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Comptes Rendus

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Modeling properties of molecules with open d-shells using density functional theory
Comptes Rendus. Chimie, Volume 8 (2005) no. 9-10, pp. 1421-1433.

Abstracts

An overview of the theory and applications of a recently proposed ligand-field density functional theory (LFDFT) is given. We describe a procedure based on DFT allowing to deduce the parameters of this non-empirical LF approach consisting of the following steps: (i) an average of configuration (AOC) DFT calculation, with equal occupancies of the d-orbitals is carried out (ii) with these orbitals kept frozen, the energies of all single determinants (SD) within the whole LF-manifold are calculated and used as a data base in a further step to provide all the Racah- and LF-parameters needed in a conventional LF-calculation. A more rigorous analysis of this approach in terms of Löwdin's energy partitioning and effective Hamiltonians is used to provide explicit context for its applicability and to set more rigorous criteria for its limitations. The formalism has been extended to account for spin-orbit coupling as well. Selected applications cover tetrahedral CrX4 (X = Cl, Br) and FeO42– and octahedral CrX63– (X = F, Cl, Br) complexes. Transition energies are calculated with an accuracy of 2000 cm–1, deviations being larger for spin-forbidden transitions and smaller for spin-allowed ones. Analysis show, that ligand field parameters deduced from experiment are well reproduced, while interelectronic repulsion parameters are calculated systematically to be by 30–50% of lower in energy. A generalization of the LFDFT theory to dimers of transition metals allows to calculate exchange coupling integrals in reasonable agreement with experiment and with comparable success to the broken symmetry approach; in addition they allow to judge ferromagnetic contributions to exchange coupling integral which have been ignored before. .

Une vue d'ensemble est apportée ici sur la théorie et les applications de la récente théorie des fonctionnelles de la densité du champ de ligand (LFDFT). Nous décrivons une procédure basée sur la théorie de la fonctionnelle de la densité (DFT), permettant de déduire les paramètres de cette approche non empirique du champ de ligand, comprenant les étapes suivantes : (i) un calcul DFT de configuration moyenne (AOC) est effectué, avec une occupation égale des orbitales d, (ii) avec ces orbitales figées, les énergies de tous les déterminants simples (SD) dans l'espace du champ de ligand sont calculées et sont utilisées comme base de données lors de l'étape suivante, pour fournir tous les paramètres Racah et champ de ligand nécessaires dans un calcul classique de champ de ligand. Une approche plus rigoureuse en termes de partition d'énergie selon Löwdin et d'hamiltonien effectif est utilisée pour fournir un contexte explicite à son applicabilité et pour mettre en place des critères plus rigoureux, dans le but de fixer les limites de la méthode. Le formalisme est étendu pour prendre aussi en compte le couplage spin–orbite. Les exemples sélectionnées comprennent les complexes tétrahédriques CrX4 (X = Cl, Br) et FeO43– (X = F, Cl, Br). Les énergies de transition sont calculées avec une précision de ±2000 cm–1, les déviations étant plus grandes pour les transitions de spin interdites et plus petites pour celles qui sont autorisées. L'analyse montre que les paramètres de champ de ligand déduits de l'expérimentation sont bien reproduits, tandis que les paramètres de répulsion inter-électronique sont calculés systématiquement de 10 à 20% plus petits. Une généralisation de la théorie de la fonctionnelle de densité du champ de ligand aux dimères des métaux de transition permet de calculer les constantes de couplage d'échange en accord raisonable avec l'expérimentation et avec un succès comparable à l'approche de la symétrie brisée ; de plus, ils permettent de juger des contributions du ferromagnétisme à la constante de couplage d'échange, qui jusqu'à présent avait été ignorée. .

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Accepted:
Published online:
DOI: 10.1016/j.crci.2005.04.003
Keywords: Density functional theory, Ligand-field theory, Spin–orbit coupling, Zero-field splitting, Magnetic exchange coupling
Keywords: Théorie de la fonctionnelle de la densité, Théorie du champ de ligand, Couplage spin–orbite, Zero-field splitting, Couplage d'échange magnétique

Mihail Atanasov 1, 2; Claude A. Daul 1

1 Département de Chimie, université de Fribourg, chemin du Musée 9, CH-1700 Fribourg, Switzerland
2 Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
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     title = {Modeling properties of molecules with open d-shells using density functional theory},
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Mihail Atanasov; Claude A. Daul. Modeling properties of molecules with open d-shells using density functional theory. Comptes Rendus. Chimie, Volume 8 (2005) no. 9-10, pp. 1421-1433. doi : 10.1016/j.crci.2005.04.003. https://comptes-rendus.academie-sciences.fr/chimie/articles/10.1016/j.crci.2005.04.003/

