"

# $

%

& '

)8(1

3EA =

The resulting spin state energies are collected in Table 6 with respect to the energy of the 3B1 ground state. It is easy to notice that the results we obtain are not better than those of Cole et al. Therefore the large error in the triplet singlet energy gap can not be ascribed fully to the methodology used. Table 6. Computed energy of the different spin states (at the LDA and GGA level) of the [Cu3O2(NH3)6]3+ model system. Energy in eV.

LDArel PW91rel BPrel 1 3B1 0 0 0 2 3B1 0.4974 0.5148 0.5293 1 3A1 0.3348 0.9848 0.3629 1 1A1 0.2015 0.3710 0.1026 2 1A1 0.8803 0.8509 0.8853 3 1A1 0.5960 2.5405 1.9353 4 1A1 2.7188 2.7046 2.9830 1 1B1 0.6696 0.4559 0.4634 2 1B1 0.8807 0.8077 0.8131

To understand the effect of the structural model on the computed spin manifold we performed two different types of calculation. First we simply used the crystallographic position of the Cu3O2N6 core and saturate the N with three H and compute at this frozen geometry the triplet-singlet gap. Next we considered the full system and included the ligand at a QM level and fully optimised the

structure using a QM/MM approach. In particular we included in the QM core the Cu3O2N6 where the link bonds have been capped by hydrogen atoms while the ligand part was modeled at the MM level using the AMBER force field. The QM core was treated at a GGA BP level. The resulting structure shows a core topology similar to that of the model system except for one structural parameter the NCuN angle. This angle is dramatically reduced (about 10°) leading to better agreement with the experimental values. On this optimised structure we computed the spin manifold at the full QM level using the SD method (always using the BP exchange-correlation functional). The results obtained for the triplet-singlet energy gap are reported in Table 7. Table 7. Computed triplet-singlet energy gap for [Cu3O2L3]3+ and related models � (T-S) [Cu3O2(NH3)6]3+ a 827 [Cu3O2(NH3)6]3+ b 25/550 [Cu3O2L3]3+ c 72 [Cu3O2L3]3+ d 157/340 Exp. 14 a optimised model geometry; BPrel level b Xray [Cu3O2N6] core structure; BPrel. The two values correspond to the two different crystallographic units.c optimised real system geometry at QM/MM level; BP level.d Xray [Cu3O2L3]3+

structure BP level. The two values correspond to the two different crystallographic units. From all these results, the following conclusions can be drawn: • neglecting the full ligand mainly has structural effects (i.e. the change of the NCuN angle)

which indirectly affects the computed spin manifold • the direct effect of the ligand on the electronic structure of the core is negligible since the results

obtained for a model of the X-ray structure (25 cm-1 and 550 cm-1) and the full X-ray structure (157 cm-1 and 340 cm-1) are practically equivalent within the error range of the methodology

• the model retains all the electronic features of the system in terms of exchange and super-exchange pathways but allows for a structural flexibility that the real system does not posses.

We conclude that the model we used influences only the triplet singlet energy gap due to structural effects while from an electronic point of view it correctly reproduces the core of the system. It also worthwhile to recall that discrepancies between the experimental and the computed values are also due to the approximations implicit in the methodology we apply and better agreement is not expected if not fortuitous. Finally, we can conclude that when dealing with complex molecular system, a QM/MM approach for structural optimisation combined with a DFT calculations for the spin manifold do yield a quantitative description of the magnetic properties of the system. 4. Concluding Remarks We have given in this work a short overview of the calculation of molecular magnetic properties using DFT. A comparison between the DFT based approach and post HF ones was reported in the case of small model systems. From our and other results available in the literature, we can conclude that the prediction of molecular magnetic properties with chemical accuracy is still a difficult task for DFT. The problem we have to face is the accurate calculation of multiplet structure with very small splitting. Hence, the problem reduces to the proper treatment of near degeneracy correlation. We believe that progress in this field will depend upon the development in the future of new functionals, which will include this interaction in some way. Probably, many-body wave-function theory will be needed for this purpose. In this respect, we point out Density Matrix Renormalisation Groups [71] applied to quantum chemistry might play a key role in the near future. Nevertheless, we have shown in this review that the prediction of magnetic exchange coupling constants: Jex is feasible for fairly large systems, provided that |Jex| exceeds 100 cm-1, i.e. for large

