Group-theoretical approaches in molecular magnetism: Metal clusters

Inorganica Chimica Acta 361 (2008) 3746–3760

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Review

Group-theoretical approaches in molecular magnetism: Metal clusters Boris Tsukerblat * Department of Chemistry, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

a r t i c l e

i n f o

Article history: Received 25 February 2008 Accepted 3 March 2008 Available online 10 March 2008 Dedicated to Professor Dante Gatteschi—to highlight his exceptional achievements and kudos. Keywords: Molecular magnetism Irreducible tensor operators Spin-symmetry Point symmetry Exchange interaction Magnetic anisotropy Mixed-valence Vibronic interaction Jahn–Teller effect Spin rings Polyoxometalates

a b s t r a c t In this article I review the application of the group-theoretical approaches to a wide area of molecular magnetism dealing with metal clusters. The following main aspects are discussed: (1) irreducible tensor operator (ITO) approach that is based on the so-called ‘‘spin-symmetry”. Use of this approach in molecular magnetism has given a revolutionary impact on the evaluation of the energy levels, thermodynamic and spectroscopic properties of high-nuclearity metal clusters. ITO approach facilitated development and applications of the isotropic and anisotropic spin-Hamiltonians and the study of the magnetic anisotropy in clusters containing orbitally degenerate metal ions; (2) group-theoretical classification (assignment) of the exchange multiplets based on both spin-symmetry and point symmetry that allows to analyze the non-Heisenberg forms of the exchange interaction and magnetic anisotropy in general terms, establishes selection rules for magnetic resonance transitions and facilitates computation of spin levels. This approach allows also to reveal the selection rules for the active Jahn–Teller coupling and to clear understand the interrelation between spin frustration and structural instabilities; (3) group-theoretical classification of the delocalized electronic and electron-vibrational states of mixed-valence compounds in terms of spin and point symmetries (including delocalization of the electronic pair) that essentially reduces the time of calculations and provides direct assess to the selection rules for different kinds of transitions. This becomes crucial in the dynamical vibronic problems inherently related to mixed-valency even for the truncated basis sets when the calculations become hardly executable not only in the case of strong vibronic coupling but even provided that the vibronic coupling is moderate. The proposed approach includes the design of the symmetry adapted vibronic basis and can enormously extend computational abilities in the dynamical problem of mixed-valency. Ó 2008 Elsevier B.V. All rights reserved.

Boris Tsukerblat obtained his scientific degrees of ‘‘Candidate of Sciences” (Ph.D.) in theoretical physics from the Kazan State University (Russia) in 1967 and ‘‘Doctor of Sciences” in 1975 from the University of Tartu (Estonia). He got a title of Full Professor in 1987 (Moscow). From 1967 to 1998 he headed a molecular magnetism group at the Institute of Chemistry and then till 2002, at the Institute of Applied Physics of the Academy of Sciences of Moldova in Kishinev. He is a Corresponding Member of this Academy (from 1995). He supervised more than 20 Ph.D. theses. In 2002, he became a Full Professor of the Chemistry Department of Ben-Gurion University of the Negev in Israel (Beer-Sheva). His scientific interests are focused on moleculebased magnetic materials (exchange interactions, single-molecule magnets, mixed valency, double exchange), vibronic interactions and Jahn-Teller effect in molecules and crystals, group-theoretical approaches in molecular magnetism and Jahn-Teller effect. He has authored and coauthored more than 300 research papers, 15 major review articles and 4 books on group theory, molecular magnetism and vibronic interactions.

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3747 The irreducible tensor operators approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3748 Group-theoretical classification of the exchange multiplets in spin-clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3749

* Tel.: +972 8 647 93 61; fax: +972 8 647 29 43. E-mail address: [email protected] 0020-1693/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ica.2008.03.012

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4.

5. 6. 7. 8.

3.1. General remarks about approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Some examples of trinuclear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Symmetric trimer d1–d1–d1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Hexanuclear spin rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Biprismatic V6 cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Octanuclear Fe(III) cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Group-theoretical classification of the delocalized states in mixed-valence systems . . . . . . 4.1. Mixed valency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Trimeric system with one delocalized hole (d1–d1–d0) . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Trimeric system with one electron delocalized over spin sites (d1–d1–d2) . . . . . . . . . 4.4. Systems with partial electron delocalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Site-symmetry approach for the delocalized electronic pair . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry adapted bases for the vibronic problems in mixed-valence clusters . . . . . . . . . . 6.1. Trinuclear mixed-valence systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem of orbital degeneracy in the exchange and double exchange . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Contemporary molecular magnetism pioneered by Dante Gatteschi originates from classical magnetochemistry and represents an interdisciplinary field of science that incorporated basic concepts of physics, chemistry and material sciences. The objects of molecular magnetism include molecular metal clusters, i.e., molecular assemblies consisting of a finite numbers of exchange coupled ions, represent the so-called class of zero-dimensional magnets [1–15]. These systems are of current interest in many areas of research and applications, like molecular magnetism and biochemistry and have perspective applications as single molecular magnets (SMM) [1,2,8–10] and multifunctional nanomaterials. As it was recently demonstrated, coexistence of ferromagnetism and metallic conductivity can be reached in one molecular material [16,17]. One of the most important trend in the spin-crossover area is focused on the molecule-based multifunctional materials [18,19]. Organic molecules of increasing sizes and large numbers of unpaired electrons are also being explored as building blocks for molecular-based magnets [20,21]. The modern trend in molecular magnetism is focused on the possibility to use molecular clusters as magnets of nanometer size, which exhibit magnetic bistability and quantum tunneling of magnetization at low temperatures. The first 10 years of activity summarized in the review article of Gatteschi and Sessoli [3] showed that the fundamentals of the field are established and the SMMs are expected to provide important nano-technological applications as the memory storage units of molecular size and as a novel route to a spin-based implementation of quantum information processing [22–25]. The understanding of the magnetic and spectroscopic properties of nanoscopic objects between molecules and bulk magnets require special theoretical concepts. In this paper we will review the theoretical tools of molecular magnetism based on symmetry concepts. First, we shortly summarize the background of the irreducible tensor operators (ITO) approach and give some selected examples illustrating the efficiency of the ITO technique in different kinds of problems related to the field of the exchange interactions in metal clusters (Section 2). For a long time the treatment of the metal clusters has been restricted to comparatively simple systems comprising a reduced number of magnetic centers. Gatteschi with coworkers (Gatteschi and Pardi [26], Bencini and Gatteschi [4]) introduced in molecular magnetism a very efficient approach based on the use of the ITOs of the full rotation group [27–29]. The ITO approach has given a revolutionary impact on the consideration of high-nuclearity clusters including mixed-valence clusters and orbitally degenerate systems. ITO approach allows to