Version originale du texte intégral

1 Introduction

When describing electronic structures of transition metal complexes one goes a different way from the point of view of quantum chemistry and experimental spectroscopy. Experimentalists make use of ligand field parameterized (effective) Hamiltonians, or of the spin-Hamiltonian when interpreting optical or ESR spectra, or they apply the Heisenberg exchange operator in the case of magnetic exchange coupling. Empirical models of that kind have been therefore tools for description, rather than tools for prediction of ligand field properties. A quantum chemist solves more or less rigorously numerically the Schrödinger equation (ab initio) or the Kohn–Sham equations (density functional theory, DFT) and is able to make predictions as well. However, numerical results are sometimes not easy to interpret or analyze and the bridge between the ab-initio approach and chemical intuition is not always transparent. DFT became increasingly popular in recent time. As manifested by the groups of Baerends, Ziegler [1,2] and Daul [3] it is able to predict both ground and excited states of TM complexes. Recently, a new approach has been developed in our group [4]. It is based on a multi-determinant description of the multiplet structure originating from the well defined dn configurations of a TM in the surrounding of coordinating ligands by combining the CI and the DFT approaches. In doing so, both dynamical (via the DFT exchange-correlation potential) and non-dynamical (via CI) correlation is introduced, the latter accounting for the rather localized character of the d-electron wavefunction. The key feature of this approach is the explicit treatment of near degeneracy effects (long-range correlation) using ad hoc configuration interaction (CI) within the active space of Kohn–Sham (KS) orbitals with dominant d-character. The calculation of the CI-matrices is based on a symmetry decomposition and/or on a ligand field analysis of the energies of all single determinants (SD, micro-states) calculated according to DFT for frozen KS-orbitals corresponding to the averaged configuration, eventually with fractional occupations of the d-orbitals. This procedure yields multiplet energies with an accuracy within 2000 cm–1. Currently, the procedure has been extended to spin-orbit coupling [5] and allows to also treat Zero-Field Splitting (ZFS) [6], Zeeman interactions and Hyper-Fine Splitting (HFS) [7] and magnetic exchange coupling [8] as well.

In this account a more rigorous analysis of our approach utilizing Löwdin's energy partitioning [9] and effective Hamiltonians theory [10,11] is used, providing explicit context for its applicability and allowing to set up more rigorous criteria for its limitations. We will then briefly sketch the mathematical procedure behind our approach. An extension to spin-orbit coupling will be also given. Selected applications cover tetrahedral CrX4 (X = Cl, Br) and FeO42– and octahedral CrX63–(X = F, Cl, Br) complexes. In addition the spin-orbit coupling constant of Cr(acac)3 deduced from a DFT-ZORA calculation will be applied to calculate the ground and excited state zero-field splitting.

A generalization of the LFDFT theory to dimers of transition metals allows to calculate exchange coupling integrals in reasonable agreement with experiment and with comparable success to the broken symmetry approach; in addition they allow to judge ferromagnetic contributions to exchange coupling integrals, the latter are usually being ignored. An illustration of how the procedure works will be given for hydroxo-dimers of Cu2+.

2 Partitioning technique, effective Hamiltonians and the ligand field approach

Let us consider a system consisting of transition metals and ligands, which can be bridging or terminal (Fig. 1). In electronic structure calculations, one usually immediately recognizes antibonding molecular orbitals (MO's), as being dominated by metal |nd〉 functions, which are partly filled and bonding orbitals dominated by ligand AO's which are fully occupied. Following Löwdin [9] we can write down the Schrödinger equation H ψ = E ψ in a discrete representation based on the use of a complete orthonormal set Φ = {Φk} and introduce the Hamiltonian matrix H = [............Hkl............] and the column vector c = [ck......] using the relations:(1)

Hkl=Φk|H|Φl,ck=Φk|Ψ
(2)
Ψ=kckΦk
(3)
Hc=Ec

Fig. 1

Partioning scheme for LFDFT approach of a pair of transition metals joined by bridging ligands.

Let us subdivide the system into two parts, one build of from metal nd-orbitals (d) and another composed of valence metal (n + 1) s and (n + 1) p and ligand functions (v).

Then the eigenvalue problem (3) can be represented in a form given by:(4)

[HddHdvHvdHvv][cdcd]=[E1d00E1v][cdcv]=E[1d001v][cdcv]
with 1d and 1v the identity matrices of dimension Nd × Nd and Nv × Nv, respectively.