and moderately large coupling. Moreover, we have shown that DFT applied to molecular magnetism does accurately describe magneto-structural correlation and yields an illuminating insight into the exchange pathways involved. References: [1] a) Caballol, R.; Castell, O.; Illas, F.; Moreira, I. de P. R.; Malrieu; J. Phys. Chem. A

1997, 101, 7860. b) Ceulemans, A.; Heylen, G. A.; Chibotaru, L. F.; Maes, T. L.; Pierloot, K.; Ribbing, C.; Vanquickenborne, L. G.; Inorg. Chim. Acta 1996, 251, 15. c) Molecule-Based Magnetic Materials: Theory, Techniques, and Applications: Turnbull, M.M.; Sugimoto, T.; Thompson, L.K., Eds. ACS Symposium Series 644, Washington DC, 1966.

[2] a) Awaga, K.; Sugano, T.; Kinoshita, M.; Solid State Commun. 1986, 57, 453. b) Awaga, K.; Maruyama, Y. J. Chem. Phys. 1989, 91, 2743. c) Veciana, J.; Rovira, C.; Armet, O.; Domingo, V. M.; Crespo, M. I.; Palacio, F. J. Am. Chem. Soc. 1991, 113, 2552. d) Rajca, A. Utamapayana, S. J. Am. Chem. Soc. 1993, 115, 2396.

[3] a) Benelli, C.; Caneschi, A.; Gatteschi, D.; Pardi, L.;in Magnetic Molecular Material: Gatteschi, D., Kahn, O., Miller, J. S., Palacio, F., Eds. Kluwer: Dordrecht, The Neatherlands, 1991: p. 233. b) Caneschi, A.; Gatteschi, D.; Rey, P.; Prog. Inorg. Chem. 1991, 39, 331. c) Dei, A.; Gatteschi, D.; Inorg. Chim. Acta 1992, 813.

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Part III, A DFT based Ligand Field model for Magnetic Exchange Coupling in Transition Metal Dimmer Complexes: Principles Abstract A DFT based ligand field model for magnetic exchange coupling in homonuclear transition metal dimmer complexes is presented. It is based on a model of localized d-electrons and on a procedure allowing to express its parameters in terms of exchange and Coulomb integrals from DFT calculations of a homonuclear dimmer complex. We describe a recipe to relate the latter integrals with the manifold of all Slater determinants of the active space originating from the magnetic electrons of the dimmer. The method is applied to d9 and d1 model clusters involving CuII and Ti2Cl9