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avoid evaluation of the spin functions of the system and to construct directly the matrix of the spin-Hamiltonian which takes into account all kinds of the exchange terms, or more common effective Hamiltonian containing orbitally-dependent exchange interactions (see Refs. [13,14,30–35]). Then (Section 3) we consider exchange clusters consisting of orbitally non-degenerate ions for which spin–spin interactions are the most important (hereunder, spin-clusters) while the orbital magnetic contributions are small and act as anisotropic corrections. We show how the exchange multiplets within the isotropic Heisenberg–Dirac–Van Vleck (HDVV) scheme can be correlated with the exact terms of the system (full spin and irreducible representation of the point group) [13,14,30]. Using selected examples we illustrate also how the symmetry assignment of the spin multiplets allows to select active non-Heisenberg terms in the exchange Hamiltonian and to reduce the matrices of the HDVV Hamiltonian that is crucially important for the treatment of high-nuclearity spin-clusters. Group-theoretical classification allows also to directly use the pseudoangular momentum representation and in this way to establish the selection rules for the EPR transitions and exact rules for crossing/anticrossing of the magnetic sublevels. This theoretical tool allows also to find the conditions under which the spin-vibronic Jahn–Teller coupling is active and consequently one can expect manifestations of structural instability in a spin-cluster. This is shown to be especially important for the spin-frustrated systems whose magnetic properties are essentially affected by even small structural distortions (static or/and dynamic) [36]. In Section 4 the group-theoretical classification is applied to mixed-valence systems with the delocalized electrons and holes and to the systems with partial electron delocalization [14] in which localized and itinerant electrons coexist and interact. In Section 5 the symmetry aspects of the dynamical electron-vibrational (vibronic) problem in mixed-valence (MV) compounds are discussed. It is shown how to construct the symmetry adapted vibrational and electron-vibrational states [37]. This can to essentially reduce the sizes of the enormously large matrices that appear while solving the dynamical vibronic problems. The procedure is exemplified by classification and evaluation of the vibronic states for a trimeric mixed-valence system with a delocalized electron. Finally, in Section 7 we shortly mention application of ITO in the problem of orbital degeneracy in the exchange and double exchange when the HDVV model in invalid even as an approximation [35,38]. The aim of the paper is to illustrate in an easily accessible way how the symmetry concepts can be applied to the study of the molecular magnets and what are the main advantages one can gain. Consequently, the main emphasis is put on the computational

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aspects and from this point of view the systems are chosen rather to illustrate the approaches and methodology than to give systematic reviews of the particular kinds of materials (the last is partly filled up by references).

2. The irreducible tensor operators approach In the metal clusters comprising orbitally non-degenerate ground terms the leading contribution to the exchange interaction is represented by the isotropic HDVV spin-Hamiltonian H0 ¼ 2

X

J ij S i  S j ;

ð1Þ

i
where Jij is the set of the many-electron exchange parameters, Si are the spin operators of the full electronic shells of the constituent ions and the summation is extended over all pairwise interactions. Physical nature of the exchange parameters and mechanisms of the exchange have been widely discussed (see, for example [4,32,33]). The dimension of the full Hilbert space for the system of N ions with the identical spins Si is (2Si + 1)N that is rather large even for the relatively simple systems. The total spin S of the system is always a good quantum number (since all scalar products Si  Sj commute with S2) but only in some restricted cases of high symmetry the energy levels can be explicitly expressed in terms of full and intermediate spin quantum numbers [2,13,39]. In general, the states with the same values of S are mixed accordingly to the general rules of quantum mechanics (as they belong to the same irreducible representations D(S) of the rotation group) and therefore the solution of the eigen-problem requires to construct rather large matrices of the exchange Hamiltonian with their subsequent diagonalization. The ITO approach allows to avoid practically unfeasible evaluation of the many-spin wave-functions and allows to directly evaluate the matrix elements of all kinds of isotropic and anisotropic exchange interactions within a specified basis sets of spin functions. The main advantage of ITO technique is that it deals with the symbolic representation of the spin functions in terms of relevant quantum numbers specified by spin coupling scheme. Applications of ITO in molecular magnetism are reviewed in Refs. [4,14,30,34]. Let us consider a spin cluster of arbitrary topology formed by an arbitrary number of magnetic sites, N, with local spins S1, S2, . . . , SN which, in general, can have different values. In this Section the orbital angular momentum contributions are assumed to be quenched by the local crystal fields acting on the constituent metal ions. A successive spin coupling scheme is adopted: S1 þ S2 ¼ e S2; e S 2 þ S3 ¼ e S3 ; . . . ; e S N1 þ SN ¼ S; ð2Þ where e S 2 ¼ S12 ; e S 3 ¼ S123 , etc. are the intermediate spins in the adopted spin coupling scheme and S is the full spin of the system. The spin states can be labeled as jS1 S2 ðe S 2 ÞS3 ðe S 3 Þ . . . SN1 ðe S N1 ÞSN SMi  jðe SÞSMi; ð3Þ e where ð SÞ denotes the full set of intermediate spins, ðe SÞ  ðe S2 ; e S3; . . . ; e S N1 Þ. All kinds of exchange interactions (isotropic and anisotropic) and single-ion anisotropic terms can be incorporated into the generalized spin-Hamiltonian operating in the spin-space, Eq. (2), of the entire system [13,14]: X X Hef ¼ ~ k ~ ~ k1 k2 ...kN k 1 2 ...kN 1



X

ðkÞ ~ ~ ~ ~ C ðkÞ q ðk1 k2 ðk2 Þ . . . kN1 ðkN ÞÞT q ðk1 k2 ðk2 Þ . . . kN1 ðkN ÞÞ;

ð4Þ

kq

~2 Þ . . . kN1 ðk ~N ÞÞ are the semiempirical parameters where C qðkÞ ðk1 k2 ðk ðkÞ ~ ~ and T q ðk1 k2 ðk2 Þ . . . kN1 ðkN ÞÞ is the component q of the complex ITO of the rank k composed from ITOs acting in the spin-space of the individual spins:

~

~

~

T qðkÞ ð  Þ ¼ f. . . fSðk1 Þ  Sðk2 Þ gðk2 Þ  Sðk3 Þ . . . gkN1  SðkN Þ gðkÞ q ;