Collecting terms with same dimension, we get:(5.1)

[Hdd1dd]cd+Hdvcv=0
(5.2)
Hvdcd+[Hvv1v]cv=0

One can easily express the column vector cv in terms of cd (Eq. (6)), and after substitution into Eq. (5.1), one obtains a Hamiltonian completely restricted to the d-subspace:(6)

cv=(HvvE1v)1 Hvdcd
(7)
(HddEId)cd+Hdv(HvvEIv)1 Hvdcd=0

We thus arrive at a pseudo-eigenvalue equation (Eq. (8)), with the explicit form of Hddgiven by Eq. (9). No approximation is inherent in Eq. (9). The representation given by Eq. (9) provides:(8)

Hddcd=EIdcd
(9)
Hdd=Hdd+Hdv(HvvEIv)1 Hvd
the physical background of ligand-field theory. The matrix Hdd´ represents the purely electrostatic effect of the metal d-orbitals by the surrounding ligand nuclei and the valence electron distribution excluding the d-electrons. It is subject of the usual description in terms of a ‘crystal-field theory’ applied to TM impurities in crystals. Since orbitals of the subsystems d and v are orthogonal to each other, the Hdd´ matrix also incorporates important exchange (Pauli) repulsion terms; these have been shown to be proportional to the squares of corresponding overlap integrals, allowing one to formulate the ligand field as a pseudopotential [12]. The second term in Eq. (9) is energy dependent. For diagonal Hdd, perturbation expansions (which presuppose |Hvv – Hdd|>> |Hvd|) allow us to identify E with the corresponding diagonal element of Hdd´. It is this second term that reflects metal–ligand covalency (charge-transfer) and is the subject of parameterization by the angular overlap model (AOM) [13,14]. Earlier analysis based on Eq. (9) have been purely theoretical, attempting to place a correct context and limits of applicability of ligand-field approach within the main body of quantum chemistry [15]. In Section 3, we describe a practical scheme allowing to deduce the matrix Hdd´ from DFT and to apply it directly to the calculation of dn electronic multiplets.

The one-electron representation of Eq. (9) can easily be extended to systems with more than one TM. The Hdd matrix for such cases contains terms that account for d-electron delocalization (via the second term in Eq. (9)) from one metal to another – indirect (via the bridging ligands) or direct (via corresponding off-diagonal terms of Hdd´) [15]. It gives rise to magnetic exchange coupling. This will be the topic of Section 4.

Finally, the one-electron scheme given by Eq. (9) can be extended to the many electron states resulting from the redistribution of all the d-electrons within the active d-orbital subspace. Their treatment requires two-electron repulsion integrals in addition to the one-electron ones. The general scheme we describe in Section 3 allows to also deduce these intergrals from DFT.

3 The ligand-field density functional theory (LFDFT)

The current DFT software includes functionals at the Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) levels. The former approximation is well adapted for molecular structure calculation: M–L bond lengths are usually accurate to ±0.02 Å but bond energies are too large. The latter approximation, however, is roughly twice as expensive in computer time and yields M–L bond energies accurate to ±5 kcal mol–1. Recently a new generation of functionals called meta-GGA emerged. These functionals are more accurate but also more expensive and their implementation in computer codes is not yet generalized. It is generally accepted that all these functionals (LDA and GGA) describe well the so-called dynamical correlation, however, none of them includes near degeneracy correlation. In the method described next we address this problem specifically and include CI of valence electrons on the d-orbitals. The calculation scheme we developed includes three steps as described next.

We assume that we know the molecular geometry, either from a first principle geometry optimization or from X-ray data. Moreover, for the sake of simplicity, let us focus the following description to open d-shells: the extension to open f-shells is similar. The first step consists in a spin-restricted, i.e. same orbitals for same spin, Self-Consistent Field (SCF) DFT calculation of the average of the dn configuration (AOC), providing an equal occupation n/5 on each MO dominated by the d-orbitals. The KS-orbitals which we construct using this AOC are best suited for a treatment in which, interelectronic repulsion is – as is done in LF theory, approximated by atomic-like Racah parameters B and C. The next step consists in a spin-unrestricted calculation of the manifold of all Slater Determinants (SD) originating from the dn shell, i.e. 45, 120, 210 and 252 SD for d2,8, d3,7, d4,6, and d5 Transition Metal (TM) ions, respectively. These SD-energies are used in the third step to extract the parameters of the one-electron 5 × 5 LF matrix dμ|hLF|dv as well as Racah's parameters B and C in a procedure, which we describe below. Finally, we introduce these parameters as input for a LF program allowing to calculate all the multiplets using CI of the full LF-manifold utilizing the symmetry as much as possible. We should note that in classical LF theory, it is only the LF matrix which carries information about the symmetry and the actual bonding in the complex, thus providing useful chemical information.