3- and has been extended to cases with more than one unpaired electron per site. 1. Introduction The electronic spectra and magnetic properties of transition metal (TM) complexes are governed by the shape and energies of the molecular orbitals with dominant d-character i.e. of d-orbitals modified by overlap with the surrounding ligands. The ground state and excited state multiplets are usually adequately described by ligand field theory. When joined by bridging ligands, two or more TM ions interact weakly with each other to give rise to exchange split multiplets and to magnetic exchange coupling. For ions without orbital degeneracy in their ground state, the exchange coupling of the spin momenta on the two metallic centres with ground state spins S1 and S2 are adequately described by the phenomenological Heisenberg-Dirac-van Vleck spin Hamiltonian HS = –J12S1.S2 (1) The eigenvalues of this Hamiltonian within the basis of the spin-functions S=S1–S2…S1+S2 are easily obtained from eq. 2 E(S) = – (J12/2)[S(S+1)–S1(S1+1)–S2(S2+1)] (2) The exchange integral J12 can be deduced from experiments, either according to Landé’s interval rule (eq.3) directly (spectroscopy, inelastic neutron scattering), ES-1 – ES = S·J12 (3) - or by the interpretation and fit to magnetic susceptibility data based on eq.(2). The calculation of the J12, has been and still is a great challenge for theoretical chemists [1]. Owing to the localized nature of the d-electrons, calculations of J12 seem to be most adequately done starting from localized MOs, utilizing the concept of magnetic orbitals as described first by P.Anderson in his superexchange model [2]. The elaboration of this formalism using MO theory has lead Hay, Thibeault and Hoffman to express the transfer (hopping) integral in Andersons’ theory in terms of the energy separation between the two MO’s resulting from the overlapping localized magnetic orbitals [3]. More sophisticated calculations of the exchange constant based on ab-initio methods by de Loth et al [4], Broer and Maaskant [5] and more recently using the difference dedicated CI [6-7] and CASSCF and CEPA methods [8] are restricted to smaller systems, due to the computational demand, or to coordination geometries truncated in a somewhat unrealistic way [8]. The advances of applying Density Functional methods when calculating J12 evolved in two different

approaches. The first one, uses the broken spin symmetry approach by Noodleman et all [9-10] in which case the value of J12 is obtained from energy difference: (eq. 4)

!

J =E BS( ) "E HS( )

2S1S2

(4)

In this equation the overlap between the two magnetic orbitals has been neglected. It represents the energy difference between the total energies of two spin-unrestricted DFT calculations: one for the high-spin state, yielding the energy E(HS) and one for a MS=0 determinant with energy E(BS) obtained in spin polarized DFT calculations by breaking the symmetry, i.e. using different orbitals for α and β spins. It has been shown that this approach allows to account for a large part of the electronic correlation [11]. The resulting values of J12, exceed experimental values typically by a factor of 2. Possible reasons for this overestimation has been discussed [12] and some improvements proposed [13]. Another procedure to calculate J12 is based on a single determinant approach [14]. It consists in the following recipe. Let us assume that two semi-occupied orbitals l1 and l2 on M1 and M2 couple to yield an in-phase (a) and an out-of-phase (b) MO (eq.5).

!

a =1

2dl1 + dl2( )

b =1

2dl1 " dl2( )

(5)

Six micro-states or Single Determinants (SD) are possible. Two are doubly occupied

!

a+a" ,

!

b+b"

and four are singly occupied

!

a+b" ,

!

a+b

+ ,

!

a"b

+ ,

!

a"b" . The doubly occupied SD correspond to

closed shells and are spin singlets, whereas the singly occupied SD correspond to a singlet and to a triplet. The two SD with MS=0:

!

a+b" and

!

a"b

+ , are mixed states belonging to a singlet and to a

triplet. The energies of all these determinants can be easily calculated from DFT. Let us denote their energies respectively by

!

E1 = E a+a"( ),

!

E2 = E b+b"( ),

!

E3 = E a+b

+( ) = E a"b"( ) ,

!

E4 = E a+b"( ) = E a

"b

+( ).

To obtain these energies, a two-step calculation scheme is applied. First a spin restricted calculation with a so-called Average Of Configuration (AOC) occupation (…a1b1) is carried out yielding a corresponding set of MOs {… a, b, …}. In a second step, these Kohn-Sham orbitals are kept frozen in order to evaluate the four SD-energies E1, E2, E3 and E4 (spin-polarized DFT) without further SCF iterations. We note that the E4-E3 difference equals the exchange integral [abab] which is also the quantity accounting for the mixing (1:1 in the limit of a full localization) between the a+a- and b+b- functions. This leads to matrix (eq.6) which after diagonalization yields the eigenvalues E– and E+ and the singlet-triplet energy separation E– – E3, i.e. J12

!"

#$%

&

'

'

234

341

E)E(E

)E(EE (6)

Applications to various systems have been reviewed [15].