ð5Þ

~2 ¼ k12 , etc. It is In Eq. (5)  is the symbol of the tensor product k supposed that Sðki Þ  Sðki Þ ðiÞ acts in the subspace of the ion i. First rank tensor operators Sð1Þ q are related to the cyclic combinations of spin operators: 1 ð1Þ ð1Þ S0 ¼ Sz ; S1 ¼  pffiffiffi ðSx  iSy Þ: ð6Þ 2 The inter- and intracenter interactions as well as the local site operators can be extracted from the general expression by substituting the due sets of the symbols and parameters C ðkÞ q ð  Þ in terms of the initial parameters of the exchange Hamiltonian. The matrix elements of the effective Hamiltonian can be evaluated by the use of the Wigner–Eckart theorem to the complex ITO T ðkÞ q ð  Þ: hSkT ðkÞ kS0 i 2k SM 0 0 hSMjT ðkÞ ð7Þ q jS M i ¼ ð1Þ C S0 M 0 kq pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2S þ 1 ðkÞ 0 where C SM S0 M 0 kq are the Clebsch–Gordan coefficients [28] and hSjT q jS i is the so-called reduced matrix element. Then one should apply the successive decoupling procedure (for a review see Refs. [4,13,30]). The result for the reduced matrix element of the effective Hamiltonian within the basis set, Eq. (3) is given by ~2 Þ . . . kN1 ðk ~N ÞÞkðe hðS~0 ÞS0 kT ðkÞ ðk1 k2 ðk SÞSi ¼ hSN kSðkN Þ kSN i 9 8 ~ ~ > = < ki kiþ1 kiþ1 > N 1 Y 1=2 ~ ½e e0 ð8Þ  f½k hSi kSðki Þ kSi i e S 0i Siþ1 e S 0iþ1 ; iþ1 S iþ1 ½ S iþ1 g > > ; : i¼1 Si Siþ1 e S iþ1   ... where ...... are the 9j-symbols, [k] = 2k + 1, and the single-site reduced matrix elements hS0i jSðki Þ jSi i can be found in the textbooks [28]. Along with the HDVV terms the exchange Hamiltonian contains anisotropic contributions, that can be written in the tensorial form P as ij S i Dij S j . Among these contributions one should mention anisymmetric (AS) exchange introduced by Dzyaloshinsky [40] by using phenomenological symmetry conditions and Moria [41] from the microscopic point of view by the inclusion of spin–orbit coupling in the Anderson’s theory of superexchange: X HAS ¼ Dij ½S i  S k ; ð9Þ i;k

where Dij = Diji are the antisymmetric vector parameters. Symmetric part of the anisotropic exchange and local anisotropy are also important in the description of the magnetic properties and magnetic resonance. Although anisotropic contributions are small within the scope of the physical background of HDVV model, they can play an essential role in symmetric systems possessing degenerate multiplets (Section 3). Along with the bilinear terms biquadratic exchange have been recently demonstrated to play a significant role in the pattern of spin levels [42,43]. Numerous examples of the evaluation of spin levels in highnuclearity clusters are given in [2,4]. Here we mention the famous V15 cluster containing 15 spins 1/2 [44] and exhibiting unusual layered magnetic structure. The works of Gatteschi et al. [45–47] on the evaluation of the energy pattern of this system and proposed three-spin model influenced all subsequent works (more than 60 papers) in this area. Use of both, spin-symmetry and point symmetry allows to further reduce the dimensions of the sizes for the energy matrices. In this way a computation of the energy spectrum for the octanuclear high-spin Fe(III) cluster (Si = 5/2) proved to be feasible [48]. An efficient approach based on the permutational symmetry (interchange of spin sites) has been developed in Ref. [49]. The ITO technique has been used to design the MAGPACK software [50,51], a package to calculate the energy levels, bulk magnetic properties, and inelastic neutron scattering spectra of

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high-nuclearity spin-clusters that allows to model the properties of nanoscopic magnets. Finally, one should mention that the ITO approach for the point groups is well developed [52] and also applied to molecular magnetism problems (Section 7).

3. Group-theoretical classification of the exchange multiplets in spin-clusters 3.1. General remarks about approach A concept of the group-theoretical classification (assignment) of the exchange multiplets appeared as a tool to elucidate the nature of so-called ‘‘accidental” degeneracies of the spin states in symmetric exchange clusters [13,14]. The degeneracies in high symmetric metal clusters in the HDVV model exceed those allowed by the actual point symmetries and therefore they evidence that the theoretical models require an additional examination. More deep insight on the problem of degeneracy and consequently, zero-field splitting of spin multiplets (in particular, in view of Kramers theorem) can be made on the basis of the group-theoretical consideration of the exchange coupling scheme that sheds light on the nature of the excessive degeneracies in the energy pattern of the exchange coupled systems. Let us note the obvious fact that HDVV Hamiltonian (that is a model Hamiltonian) operates in the spin-space only and therefore the energy levels are enumerated by the values of the total spin while each eigen-state according to Wigner theorem, must be characterized by the corresponding total spin as well as by an irreducible representation (irrep) of the cluster point group. The group-theoretical classification procedure makes it possible to establish one-to-one correspondence between the states D(S) belonging to HDVV model and spin–orbit multiplets (terms) SC that are the exact eigen-states of the first principle Hamiltonian for a polynuclear many-electron cluster. Special cases when D(S) are n-fold repeated and form accidentally degenerate nD(S) manifolds are also involved. A detailed description of the procedure of the group-theoretical classification based on the analysis of permutation (or unitary) symmetry is given in Ref. [13,14]. Using relatively simple examples here we describe a more simple approach firstly applied to mixed-valence systems (see [30] and references therein).

Fig. 1. Homo- and hetero-nuclear trimeric carboxylates [M3O(CH3COO)6(H2 O)3]+ (metal ions – full circles).

3.2. Some examples of trinuclear systems

change problem to be tractable in the full Hilbert space of 15 spins (215 = 32 768 states) due to the fact that the full matrix is blocked according to full spin values (maximal size of the block is 2002 that relates to S = 3/2). The energy pattern of V15 shows that ground state with the total spin S = 1/2 and first excited state with S = 3/ 2 are well isolated from the higher states thus justifying an effective triangle model that can be applied at low temperatures when the excited levels are sparsely populated. Scheme of spin arrangement in V15 (Fig. 2) shows that at low temperature the spins of the external hexagons are paired while the spins of the central triangle are frustrated. The three-spin model proposed and substantiated by Gatteschi et al. on the basis of the exact diagonalization of the 15-spin HDVV Hamiltonian [45,46], gives accurate and descriptive results for the low lying set of levels and allows to discuss adequately the system behavior at low temperatures (T < 20 K) when the spins of hexagons are paired. This model has been employed in all subsequent magnetic and spectroscopic studies of the V15 cluster including modeling of the Jahn–Teller properties [36,64,65]. Hereunder we will illustrate the method of classification on selected simple examples.