In order to establish a link between ligand field theory and the energy of each SD mentioned earlier we need to introduce an effective LF-Hamiltonian hLFeff together with its five eigenfunctions φi {hLFeffφ1=ɛiφi,i=1,...,5} which are in general linear combination of the five d-orbitals:(10)

[φ1φ2φ3φ4φ5]=C[dxydxzdyzdx2-y2dz2]
and where C is an orthogonal 5 × 5 matrix. Using this definition we can express the energy of each SD in terms of φi|hLFeff|φi, the diagonal elements of the ligand-field splitting operator and electrostatic Coulomb and exchange integrals:(11)
E(SDkφ)=E(det|φi(k,1)σi(k,1)φi(k,2)σi(k,2)...φi(k,n)σi(k,n)|)=E0+ikφi|hLFeff|φi+i<j(JijKijδσiσj)

The SDkφ are labeled with the subscript k = 1, …, (10-n-) and where the superscript φ does refer to eigenfunctions of the ligand-field Hamiltonian hLF. The summation ik of ligand-field splitting matrix elements φi|hLFeff|φi specify the occupation of the level φi, while Jij=φiφi|φjφj and Kij=φiφj|φiφj denote Coulomb and exchange integrals; σi are spin functions and E0 represents the gauge origin of energy. This expression does only involve φi|hLFeff|φi the diagonal matrix elements of hLFeff. In order to obtain dμ|hLFeff|dν, the full matrix representation of hLFeff, we make use of the general observation that the KS-orbitals and the set of SD considered in Eq. (11) convey all the information needed to setup the LF matrix. In Ref. [4], we give a justification for this.

Thus, let us denote KS-orbitals dominated by d-functions which result from an AOC dn DFT-SCF calculation with a column vectors Vi. From the components of the eigenvector matrix build up from such columns one takes only the components corresponding to the d functions. Let us denote the square matrix composed of these column vectors by U and introduce the overlap matrix S:(12)

S = U UT

Since U is in general not orthogonal, we use Löwdin's symmetric orthogonalization scheme to obtain an equivalent set of orthogonal eigenvectors (C):(13)

C=S12U

We identify now these vectors as the eigenfunctions of the effective LF-Hamiltonian hLFeff, we seek as(14)

φi=μ=15cμidμ

Thus, the fitting procedure described below will enable us to estimate hii=φi|hLFeff|φi= and hence the full representation matrix of hLFeff as(15)

hμν=dμ|hLFeff|dv=i=15cμihiicvi
In order to calculate the electrostatic contribution (second term in Eq. (11)), it is useful to consider the transformation from the basis of SDkφto the one of SDμd. Using basic linear algebra, we get:(16)
|SDkφ=μTkμ|SDμd
where Tkμ=det|ci(k,l:n),j(μ,l:n)| i.e. the determinant of a n × n sub-matrix of Cσ, i.e. the direct product between C and [1001]:(17)
|ci(k,1),j(µ,1)ci(k,1),j(µ,2)...ci(k,1:n),j(µ,n)ci(k,2),j(µ,1)ci(k,2),j(µ,2)...ci(k,2),j(µ,n)[...][...]...[...]ci(k,n),j(µ,1)ci(k,n),j(µ,2)ci(k,n),j(µ,n)|
with the indices of the spin orbitals φi(k,1)σi(k,1),φi(k,2)σi(k,2),...,φi(k,n)σi(k,n) and dj(μ,1)σj(μ,1),dj(μ,2)σj(μ,2),...,dj(μ,n)σj(μ,n), respectively. Note that these indices are in fact a two-dimensional array of (number of SD) × (number of electrons or holes) integers. Finally, the energy of a SD in Eq. (11) can be rewritten as:(18)
Ek=E(SDkφ)=iφi|hLF|φi+μ,νTkμTkνSDμd|G|SDνd
where G = 1/r12, i.e. the electrostatic repulsion of all electron pairs in the LF manifold. The matrix elements in the second term of Eq. (18) are readily obtained using Slater's rules and the resulting electrostatic two-electron integrals < ab|cd > in terms of Racah's parameters.

X=(h11,...,h55,B,C)Having obtained energy expressions for each SDkφ: φi|hLF|φi, B, C and E0 are estimated using a least-square procedure. Using matrix notation, we obtain an overdetermined system of linear Eq. (19), with the unknown parameters stored in and given by:(19)

E=AX
(20)
X=(ATA)1ATE

It is worthwhile to note that the KS-eigenvalues ɛi of the orbitals with dominant d character are almost equal to the ligand-field parameters obtained in the fitting procedure, i.e.:(21)

ɛiKSE0+φi|hLF|φi

Thus, we conclude this section with the statement that the separation between KS-eigenvalues of orbitals with dominant d-character are good approximations for the ligand field splitting parameters.

For cases where the fine-structure is sought it is now easy to include spin–orbit coupling and calculate:(22)

SDkd|ζdi=1nisi|SDk'd
where ζd is the spin–orbit coupling constant, whose value is easily obtained either from a ZORA [16] calculation [5] or from the radial part of the d-orbitals as:(23)
dμ|ζ(r)|dνkorb_redRnd|r3|Rnd
where korb_red is an orbital reduction factor equal to the population on the metal Ao's. Thus, properties involving spin–orbit coupling are obtained in adding (22) to the full LF-Hamiltonian H0 + Hligand field + Helect_rep + Hspin–orbit and calculating the sought properties from its eigenfunctions.