Recently, a new DFT based ligand field model (LF-DFT) has been proposed [16-17]. In this approach inter-electronic repulsion and ligand field parameters have been adjusted to the energies of all Slater determinants resulting from a dn configuration, with a remarkable consistency between DFT and calculated SD energies. In a second step, a full CI within the ligand field domain of SD has been carried out to account for dynamical (via DFT) and non-dynamical (via the CI) correlations within the ligand field model. In this note we extend the formalism to dimmer systems, with special emphasise on the ground state exchange coupling. However, the procedure is equally well suited to exchange interactions in the excited states, as well, opening a good perspective for spectroscopic applications. The main idea is to obtain exchange and Coulomb repulsion parameters from the manifold of all SD determinants for a dimmer, resulting from electron replacements in the MOs which carry magnetic electrons an to express further these quantities in terms of parameters related to the magnetic orbitals localized on one centre. Thus the CI problem can be set up in a way similar to the LF-DFT for single centre but in addition, inter atomic exchange and charge transfer from one centre to another are taken into account. In this way, local multiplets (as in LF-DFT) and, in addition, their exchange-coupled components can be calculated on the same footing. After a brief computational part (Section 2) we will describe our approach considering a M1-M2 dimmer which contains a single electron (hole) per centre (Section 3). An analysis of J12 in terms of the underlying ferro- and antiferro-magnetic exchange coupling pathways, which is not easy using other methods [14] will be demonstrated. Finally, the performance of the up-to-date functionals as to their ability to describe exchange coupling phenomena will be discussed. 2. Computational details All DFT calculations have been performed using the ADF program package, version 2002.02 [18-20]. The approximate SCF KS one-electron equations are solved by employing an expansion of the molecular orbitals in a basis set of Slater-type orbitals (STO). All atoms were described through triple-ς STO basis sets given in the program database (TZP basis) and the core-orbitals up to 3p for the TM (“large core”) and up to 1s (for O,C,N) were kept frozen. In order to study the sensitivity of our results with respect to the adopted functionals, we used both, the gradient corrected (GGA) as well as the local density approximation (LDA). In the latter case, the exchange-correlation potential and energies have been computed according to the Vosko, Wilk, and Nusair’s (VWN) [21] parameterization of the electron gas data. We also checked functionals with a correct asymptotic behaviour, such as the gradient regulated asymptotic correction (GRACLB, [22]). For the generalized gradient approximation (GGA) we used the functional of Perdew et al [23], which uses gradient correction to the local expression both of the exchange and the correlation energies. SD energies have been obtained using spin-unrestricted non-SCF DFT calculations based on average-of-configuration electron density. This is routinely accomplished in ADF code making use of “SLATERDETERMINANTS” keyword followed by occupation numbers defining each SD. 3. A DFT based LF model for magnetic exchange coupling We illustrate the essential of our approach using a dimmer with two equivalent magnetic orbitals dl1 and dl2 per site (eq.5). These are composed of 3d-functions of the TM (d) and tails on the ligands (l) and we assume these functions to be Löwdin orthogonalized to each other justifying use of eq.5. Within the Anderson exchange model, dl1 and dl2 are singly occupied in the ground state giving rise to a triplet and a singlet with wave functions ψT and ψS (eq.7, 8). There are two further singlet states

!

"SCT and

!

"S

' CT arising when either of the two magnetic electrons is transferred to the other magnetic orbital, i.e.

!

"T = dl1+dl2

+; dl1

#dl2#;

1

2dl1

+dl2# + dl1

#dl2

+( ); (7)

!