Numerous examples of the trimeric systems are represented by the homo- and heteronuclear carboxylates of the type of [M3O(CH3COO)6(H2O)3]+ based on different trivalent metal ions, M3 = Cr3, Fe3Rh3, Cr2Fe, CrFe2 (Fig. 1). The constituent ions are supposed to have orbital singlets as the ground terms in octahedral crystal field (4 A2 ðt 32 Þ for CrIII, etc.) so that the HDVV model is applicable (this does not refer to the CoII based compounds with strong unquenched orbital contribution, see Ref. [53]). The second important system of this type that can be mentioned is the cluster anion present in K6 ½VIV 15 As6 O42 ðH2 OÞ  8H2 O (hereafter V15 cluster). This famous system was discovered almost two decades ago [44] and since that time has attracted continuous and increasing attention as a unique molecular magnet based on a unique structure [54–65]. Studies of the adiabatic magnetization and quantum dynamics of the V15 cluster with S = 1/2 ground state showed that this system exhibits the hysteresis loop of magnetization of molecular origin and can be referred to as a mesoscopic system [54]. The molecular cluster V15 has a distinct layered quasispherical structure within which 15 VIV ions (Si = 1/2) are placed in a large central triangle sandwiched by two hexagons [45] possessing D3 symmetry. The use of ITO makes the HDVV ex-

Fig. 2. Schematic structure of the metal network of V15 cluster, dominant exchange pathways and pictorial representation of spin arrangement in the ground state and frustration effect in the central triangle.

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Table 1       The characters of transformations in C3v for the basis functions ama 12 bmb 12 cmc 12  C3v

b E

b3 2C

3^ rt

A2 A2 + E

1 3

1 0

1 1

b CðM¼3=2Þ ð RÞ b CðM¼1=2Þ ð RÞ

3.3. Symmetric trimer d1–d1–d1 Let us consider the most simple case, a trimeric system of C3v symmetry with S1 = S2 = S3 = 1/2 that allows to illustrate the procedure of the group-theoretical classification and the main conclusion one can draw from this approach. The exchange interaction in a symmetric (belonging to any trigonal point group) trinuclear system can be described by the following HDVV Hamiltonian: H0 ¼ 2J 0 ðS 1 S 2 þ S 2 S 3 þ S 3 S 1 Þ;

ð10Þ

where S1, S2 and S3 denote the spin operators on the sites 1, 2 and 3. The following spin coupling scheme for three spins, S1S2(S12)S3S  (S12)S is assumed with S12 being the intermediate spin (S12 = S1 + S2). The energy levels e0(S) are expressed as e0 ðSÞ ¼ J 0 ½SðS þ 1Þ 

3 X

Si ðSi þ 1Þ :

ð11Þ

i¼1

As one can see they do depend upon the full spin S and are independent of S12. This leads to the problem of the 4-fold ‘‘accidental” degeneracy of two S = 1/2 doublets for all trigonal systems composed of half-integer spins that is in a seeming contradiction with the Kramers theorem and last years this issue became a subject of the discussion regarding the model for V15. The ‘‘accidental” degeneracies occur in the high-nuclearity symmetric systems and a question appear about physical nature of these degeneracies, especially with regard to magnetic resonance when even small splittings and anisotropic interactions are crucially important. The coupling scheme for three spins Si = 1/2 can be expressed as Dð1=2Þ  Dð1=2Þ  Dð1=2Þ ¼ 2Dð1=2Þ þ Dð3=2Þ

ð12Þ

that means that this system has two levels with total spin S = 1/2 (S12 = 0 and S12 = 1) and one S = 3/2 level. The determinant wave-functions of a tri-center system satisfying the Pauli principle can be expressed in terms of Slater determinants jama ðSa Þbmb ðSb Þcmc ðSc Þj, where ama ðSa Þ is a multielectron (in general) spin–orbital function of the ion a with spin Sa and spin projection

     ma. Let us consider the determinant state a1=2 12 b1=2 12 c1=2 12  with maximum total spin projection M = ma + mb + mc = 3/2 that definitely belongs to maximal total spin S = 3/2. Point group C3v includes irreps: A1(basis z), A2(basis Lz), E(basis x, y or Lx, Ly) or C1,C2,C3 (in the Bete’s notations). Operations of C3v induce permutations of the localized orbitals, for example:               b 3 a1=2 1 b1=2 1 c1=2 1  ¼ b1=2 1 c1=2 1 a1=2 1  C  2 2 2   2 2 2        1 1 1  ¼ a1=2 b1=2 c1=2 : ð13Þ 2 2 2  b3, One can thus conclude that the character of the rotation C ðM¼3=2Þ b v ð C 3 Þ ¼ þ1. The states with M = 1/2 are represented by three determinants with one spin reversed that belong to both full spin states S = 3/2 and S = 1/2. Applying symmetry operations one can b find the characters of the reducible representations CðM¼3=2Þ ð RÞ b for both sets (Table 1) and then simply get the irreps and CðM¼1=2Þ ð RÞ b ¼ CðS¼3=2Þ ð RÞ b of C3v. It is obviously that CðM¼3=2Þ ð RÞ and b ¼ CðM¼1=2Þ ð RÞ b  CðS¼3=2Þ ð RÞ. b Therefore one can finally find CðS¼1=2Þ ð RÞ the following one-to-one correspondence between the HDVV multiplets nD(S) and the exact terms SC of the system that symbolically can be expressed as 2Dð1=2Þ !2 E;

Dð3=2Þ !4 A2 :

ð14Þ

From the group-theoretical classifications of the exchange multiplets one can draw the following conclusions: (1) The degeneracy with respect to the intermediate spin within the spin coupling scheme in the manifold (S12)S = (0)1/2, (1)1/2 (that looks like ‘‘accidental” in HDVV scheme) is associated with the exact orbital degeneracy in the multielectron triangular system so that the ground term is the orbital doublet, i.e. term 2E of C3v point group, meanwhile the excited one is the orbital singlet 4A2. The orbital degeneracy of two S = 1/2 levels is inherently related to spin frustration in the ground state (leading to a special triangular spin arrangement and spin density distribution) and is peculiar to all symmetric triangular spin-system with half-integer spins [13,14]. (2) Any isotropic forms of the exchange interactions cannot remove the degeneracy of the (S12)S = (0)1/2, (1)1/2 manifold. According to the conventional selection rules, 2E in C3v is split (in accordance with the Kramers theorem) by spin–orbital interaction that acts as a first order perturba-

Fig. 3. Energy levels of a symmetric d1–d1–d1 spin triangle and the Zeeman splitting in the magnetic field HkC3 axis with the indication of the quantum numbers in the pseudoangular momentum representation.