From the splitting (due to the combined effect of spin-orbit splitting and perturbations of symmetry lower than Oh), say of the 3A2 and 4A2 of hexa-coordinate ground states of Ni(II), d8 and Cr(III), d3, it is possible to obtain the ZFS D-tensor using a conventional spin-Hamiltonian approach:(24)

HZFS=SDS=D(S^z223)+E(S^x2S^y2)
and equating the energies of the spin–orbit components of the 3A2 or 4A2 to the eigenvalues of this spin Hamiltonian. An application of this approach is presented in Section 6.3.

4 LFDFT and magnetic exchange coupling

The approach of Section 3 has been extended to the calculation of the exchange coupling constant of Heisenberg Hamiltonian (Eq. (25)) in the case of pairs of TM joined by bridging ligands [8]. Taking a TM pair with S1 = S2 = 1/2 spins, the singlet–triplet separation is given by Eq. (26), implying a positive and negative values of J12 for ferromagnetic and antiferromagnetic constants, respectively:(25)

HS=J12S1.S2
(26)
ES=0ES=1= J12
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. (27)).(27)
a=12(dl1+dl2)b=12(dl1dl2)

Six micro-states or SD are possible. Two are doubly occupied |a+a|, |b+b| and four are singly occupied |a+b|, |a+b+|, |ab+|, |ab|. 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 |ab+|, 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:(28)

E1=E(|a+a|),
E2=E(|b+b|),
E3=E(|a+b+|)=E(|ab|),
E4=E(|a+b|)=E(|ab+|),

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 E4E3 difference equals the exchange integral [ab|ab] 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. (29)) which after diagonalization yields the eigenvalues E and E+ and the singlet-triplet energy separation EE3, i.e. J12:(29)

[E1(E4E3)(E4E3)E2]

Within the Anderson exchange model [17], dl1 and dl2 are singly occupied in the ground state giving rise to a triplet and a singlet with wave functions ψT and ψS (Eqs. (30, 31)). There are two further singlet states ψSCT and ψS'CT arising when either of the two magnetic electrons is transferred to the other magnetic orbital, i.e.:(30)

ψT=|dl1+dl2+|;|dl1dl2|;12(|dl1+dl2|+|dl1dl2+|);
(31)
ψS=12(|dl1+dl2||dl1dl2+|)ψSCT=12(|dl1+dl1|+|dl2+dl2|);ψSCT=12(|dl1+dl1||dl2+dl2|);
with ψS being by 2 K12 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. (32)):(32)
K12=dl1(1)*dl2(1)1r12dl1(2)*dl2(2)dV1dV2=[dl1dl2|dl1dl2]
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 ψSCT. Its energy, denoted with U equals the difference between the Coulomb repulsions of two electrons on the same center |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. (33)), i.e.:(33)
U=U11U12

U is again a positive and large quantity (typically 5–8 eV). The interaction matrix element between ψS and ψSCT (Eq. (34)) reflects the delocalization 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.:(34)

<ψS|H|ψSCT>=2T12=2(t12+[dl1dl1|dl1dl2])

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. (35)) describes the combined effect of these two opposite interactions:(35)

ψSψSCT[2K122T122T12U+2K12]

Perturbation theory yields Eq. (36) for the (ESET)P energy separation, i.e. J12 . This allows us to decompose J12 into a ferromagnetic (J12f) and an anti-ferromagnetic (J12af) part.(36)

(ESET)P=J12P=J12f+J12af=2K124T122U

As has been pointed out already in [18], the parameters K12, U and T12 can be expressed in terms of the Coulomb integrals (Jaa, Jbb and Jab), exchange integral Kab and of ɛ(b) – ɛ(a), the KS-orbital energy difference. Eqs. (37–39) below, resume these relations:(37)

K12=14(Jaa+Jbb2Jab)=14(E1+E22E4)
(38)
U=U11U12=2Kab=2(E4E3)
(39)
T1212{ɛ(b)ɛ(a)}=14(E2E1)

We like to point out that these expressions are furthermore related with the energies of the SD |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. (40–43).(40)

E1=2ɛ(a)+Jaa
(41)
E2=2ɛ(b)+Jbb
(42)
E3=ɛ(a)+ɛ(b)+JabKab
(43)
E4=ɛ(a)+ɛ(b)+Jab

Thus, Eqs. (37–39) 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. Such empirical estimates of K12, U and T12 can be deduced by a fit to magnetic and spectroscopic data (valence-bond CI approach (VBCI), Sawatzky [19,20], Solomon [21]). 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 dimmer. An application of the approach for calculation exchange integrals in bis-hydroxo bridged Cu(II) dimers is given in Section 6.4.

5 Computational details

All DFT calculations have been performed using the ADF program package [22–25] (program release ADF2003.01). 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 data base (basis set TZP) and the core-orbitals up to 3p for the TM and up to 1s (for O, N), 2p (Cl) and 3d (Br) were kept frozen. We used the local density approximation (LDA), where exchange-correlation potential and energies have been computed according to the Vosko, Wilk and Nusair's (VWN) [26] parameterization of the electron gas data.