"S =1

2dl1

+dl2# # dl1

#dl2

+( )

"SCT =

1

2dl1

+dl1# + dl2

+dl2#( ); "

S

' CT=1

2dl1

+dl1# # dl2

+dl2#( );

(8)

with ψS being by 2K12 at a higher in energy than ψT. We take the energy of the latter state as reference {E(ψT)=0}. K12 is the classical Heisenberg exchange integral eq. (9)

( ) ( ) ( ) ( ) == !! 212

*

1

12

2

*

11222

111 dVdVdldlr

dldlK [dl1dl2|dl1dl2] (9)

which is always positive. It reflects the exchange stabilization of the triplet over the singlet due to gain in potential energy connected with the spatial extension of the Fermi (exchange) hole (potential exchange). The ψS function can mix with the charge transfer state ψS

CT. Its energy, denoted with U equals the difference between the Coulomb repulsions of two electrons on the same centre,

!

dl1+dl1"

or

!

dl2+dl2" (U11=[dl1dl1|dl1dl1]=U22=[dl2dl2|dl2dl2]) and when they are on different centers

(U12=[dl1dl1|dl2dl2]), with respect to the ground state configuration (eq.10), i.e. U=U11–U12 (10) U is again a positive and large quantity (typically 5-8 eV). The interaction matrix element between ψS and ψS

CT (eq.11) reflects the delocalisation of the magnetic electrons due to orbital overlap, the quantity t12 being referred to as the transfer (hopping) integral between the two sites, i.e. <ψS|H|ψ CT

S>=2T12=2(t12+[dl1dl1|dl1dl2]) (11)

Calculations show that T12 = t12 in a very good approximations, differences being generally less than 0.002 eV. This term tends to lower the singlet- over the triplet-energy and is intrinsically connected with the gain of kinetic energy (kinetic exchange). The interaction matrix (eq.12) describes the combined effect of these two opposite interactions.

!

"S "SCT

2K12 2T12

2T12 U

#

$ %

&

' ( (12)

Perturbation theory yields eq. 13 for the (ES-ET)P energy separation, i.e. J12 . This allows us to decompose J12 into a ferromagnetic (

!

J12f ) and an anti-ferromagnetic (

!

J12af ) part.

( )U

TKJJJEE affP

PTS

2

12

12121212

42 !=+==! (13)

As has been pointed out already in [3], the parameters K12, U and T12 can be expressed in terms of the Coulomb integrals (Jaa, Jbb and Jab), exchange integral Kab and of ε(b)–ε(a), the KS-orbital energy difference. Eqs. 14,15 and 16 below, resume these relations:

( ) ( )42112

24

12

4

1EEEJJJK

abbbaa!+=!+= (14)

!

U =U11 "U12 = 2Kab = 2 E4 "E3( ) (15)

( ) ( ){ } ( )1212

4

1

2

1EEbbT !=!" ## (16)

We like to point out that these expressions are furthermore related with the energies of the single determinants |a+a-|, |b+b-|, |a+b+|, |a+b-| (i.e. E1, E2, E3 and E4 respectively). In deriving these expressions we made use of eqs (17-20). E1 = 2ε(a) + Jaa (17) E2 = 2ε(b) + Jbb (18) E3 = ε(a) + ε(b) + Jab - Kab (19) E4 = ε(a) + ε(b) + Jab (20) Thus, eqs. (14) - (16) allow us to obtain K12, U and T12 directly from DFT data and to compare them with the corresponding empirical values checking the consistency of the current functionals. Empirical estimates of K12, U and T12 can be deduced independently by a fit to magnetic and spectroscopic data (valence-bond CI approach (VBCI), Sawatzky [24], Solomon [25]). We get therefore a model of localized magnetic orbitals, whose parameters are readily obtained from the DFT SD energies E1,E2,E3 and E4 of the dimer.

Fig.1 Correlation diagrams between the energies of single determinants from DFT and the resulting multiplets of relevance for the magnetic exchange coupling in (Cu2+)2, Cu2(ac)4(H2O)2 and Ti2Cl9

3- model clusters.