B. Tsukerblat / Inorganica Chimica Acta 361 (2008) 3746–3760

tion. Two resulting Kramers doublets can be specified as a pair of complex conjugated double-valued irreps A1 þ A2  2A and E. In the framework of spin-Hamiltonian approach spin–orbital interaction is equivalent to AS exchange that arises from the combined action of spin–orbital coupling and off-diagonal isotropic exchange. AS exchange that acts as a first order perturbation in orbital multiplets was shown to result in a strong magnetic anisotropy [63] and special shape of the steps in magnetization versus field in V15 observed experimentally [54]. In this sense the group-theoretical assignment allows to discuss the magnetic anisotropy in general terms. (3) The orbital singlet 4A2 (S = 3/2 level) is split by second order spin–orbital interaction that is equivalent to the application of the conventional zero-field Hamiltonian HZFS ¼ D0 ½b S 2  ð1=3ÞSðS þ 1Þ . Fig. 3 illustrates the energy pattern of an antiferromagnetic cluster (for example, V15 within three-spin model) exhibiting two spin levels with S = 1/2 and S = 3/2 split by AS exchange in the first and second order perturbation correspondingly. (4) Assignment of spin multiplets to the orbitally degenerate terms allows to employ the pseudoangular momentum representation and in this way to derive exact selection rules in magnetic resonance transitions. This was considered in detail in [61] with regard to the discussion of the experimental data on EPR in V15 cluster. Within the pseudoangular momentum representation the S = 1/2 basis j(0)1/2, ± 1/2i,j(1)1/2, ± 1/2i of the irreducible representation E can be related to two projections ML = +1 and ML = 1 of the fictitious orbital angular momentum L = 1, the basis functions uLML ðS; M S Þ  uML ðS; M S Þ can be found as the circular superpositions of spin function with two intermediate spins (S12 = 0 and S12 = 1) and total spin S = 1/2 [60]: pffiffiffi u1 ð1=2; 1=2Þ ¼ 1= 2ðjð0Þ1=2; 1=2i  ijð1Þ1=2; 1=2iÞ;

pffiffiffi u1 ð1=2; 1=2Þ ¼ 1= 2ðjð0Þ1=2; 1=2i  ijð1Þ1=2; 1=2iÞ:

ð15Þ

Using this conception one can introduce the functions US(MJ) belonging to a definite full spin S and projections MJ = ML + MS of the full pseudoangular momentum. The quantum numbers so far introduced correspond to the Russel–Saunders coupling scheme in axial symmetry. The level with S = 3/2 is an orbital singlet corresponding thus to ML = 0, the components are labeled as u0(3/2,MS)  U0(MJ) with MS = ±1/2 and MS = ±3/2, so that MJ = ±1/2 and ±3/2. The assignment of the levels based on this concept are presented in Fig. 3 that shows also the Zeeman splitting in parallel field HkC3 and the allowed EPR transitions obeying the selection rules in pseudoangular momentum scheme (for the details see [60]). (5) Pseudoangular momentum scheme allows to establish the exact rules for the crossing/anticrossing Zeeman levels in parallel field. In fact the levels with different MJ arising from HDVV multiplets S = 3/2 and S = 1/2 undergo to exact crossing, while the levels with the same MJ exhibit anticrossing. This conclusion is helpful in the modeling of static and dynamic magnetization. (6) Finally, knowledge of the irreps of the point group to which the exchange multiplets belong allows to establish the selection rules for the vibronic Jahn–Teller coupling. In fact, the vibrations mode of the symmetry Cv is active if the symmetric part [C  C]sym of the direct product C  C contains irrep Cv [30,66–68]. In the case under consideration [E  E]sym = A1 + E so that the double degenerate mode Cv = E is active in the Jahn–Teller effect (A1 is the totally sym-

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metric vibration that is irrelevant to the Jahn–Teller effect). This means that in the ground doublet 2E the trimeric system of C3v symmetry can be subjected to the Jahn–Teller distortion (Jahn–Teller instability). In the distorted configurations (isosceles triangle) the degeneracy is removed so that spin frustration is eliminated. The JT coupling is shown to be competitive to the AS exchange so that the increase of the vibronic coupling decrease the magnetic anisotropy of the system. On the other hand AS exchange tends to suppress to Jahn–Teller effect so that strong (as compare to the Jahn–Teller coupling) AS exchange restores a symmetric configuration of spin sites. This is demonstrated in [36] by the theoretical modeling of the field dependence of magnetization that clearly exhibits crucial role of the Jahn–Teller coupling in spin-frustrated systems.

3.4. Hexanuclear spin rings Here we illustrate the approach so far discussed by the analysis of the multiplets of the hexanuclear spin rings that can be exemplified by the two novel polyoxotungstates, (n-BuNH3)12[(CuCl)6–(AsW9O33)2]  6H2O and (n-BuNH3)12[(MnCl)6(SbW9O33)2]  6H2O which have recently been synthesized and characterized in Ref. [69] and have been discussed in [70]. These complexes are D3dsymmetric, and six 5-fold coordinated metal ions form approximately equatorial hexagons Cu12þ and Mn12þ (hereafter Cu6 and 6 6 Mn6) as shown in Fig. 4 [69]. The magnetic behavior of these compounds [69] provides a clear evidence for the ferromagnetic character of the exchange. The HDVV Hamiltonian including nearest neighbor interaction is the following: H ¼ 2J 0 ðS 1 S 2 þ S 2 S 3 þ S 3 S 4 þ S 4 S 5 þ S 5 S 6 þ S 6 S 1 Þ:

ð16Þ

In Eq. (16) the parameter J0 refers to the first-neighbor exchange interactions, Si = 1/2 for Cu2+ ions and Si = 5/2 for Mn2+. Evaluation (with the use of MAGPACK, [51]) of the energy patterns is given in Ref. [70]. The six spin coupling scheme for Cu6 ring gives the following full spin states: Dð1=2Þ  Dð1=2Þ  Dð1=2Þ  Dð1=2Þ  Dð1=2Þ  Dð1=2Þ ¼ 5Dð0Þ þ 9Dð1Þ þ 5Dð2Þ þ Dð3Þ :

ð17Þ

Point symmetry assignment [70] of spin multiplets n(S)D(S) shows that they correspond to the sets of the irreps C of C6v point group. Using the procedure so far described one gets the following correlation for the Cu6 cluster:

Fig. 4. Scheme of Cu6 and Mn6 rings.