6 Applications

6.1 Tetrahedral d2 CrX4 (X = Cl, Br)

Tetrahedral d2 complexes possess a 3A2(e2) ground state as well as 3A23T2 and 3A23T1, e → t2 singly excited states. They give rise to broad d–d transitions in the optical spectra. In addition, spin-flip transitions within the e2 configuration lead to sharp line excitations. Multiplet energies from LDA agree within a few hundred cm–1 with experimental data. In particular the 3A23T2 transition energy and thus 10Dq nicely agrees with experiment as is seen from inspection of Table 1. Experimental transition energies for CrCl4 and CrBr4 as well as values of B, C and 10Dq deduced from a fit to experiment for CrCl4 are also listed.

Table 1

Electronic transition energies of CrX4, X = Cl and Br, with geometries optimized using LDA functional and calculated using values of B, C and 10 Dq from least square fit to DFT energies of the Slater determinants according to the method described in Section 2

Cr Cl4CrBr4
TermThis workLF-fitExp.aThis workExp.a
3A2(e2)00000
1E(e2)6542608963736666
1A1(e2)11 11410 58610 69810 869
3T2(e1t2 1)7008701072506163
3T1(e1t2 1)10 31610 44010 0009269
1T2(e1t2 1)13 45412 99112 00012 434
1T1(e1t2 1)15 07414 71814 037
1A1(t22)32 09930 59930 120
1E(t22)21 12120 71619 271
3T1(t22)16 03316 22916,66614 42413 258
1T2(t22)21 21720 82219 373
R(M–X)2.1042.264
B355376347
C190315791855
10 Dq700872506162
S.D.0.0300.030

a P. Studer, Thesis, University of Fribourg, 1975.

6.2 Octahedral CrIII d3 complexes

In Table 2 we list the predicted (this work), adjusted (LF fit to exp.) and observed (Exp.) multiplet energies for CrX63– (X = F, Cl, Br) complex ions. We used a LDA functional to calculate the CrIII–X bond lengths and we compare these results with energies from a LF-calculation utilizing values of B, C and 10 Dq obtained from a best fit to spectra from experiment. Bond lengths are too long while values of 10 Dq are too small compared to experiment. The situation improves if instead of optimized, experimental bond lengths are taken for the calculation. Even in this case, spin-forbidden transitions come out by 3000–4000 cm–1 too low in energy compared to experiment. Clearly, in this example of highly charges species, our prediction is much less accurate. In order to improve the quality of the prediction we obviously need to consider the environment of the CrX63– chromophore by adding an appropriate embedding potential to the KS-hamiltonian. Already the use of experimental bond lengths does significantly improve the precision of our calculation as mentioned before. A full analysis of this problem is given in [27].

Table 2

Electronic transition energies of CrX63–, X = F, Cl, Br with geometries optimized using LDA functionals calculated using values of B, C and 10Dq from least square fit to DFT energies of the Slater determinants and to experiment. The values of (10Dq)orb as deduced from the eg – t2g KS-orbital energy difference taken from the …t2g1.8eg1.2 SCF KS-energies are also listed. Experimental transition energies are also listed

TermCrF63–CrCl63–CrBr63–
This workLFT fit to exp.Exp.This workLFT fit to exp.Exp.This workLFT fit to exp.Exp.
4A2g(t2g3)000000000
2Eg(t2g3)12 49715 80216 300a10 75614 42614 430b10 33313 90013 900b
2T1g(t2g3)13 04416 46116 300a11 18014 87310 69414 348
2T2g(t2g3)18 62823 26023 000a15 91821 03715 18520 281
4T2g(t2g1eg1)13 56915 29815 200a10 91112 80012 800b981612 40012 400b
4T1g(t2g1eg1)19 44322 26221 800a15 61818 19818 200b13 99217 70017 700b
2A1g(t2g1eg1)24 07128 70920 05625 35118 70924 459
2T1g(t2g1eg1)26 34831 47321 87827 42120 31626 503
2T2g(t2g1eg1)25 95930 97021 56827 07920 04726 159
2Eg(t2g1eg1)27 81933 34123 14729 09821 53028 126
4T1g(t2g1eg1)30 33934 63635 000a24 37528 45521 86127 643
R(M–X)1.957-1.933c2.4192.335d2.5882.47e
B605734484550427543
C269434922403345023953296
10Dq13 59815 29710 91112 800981612 400
SD0.1130.1050.113
(10Dq)orb13 92810 7759622

a K3CrF6: G.C. Allen, A.M. El-Sharkawy, K.D. Warren, Inorg. Chem. 10 (1971) 2538.

b Cs2NaYCl[Br]6: Cr3+, R.W. Schwartz, Inorg. Chem. 15 (1976) 2817.

c K. Knox, D.W. Mitchell, J. Inorg. Nucl. Chem. 21 (1961) 253.

d Estimated for Cs2NaCrCl6 and Cs2NaCrBr6, F. Gilardoni, J. Weber, K. Bellafrouh, C. Daul, H.-U. Guedel, J. Chem. Phys. 104 (1996) 7624.

e estimated for Cs2NaCrCl6 and Cs2NaCrBr6, F. Gilardoni, J. Weber, K. Bellafrouh, C. Daul, H.-U. Guedel, J. Chem. Phys. 104 (1996) 7624.