Table 1 Energies of Slater determinants for dimmers of Cu(d9) and of Ti2Cl9

3-, calculated singlet(S)-triplet(T) splitting ES-ET, the value (ES-ET)P given by perturbation theory (=2K12-4t12

2/U) and the one deduced from a broken symmetry calculation (ES-ET)BS as well as available from experiment (exp.). Values of the transfer (hopping) integral (t12), the Heisenberg exchange integral (K12) and the effective transfer energy (U=U11-U12) are also included. LFDFT data in columns 1 and 3 for Cu2(ac)4(H2O)2 are calculated using LDA-GGA(PW91) functional with SD energies obtained without and with SCF interactions, while results in column 2 refer to a GRACBL functional.

Parameter (Cu2+)2 c Cu2(ac)4(H2O)2 a Ti2Cl9

3- b E1(a+a-) -0.784 -0.363 -0.367 –0.372 -0.040 E2(b+b-) 0.849 0.404 0.406 0.390 0.676 E3(a+b+) -1.495 -0.450 -0.454 -0.504 -0.547 E4(a+b-) -0.019 -0.009 -0.009 -0.010 0.265

J12=ES-ET -0.160 -0.114 -0.116 -0.110 -0.023 J12(p)=(ES-ET)P -0.173 -0.132 -0.139 -0.126 -0.027

J12(p)f 0.052 0.030 0.028 0.020 0.052 J12(p)af -0.225 -0.162 -0.167 -0.146 -0.079

(ES-ET)BS -0.201 -0.147 -0.069 (ES-ET)exp - -0.037 -0.065

K12 0.026 0.015 0.014 0.010 0.026 t12 0.408 0.189 0.193 0.190 0.179 U 2.953 0.882 0.890 0.988 1.623

a Cu-Cu, Cu-O(ac) and Cu-O(H2O) distances 2.525, 2.038, 2.355 Å (DFT geometry optimized); b distances Ti-Ti, Ti-Clbridge, Ti-Clterminal 3.503, 2.596, 2.434 Å and Ti-Clbridge-Ti angle 84.9o (DFT geometry optimized); c Cu-Cu distance 2.525 Å taken as in a) with 8 point charges (-0.5) at the vertices of a cube. In Table 1 and Figure 1 we present results obtained for (Cu2+)2,Cu(ac)4(H2O)2 (ac=acetate) and Ti2Cl9

3- with unpaired electrons on d-orbitals of dz2, dx2-y2 and dz2 type, respectively. For the latter two complexes a full DFT (spin-restricted) geometry optimization has been carried out beforehand. The corresponding geometries are shown in Table 1. In the hypothetical case of (Cu2+)2, the Cu-Cu distance has been chosen as for the acetate complex. We draw the attention to the rather large K12 values (120-210 cm-1); K12 have been deemed to be small by Anderson [2] and frequently neglected in theoretical studies [26]. Our results are in line with earlier ab-initio work [4] which showed that values of K12 could be rather large as also supported by experimental data. Thus, hydroxo-bridged dimmers of Cu2+, studied by Hodgson et al, exhibit values of K12 as large as 85 cm-1 for [Cu(bpy)OH]2(NO3)2 with an Cu-O-Cu bridging angle, close to 90o (95.6o) [27]. We should note that the values of J12 obtained with our present approach are smaller than the ones deduced from corresponding BS calculations, but still larger than the experimental ones. A critical result of our calculations is the small value of the charge transfer energy U in all considered cases. Typically, we expect values in the range 5-8 eV (see above). This leads to an overestimation of

!

J12af and hence to

the prediction of an antiferromagnetic coupling constants, which in cases like Cu(acac)4(H2O)2 can exceed the experimental value by a factor of 3 to 4. This result is common to the present LF-DFT approach as well as to DFT-BS calculations and does not change if a GRACLB functional, with correct asymptotic behaviour is taken into account. The possible reason for the underestimation of U by DFT might be the intrinsic failure of the up-to-date functionals to correctly describe charge separation and thus the underlying charge transfer phenomena. The point is, that in DFT