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sponding 23B2, 21A1 and 23A1 so that the eigen-problem in this case has an analytical solution. The dimensions of the SC blocks for Mn6 are much higher so that a numerical solution in this case is required. The group-theoretical assignment allows one to elucidate the character of the anisotropy in the excited states of the systems which exhibit orbital degeneracy. In this case the conventional zero-field term persist (for the levels with S P 1) but plays a secondary role in the zero-field splitting and anisotropy. It can be illustrated taking as an example the most simple case of Cu6 hexagon for which the S = 2 set involves two orbital doublets 5E1 and 5 E2. The selection rule [30] for the matrix elements of the purely imaginary operator L of the orbital angular momentum is defined by the decomposition of the antisymmetric parts fE21 g and fE21 g of the direct products E1  E1 and E2  E2. In C6v symmetry fE21 g ¼ fE22 g ¼ A2 so that only LZ component (irrep A2) of L has non-vanishing matrix elements within the E1 and E2 basis sets. This means that 5E1 and 5E2 terms are split by spin–orbital interaction and only kLZSZ part of spin–orbital coupling is active. We arrive at the conclusion that AS exchange appears and only z-component of the AS exchange split 5E1 and 5E2 spin multiplets. When orbital degeneracy is present the AS exchange acts as a first order effect with respect to the spin orbital interaction and for this reason, in general, exceeds the zero-field term that appears as a second order effect [5,34] and gives also non-zero contribution for all terms with S P 1. Again, the sublevels of the system (G is an effective parameter of AS exchange) can be labeled by the pseudoangular momentum quantum numbers (Fig. 5b). Fig. 5. Energy levels for a Cu6 ring in the first-neighbor coupling approximation (a). The terms for S = 2 and S = 3 are shown. The remaining multiplicities of spin states are indicated in parentheses. (b) Splitting by AS exchange and Zeeman splitting (HkC6) of 5E1 level; pseudoangular momentum labels are indicated.

Dð3Þ !7 B2 ; 5Dð2Þ !5 A1 þ5 E1 þ5 E2 ; 9Dð1Þ !3 A2 þ 23 B2 þ3 E2 þ 23 E1 ;

ð18Þ

5Dð0Þ ! 21 A1 þ1 B1 þ1 E1 : One can see that the energy pattern (Fig. 5a) comprises orbitally degenerate states (irreps E1 and E2 in C6v) and orbital singlets A1, A2, B1, B2. For example, nine levels with S = 1 levels involve three orbital singlets 3A2, 23B2, an orbital doublet 3E2 and two orbital doublets 3E1. Only ferromagnetic ground state S = 3 does not exhibit excessive degeneracy and is represented by the orbital singlet 7B2. The energy pattern exhibits doubly degenerate levels (irreps E1 and E2), for example, first and second S = 1 level for both Cu6 and Mn6, etc. These degeneracies arise from the point symmetry of the system in the sense that they are associated with definite terms SC of the systems and thus cannot be removed by the remaining isotropic interactions preserving hexagonal symmetry, for instance, by the next-neighbor interactions or/and biquadratic exchange. It also should be noted that the matrix of the HDVV Hamiltonian for Cu6 cluster (including also all interactions) can be blocked in symmetry adopted basis and contains only three 2  2 matrices corre-

3.5. Biprismatic V6 cluster This magnetic clusters consisting of six VIV or FeIII ions are embedded into complex polyoxometalates as described by Gatteschi et al. [71]. Metal network consists of two triangles in parallel planes with a common C3 symmetry axis so that the whole structure is prismatic (D3h symmetry) as shown in Fig. 6. In presence of antiferromagnetic interaction each triangle shows spin frustration (2E term in C3v in site-symmetry position) and therefore we are dealing with an interesting and unusual case of two interacting spin-frustrated moieties. The Hamiltonian of the system can be presented as H0 ¼ 2J 0 ðS 1 S 2 þ S 2 S 3 þ S 3 S 1 þ S 4 S 5 þ S 5 S 6 þ S 6 S 4 Þ  2JðS 1 S 4 þ S 2 S 5 þ S 3 S 6 Þ  2J 0 ðS 1 S 6 þ S 3 S 4 þ S 3 S 5 þ S 2 S 6 þ S 1 S 5 X X þ S2 S4 Þ þ Dij ½S i  S k þ Dij ½S i  S k ; ik¼12;23;31

ð19Þ

ik¼45;56;64

where the antiferromagnetic intracluster and two types of intercluster isotropic exchange interactions are taken into account as well as the AS exchange operating within the triangles (Fig. 6). Since spin frustration is inherently related to orbital degeneracy, such kind of systems are extremely sensitive to even small perturbations [36,65] as it can be observed in the case under consideration. The role of perturbation plays the intertriangle isotropic coupling and an important feature of this perturbation is that it does not lower the overall symmetry of the system. From this point of view one

Fig. 6. Scheme of the metal network and HDVV parameters in biprismatic V6 cluster.

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can expect manifestation of the AS exchange in a coupled system of two triangles. Let us focus on the case of V6 system [72]. Spin coupling scheme for six Si = 1/2 spins (64 states) is given by Eq. (12). Procedure of the group-theoretical classification is more complicated but quite similar to that described in Section 3. As a result one can obtain the following correlation between the HDVV multiplets and the irreps of D3h point group: Dð3Þ !7 A002 ; 5Dð2Þ !5 A01 þ5 E0 þ5 E00 ; 9Dð1Þ !3 A02 þ 23 A002 þ3 E0 þ 23 E00 ;

ð20Þ

5Dð0Þ ! 21 A01 þ1 A001 þ1 E0 : From this assignment one can see that the isotropic part of the Hamiltonian can be blocked, the maximal size of the submatrix is 2. Essential simplification can be reached even if the AS exchange is taken into consideration. Fig. 7 shows a fragment of the full energy pattern of the system arising from the interaction between triangles in their ground states 2E. The intra- and intertriangle isotropic interactions are antiferromagnetic and the AS exchange acts within each triangle. One can see that the system exhibit orbital doublets that are split (providing S 6¼ 0) by AS exchange acting as a first order perturbation meanwhile in the orbital singlets with S 6¼ 0 one can observe second order effect of AS exchange giving rise to a conventional zero-field splitting. Accordingly to a general statements so far discussed each orbital doublet is related to the projections ML = ±1 of the pseudoangular momentum while for the orbital singlets ML = 0. Spin–orbital coupling (AS exchange in spin-Hamiltonian) couples the states and splits them accordingly to the values of the total pseudoangular momentum MJ = ML + MS. Fig. 7 indicates also pseudoangular momentum labels that can be easily used to establish selection rules for the EPR transitions and give a qualitative insight on the magnetic anisotropy. In fact the ground state proves to be an orbital singlet 1 A01 so at lower temperature the system cam exhibit temperature independent paramagnetism (Van Vleck contribution). The orbital doublet 1E0 that looks like a diamagnetic state from the standpoint of HDVV model, but in fact, this doublet is expected to undergo to a Zeeman splitting with the orbital g-factor. This splitting is fully anisotropic in the sense that the magnetization reach maximum when the field is along C3 axis and disappears when the field is in plane. Finally, the orbital doublet 3E00 shows zero-field splitting into three doublets, MJ = 0 (MS = ±1, ML = 0), MJ = ±1 (MS = 0, ML = ±1) and MJ = ±2 (MS = ±1, ML = ±1). Therefore one can expect strong magnetic anisotropy associated with this state. The exact selection rule for the EPR transitions is MJ  M 0J ¼ 0; 1 and therefore one can qualitatively and