6.3 Application of the LFDFT to the calculation of the zero-field splitting in Cr(acac)3

In octahedral ligand fields the t2g orbital of the TM are purely π-bonding. The π-electrons of the acac-ligand lead to a significant anisotropy; as has been recognized already by Orgel [28], this anisotropy can lower the symmetry of the ligand field from Oh to D3 with clear manifestations in the spectrum. For the acac ligand, the topmost π-orbital which dominates its π-donor functions is characterized by pi-orbitals with the same sign (in-phase), the out of-phase counterparts being much lower in energy. Three such in-phase coupled functions, when combining in a complex of a D3 symmetry give rise to species of e and a2 symmetry. From these only the e-combinations interact with the TM counterpart of the same symmetry, the a1 component of t2-obital having no counterpart from the ligand and being non-bonding in this approximation. For d-orbitals of Cr which are antibonding in Cr(acac)3 this leads to a splitting of the Oh t2g level in D3 with an a1 < e orbital energy sequence even though the geometrical arrangement of the oxygen ligator is very close to octahedral. This qualitative picture has been quantified in terms of phase-coupling ligand field model [29–32], which could explain both the splitting (800 cm–1) of the 4A24T2 band in the electronic absorption spectrum and its polarization behavior and the ground 4A2 and excited state 2E zero-field splittings, 1.1–1.2 and 250 cm–1. The latter have been detected in high-resolution optical spectra and further supported by detailed optically detected magnetic resonance (ODMR) studies [32]. We applied the LFDFT method to this system and results are collected in Table 3b. LFDFT values of the spin-orbit coupling constant ς, the parameters B and C (Table 3c) and the 5 × 5 LF matrix (Eq. (33)) have been utilized in standard LF-calculation yielding multiplet energies:(44)

hLF=[dxydxzdyzdx2y2dz26981038300010383418400000418410383000103836980000012930]
and fine-structure splitting in nice agreement with experiment (Table 3a). Also the LFDFT value of 10Dq (21 300 cm–1) is not too different from experimental one (18 700 cm–1) but show the typical positive deviations. This leads to larger calculated than experimental values for the energies of the spin-allowed electronic transitions. LFDFT values for B and C (450 and 2250 cm–1) deviate from the one deduced from a direct fit to the spectrum (B = 500 and C = 3400 cm–1) in the opposite direction. As mentioned before, DFT tends to underestimate these interelectronic repulsion parameters, calculating energies of spin-forbidden transitions, which are typically 4000 cm–1 lower than experiment.

Table 3b

LFDFT ground state fine-structure levels and the energies of spin-forbidden transition of Cr(acac)3 with and without accounting for spin-orbit coupling. Data from experiment, when available, are also listed

ς = 0ς = 211Experiment [32]
4A2 0Γ56(2 A) 0.00.0
Γ4(E) 1.161.20
2E 8618Γ56(2 A) 852012 940
Γ4(E) 871313 200
2A2 10,444Γ4(E) 10 442
2E 10,676Γ4(E) 10 674
Γ56(2 A) 10 677
2A1 16,707Γ4(E) 16 701
2E 17,832Γ4(E) 17 797
Γ56(2 A) 17 890
Table 3c

LFDFT values of energies of spin-allowed transitions in comparison with experiment (ς = 0)

LFDFTExperiment [31]
4A20.0
4A121 30617 700
4E22 65518 500
4A226 74222 700
4E28 748
Table 3a

4A2 ground- and 2E excited state zero-field splittings (in cm–1) – Dg and De, respectively, of Cr(acac)3 from a LFDFT—ZORA calculation of the spin-orbit coupling constant (ς = 211 cm–1) and from experiment