calculations for dimmers, U=2Kab (eq.15) is expressed in terms of an exchange integral, which is only approximately described in DFT. The quantity U corresponds to the energy Eg, i.e. the band gap energy in insulating solids, that separates the ground state, say in a TM1-TM2 pair, and the charge transfer configuration created taking one magnetic electron from one center (TM1) and put it into the other center (TM2). As has been pointed out in [28, 29], Eg is heavily underestimated, say by about 40% in LDA. The latter one does not take into account the possible discontinuity of the exchange corelation potential (δEXC/δn) when changing number of particles. A better approximation for this quantity is given by I-A (I, A –the ionisation potential and electron affinity of TM1 and TM2, respectively). An improved scheme for calculation of U beyond the LDA approximation was developed utilising an energy(orbital) dependent (δEXC/δn) exchange-correlation potential [28,29]. This is similar to what has been made when taking self-interaction energy corrections into account [30, 31]. At the same time we should point out, that in contrast to U, the K12 energy (given by Coulomb terms) and particularly t12 are reasonably approximated by LDA-GGA. 4. Conclusions The DFT based LF model developed recently has been extended with interatomic exchange parameters, transfer (hopping) integrals (t) and charge transfer energies U to allow calculations of the exchange split local multiplets. A mixed strategy to achieve this lies in using exchange coupling theory based on localized magnetic orbitals, but DFT calculations of the dimmer allowing to extract the model parameters from the manifold of all energies of Slater determinants resulting from electron replacements within orbitals which accommodate the magnetic electrons. The approach has been extended to dimmers of TM, with equivalent magnetic sites and more than one electron per site [32]. Calculated values of J are comparable or smaller than those, resulting from BS-DFT calculations. The main feature emerging from both approaches is the anomalously low CT energy U, a result we attribute to the approximations inherent in the existing functionals (LDA, GGA) and we motivate further work in this direction. References [1] O.Kahn, Molecular Magnetism (VCH Publishers, New York, 1993). [2] P.W.Anderson, Phys.Rev. 115(1959) 2. [3] P.J.Hay, J.C.Thibeault, R.Hoffmann, J.Am.Chem.Soc. 97(1975) 4884. [4] P. de Loth, P.Cassoux, J.P.Daudey, J.P.Marieu, J.Am.Chem.Soc.103(1981), 4007. [5] R.Broer, W.J.A.Maaskant, Chem.Phys. 102(1986) 103. [6] J.Miralles, J.-P.Daudey, R.Caballol, Chem.Phys.Lett. 198(1992)555. [7] J.Miralles, O.Castell, R.Caballol, J.P.Malrieu, Chem.Phys.172(1993)33. [8] K.Fink, R.Fink, V.Staemmler, Inorg.Chem.33(1994) 6219. [9] L.Noodleman, J.G.Norman, Jr. J.Chem.Phys. 70(1979) 4903. [10] L.Noodleman, J.Chem.Phys. 74(1981) 5737. [11] L.Noodleman, E.R.Davidson, Chem.Phys. 109(1986) 131. [12] R.Caballol, O.Castell, F.Illas, I. de P.R. Moreira, J.P.Malrieu, J.Phys.Chem.A, 101 (1997) 7860. [13] E.Ruiz, J.Cano, S.Alvarez, P.Alemany, J.Comput.Chem. 20(1999) 1391. [14] A.Bencini, F.Totti, C.A.Daul, K.Doclo, P.Fantucci, V.Barone, Inorg.Chem. 36(1997) 5022. [15] I. Ciofini, C.A.Daul, A.Bencini, Recent Advances in Density Functional Methods, Vol.3 Eds. V.Barone, A.Bencini, P.Fantucci (World Scientific, Singapure, 2002, p. 106). [16] M.Atanasov, C.A.Daul, C.Rauzy, Chem.Phys.Lett. 367(2003) 737. [17] M.Atanasov, C.A.Daul, C.Rauzy, Struct. and Bonding (Springer Verlag), in press. [18] E.J.Baerends, D.E.Ellis, P.Ros, Chem.Phys. 2(1993) 42.

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