Table 2 Classification of the total spin states of FeIII 8 cluster according to irreps of D2 point group (from Ref. [48]) S

A

B1

B2

B3

Tot. deg

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

1 2 10 22 60 118 243 419 717 1088 1614 2174 2841 3401 3927 4139 4155 3704 3019 1899 703

1 6 18 50 108 225 401 690 1061 1578 2138 2799 3359 3885 4097 4122 3671 3001 1881 703

2 6 22 50 118 224 420 686 1092 1568 2184 2780 3420 3854 4170 4076 3750 2940 1960 630

2 6 22 50 118 224 420 686 1092 1568 2184 2780 3420 3854 4170 4076 3750 2940 1960 630

1 7 28 84 210 462 916 1660 2779 4333 6328 8680 11 200 13 600 15 520 16 576 16 429 14 875 11 900 7700 2666

quantitatively discuss EPR pattern using only results of the grouptheoretical classification. More detailed information can be extracted by considering separately the transitions induced by spinradiation and orbital-radiation interactions with obvious modification of the selection rules. 3.6. Octanuclear Fe(III) cluster Evaluation of the energy pattern of the octanuclear Fe(III) (Si = 5/2) cluster of D2 symmetry [48] is an impressive illustration of the efficiency of the combined application of spin-symmetry, point symmetry and group-theoretical classification of the exchange multiplets. The full dimension of the Hilbert space is very large, (2S + 1)N = 68 = 1 679 616. The matrix of HDVV Hamiltonian can be blocked accordingly to full spin values but the dimensions of the submatrices are still high (16 576 for S = 5). Group-theoretical classification gives allowed terms as indicated in Table 2 from which one can see that the maximal dimension of the matrix to diagonalized is 4170 (4170 irreps B3 with S = 5, i.e. 11B3 terms). 4. Group-theoretical classification of the delocalized states in mixed-valence systems 4.1. Mixed valency

Fig. 7. Energy scheme of V6 prismatic cluster with the indication of the point symmetry labels and pseudoangular momentum quantum numbers.

Mixed-valence (MV) compounds contain one or several electrons that can be delocalized over magnetic ions (spin cores) [73–76]. Double exchange interaction involves the coupling of two localized magnetic moments, having spin cores S0, through an itinerant extra electron that can travel forth and back between the two magnetic centers. Since the itinerant electron keeps the orientation of its spin in course of transfer, double exchange results in a strong spin polarization effect which favors a ferromagnetic spin alignment in a pair of ions. This mechanism of electron–spin interaction was first suggested by Zener [77] to explain the ferromagnetism observed in MV manganites of perovskite structure, such as ðLax Ca1x ÞðMnIII MnIV 1x ÞO3 . Now the MV oxides are an active focus of research in solid state chemistry, as they exhibit colossal magnetoresistance [78]. Anderson and Hasegawa [79] suggested a solution of the double exchange for a MV dimer deducing the spin dependence of the energy levels:

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B. Tsukerblat / Inorganica Chimica Acta 361 (2008) 3746–3760 Table 3 Characters of transformations of the basis functions of d1–d1–d0 trimer

Fig. 8. Schematic structure of the ferromagnetic MV nickel dimer [Ni2(napy)2Br2]+.

E ðSÞ ¼ 

tðS þ 1=2Þ ; 2S0 þ 1

ð21Þ

where t is the electron transfer integral, S is the total spin of the system. One can see that the energy versus full spin is linear function that is drastically different from spin dependence in the case of the exchange in localized systems. Fully delocalized NiINiII dimer [Ni2(napy)2Br2](BPh4) (Fig. 8) where napy is the bidentate ligand naphthyridine, reported in Refs. [80,81]. In fact, this compound proved to be strongly ferromagnetic with ground state S = 3/2 resulting from the double exchange coupling between S(NiI) = 1/2 and S(NiII) = 1 local spins [82]. The second classical example is represented by the fully localized dimer based on FeII(S(FeII) = 2) and FeIII(S(FeIII) = 5/2) ions [L2Fe2(l-OH)3](ClO4)  2CH3OH  2H2O [83,84] and exhibiting ferromagnetic ground state and strong intervalence absorption band. The conclusions obtained for a dimer cannot be extended to high-nuclearity systems involving localized and delocalized spins in a complicated molecular and crystal structures. The peculiarities of frustrated double exchange systems having triangular faces, like symmetric trimers, tetramers and hexagons [85–90] are especially interesting. More complicated systems [91–93] are in the focus of research because they are promising in the applications of molecular magnets in which the double exchange creates ferromagnetic ground state. A general approach to the problem of double exchange in high-nuclearity clusters containing an arbitrary number of localized spins and arbitrary number of itinerant electrons moving among spin sites have been worked out in [94]. This approach was employed to solve a very difficult problem of the electron delocalization [95] in high-nuclearity MV cluster [V18O42]4 formed by 18 oxovanadium(IV) sites [96]. While evaluating the energy levels of high-nuclearity MV clusters one faces the problem of the sizes of the energy matrices that imply the restrictions to the size of the system that can be really considered. In fact, the dimensions of the matrices to be diagonalized increase dramatically with the increase of the number of sites and the number of electrons. Application of the symmetry can essentially enlarge the ability of the approach.

C3v

b E

b3 2C

3^ rt

A2 + E A1 + A2 + 2E

3 6

0 0

1 0

spin-singlets and spin-triplets. Accordingly to the Pauli principle the basis set should be constructed from 12 bi-electron determinants:                   am 1 bm 1 ; am 1 cm 1 ; bm 1 cm 1 ; a c c b b  a 2     2 2 2 2 2  which are related to the c*, b* and a* localizations of the ‘‘hole”. For M = S = 1 (maximal spin and spin projection) one obtains three determinants:                   a1=2 1 b1=2 1 ; a1=2 1 c1=2 1 ; b1=2 1 c1=2 1   2 2   2 2   2 2  that form three-dimensional representation of C3v. Characters of this representation can be found by application of the symmetry operations, the results are collected in Table 3. Decomposing this reducible representation into the irreducible ones one finds: CðM¼1Þ ¼ CðS¼1Þ ¼ A2 þ E: Since M = 1 states belong only to S = 1 one can immediately conclude that the spin-triplet terms of the system are 3A2 and 3E. For M = 0 (and S = 1 and S = 0) we have six determinants:             a1=2 1 b1=2 1 ; a1=2 1 b1=2 1 ;    2 2 2 2  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} c-localization             a1=2 1 c1=2 1 ; a1=2 1 c1=2 1 ;  2 2   2 2  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} b-localization            1 1 1 1  b1=2   c ; b c 1=2 1=2 1=2  2 2   2 2  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} a-localization

for which we obtain characters (Table 3) and consequently the following decomposition: CðM¼0Þ ¼ A1 þ A2 þ 2E: Extracting C(S=1) from C(M=0) we find C(S=0). Finally the result of the group-theoretical classification for d1–d1–d0 trimer can be expressed as 3ðDð1=2Þ  Dð1=2Þ  Dð0Þ Þ ¼ 3Dð0Þ þ 3Dð1Þ ; 3Dð0Þ !1 A1 þ1 E;