LFDFT-ZORAExperiment
Dg1.161.20
De193260

6.4 Exchange splitting in Cu(OH)2Cu dimers

The usual pattern of an exchange coupling between pairs of TM with open shells is anti-ferromagnetic spin-alignment corresponding to a weak delocalization of unpaired spin density from one center to another center (weak covalent bond as described by the term –4 T122/U, Eq. (36)). It out weights the contribution of the first term (2 K12), the latter tending to lower exchange (Pauli) repulsion between electrons with parallel spins. It has been therefore challenging to find systems where the latter effect dominates, leading a ferromagnetic spin-alignment. This is the case if magnetic orbitals are orthogonal to each other or nearly so, a situation encountered in edge sharing square planes or octahedra with M1–X–M2 bridging angles β close to 90° [33]. This is the case in bis bipyridyl-μ-dihydroxo-dicopper (II) nitrate with a Cu–OH–Cu bridging angle of 95.6° and an exchange coupling constant J12 = 0.021 eV [34]. A DFT–LDA geometry optimization using a [(NH3)2CuOH]22+ model cluster has lead to a geometry of the bridging Cu(OH)2Cu2+ moiety very close to the experiment (Fig. 2). Unpaired electrons on Cu2+ are characterized by a dx2y2 ground state which is weakly affected by long axial contacts to NO3, which we neglect here. The LFDFT calculated exchange coupling constant J12 = 0.021 eV (Table 4a) matches perfectly the value from experiment, but deviates from the anti-ferromagnetic one given by the broken symmetry (BS) DFT approach [35] (J12BS = –0.099 eV). In Fig. 3, we compare energies of the four independent Slater determinants as given by our DFT procedure with the state energies after taking the a+ab+b configurational mixing into account. The former configuration is stabilized by localization leading to a final singlet function, but it does not cross (as different to usual cases) the triplet term T. Experimental data show [34] that J12 becomes strongly anti-ferromagnetic upon increasing the Cu–O–Cu bridging angle (β) by structural manipulations allowing one to tune this structural parameter. Thus the increase of the value of β to 104.1° in [Cu(tmen)OH]2Br (tmen = N,N,N´,N´-tetramethylethylenediamin) correlates with a large negative reported value of J12 (–0.063 eV [34]). Antiferromagnetism for this geometry is also obtained by LFDFT but the resulting value exceeds now the experimental one by a factor of 2.88 (however the BSDFT value is by 4.61 larger). The reason is that DFT leads to systematically lower values of the energy U to cause an increase of the –4T122/U term, in cases where this terms plays an important role (see [8] for other examples and an analysis).

Fig. 2

Bond distances (in Å) and bond angles (in °) from a DFT geometry optimization (spin-unrestricted, S = Ms = 1, LDA–VWN functional, non-relativistic TZP basis Cu-2p, O-1s, N-1s, frozen cores) of a [Cu(NH3)2(OH)]22+ model cluster and experimental parameters as reported from X-ray diffraction study of bis-bipyridil-μ-dihydroxo-dicopper(II) nitrate [Cu(C10H8N2)µ(OH)-NO3]2, R.J. Majeste, E.A. Meyers, J. Phys. Chem. 74 (1970) 3497.

Table 4a

Energies (in eV) of Slater determinants for the geometry-optimized (NH3)2Cuμ(OH)2Cu(NH3)22+, the calculated singlet(S)–triplet(T) splitting ESET, the value J12(p) = (ESET)P given by perturbation theory (J12(p)f + J12(p)af = 2 K12 – 4 T122/U), by the broken symmetry calculation (ESET)BS as well as the one from experiment

E1(a+a)E2(b+b)E3(a+b+)E4(a+b)
–4.434–3.798–4.692–4.238
J12 = ESETJ12(p)(ESET)BS(ESET)exp [34]
0.0210.010–0.0990.021
Fig. 3

Correlation diagram between the energies of single determinants from DFT and the resulting multiplets of relevance for the magnetic exchange coupling in a [Cu(NH3)2(OH)]22+ model cluster with a ferromagnetic spin alignment.

It is remarkable that ferromagnetic contributions to J12 seem to be realistically described by the LFDFT procedure and our results show that such terms could be indeed rather important (as large as 0.061 eV in the chosen example, see Table 4b). Such terms have been neglected in earlier studies [36] or deemed by physicists to be small [17].

Table 4b

Decomposition of J12(p) into ferro- and anti-ferromagnetic contributions to the exchange along with the transfer(hopping) integral (t12), the Heisenberg exchange integral K12(= J12(p)f/2) the effective charge, transfer energy U are also included. For bond angles and distances characterizing the bridging geometry, see Fig. 2

J12(p)fJ12(p)afK12T12U
0.121–0.1110.0610.1590.909

7 Conclusion

The model we present here is simple and easy to implement. The quality of the predictions is exceptional in regard of the low computer time consumption. Keeping in mind that time dependent (TD) DFT is restricted to closed shell molecules an is still problematic for TM complexes and difference dedicated CI approaches [37–39] could be applied with success but only to systems of a smaller size, the model presented here is probably unique to address excited states of molecules with open d- and f-shells. Moreover, the concepts used here (LF theory, Racah parameter) are familiar to all chemists and spectroscopists. Thus, the quantities involved in the calculations provide immediate insights and facilitates communication between theorists and experimentalists. On the basis of our results we can conclude that DFT provides a rigorous interpretation of the LF-parameters and leads to a justification of the parametric structure of the classical LF theory. It is remarkable to mention that a theory which was discovered three quarter of a century ago is still modern.

Acknowledgements

This work was supported by the Swiss National Science Foundation.


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