4.2. Trimeric system with one delocalized hole (d1–d1–d0) Let us consider a trigonal system with, let’s say, C3v symmetry and assume that two electrons are distributed over three orbitals a, b and c localized at the sites A, B and C (for the sake of simplicity the orbitals are supposed to be full-symmetric). One can construct the following spin–orbitals ama ðSa Þ; bmb ðSb Þ; cmc ðSc Þ with Sa = Sb = Sc = 1/2 and there are three distributions of two electrons over three sites (three places a*, b* and c* for a hole) as shown in Fig. 9 for the spins ‘‘up” configuration. The energy pattern involves

Fig. 9. Three localizations of the ‘‘hole” in MV trimer d1–d1–d0.

b CðM¼1Þ ð RÞ b CðM¼0Þ ð RÞ

3Dð1Þ !3 A2 þ3 E;

ð22Þ

where the first expression symbolizes the spin coupling scheme (the factor 3 indicates that the spin coupling is applied for each localized scheme) while the second one gives the point symmetry assignment. 4.3. Trimeric system with one electron delocalized over spin sites (d1–d1–d2) This is an example illustrating the delocalization of the extra electron over three-spin cores, i.e. the case when the concept of the double exchange is relevant (Fig. 10). It is assumed that due to strong ferromagnetic intracenter Hund coupling spin of the center with a trapped electron S* = 1 while the remaining centers have spins 1/2. The spin coupling scheme for this system can be symbolized as 3ðDð1=2Þ  Dð1=2Þ  Dð1Þ Þ ¼ 3Dð0Þ þ 6Dð1Þ þ 3Dð2Þ ;

ð23Þ

B. Tsukerblat / Inorganica Chimica Acta 361 (2008) 3746–3760

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Fig. 10. One-electron levels of d1–d1–d2 MV cluster (three localizations).

where as in the previous case the factor 3 takes into account three possible localizations and two possible spin-triplet states appear due to two possible intermediate spins in three-spin coupling scheme. The full basis set involves four-electron determinants jama ðsa Þbmb ðsb Þcmc ðsc Þj related to three possible localization of the ‘‘extra” electron. For example, for c-localization sa = sb = 1/2 and sa = 1. For M = S = 2 we have the only determinant for each localization and find a three-dimensional representation with the basis set:             a1=2 1 b1=2 1 c1 ð1Þ; a1=2 1 b1 ð1Þc1=2 1 ;    2 2 2 2       a1 ð1Þb1=2 1 c1 ð1Þ   2 b ¼ vðS¼2Þ ð RÞ b for all R b and then we find for which one gets vðM¼2Þ ð RÞ (2) 5 5 3D ? A2 + E. Continuing this procedure as in previous example one can find the following result for the group-theoretical classification of the delocalized states for d1–d1–d2 system: 3Dð2Þ !5 A2 þ5 E; 6Dð1Þ !3 A1 þ3 A2 þ 23 E; ð0Þ

1

ð24Þ

1

Fig. 11. Scheme of the magnetic sites in K6[H3KV12As3O39(AsO4)  8H2O] (from [98]).

vanadate K6 [H3KV12As3O39 (AsO4)  8H2O] contains a localized trimeric V4þ 3 unit and the mixed-valence unit consisting of one electron moving among three Si = 1/2 ions [97–99]. This system can be referred to as the system with partial electron delocalization. The scheme of the magnetic sites based on the structure of K6[H3KV12As3O39(AsO4)  8H2O] is shown in Fig. 11. Exchange (localized) trimer d1–d1–d1 (sites a, b, c) and mixed-valence d1–d0–d0-cluster (sites a, b, c) are placed in parallel planes, so the symmetry of each moiety is D3h (‘‘site-symmetry” of each moiety in the molecule) and the overall symmetry is C3v. The basis  set  of the   system   isformed by 3  24 = 48 determinants ama 12 bmb 12 cmc 12 ama 12 , etc. Applying the operations of C3v and using the approach so far discussed one can find SC-multiplets of the system with partial delocalization. For example, for maximum M = S = 2 we have the three-dimensional basis consists of the four-electron determinants: ja1=2 b1=2 c1=2 a1=2 j; ja1=2 b1=2 c1=2 b1=2 j; ja1=2 b1=2 c1=2 c1=2 j

3D ! A1 þ E: 3

One can see that the only 2  2 matrix corresponding to 2 E should be diagonalized while the remaining energies can be found directly as the diagonal matrix elements of the Hamiltonian in the symmetry adopted bases. 4.4. Systems with partial electron delocalization The named systems form a special cases of mixed-valence compounds containing localized and delocalized domains. Polyoxo-

with C(M=2) = C(S=2) = A2 + E, and hence, we find 5A2 + 5E multiplets. In the case under consideration (and this is specific for many MV systems with partial electron delocalization) the procedure can be simplified. In fact, C3v is a subgroup of D3h and so that the irreps relating to each subsystem can be decomposed accordingly to the symmetry lowering (D3h ? C3v). Then the SC-multiplets of the system can be found as the decomposition of the direct products: DðSexc Þ  DðSMV Þ and Cexc  CMV where SexcCexc and SMVCMV are the terms of the exchange and MV units (the Pauli principle in this case

Fig. 12. Two structures of polyoxometalates with delocalized electronic pairs: (a) Keggin structure of a [XW12O40]n anion; (b) Wells-Dawson structure of a [X2W18O62]n [102].

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Fig. 13. Electronic pair in Keggin structure: (a) idealized metal network; (b) five types of electronic pairs in Keggin structure looking down from C3 axis of the cube (from [104]).

does not imply any restrictions for the resulting states). Finally the result can be expressed as ð4 A2 þ 4 EÞ  ð2 A1 þ 2 EÞ ¼ 1 A1 þ 1 A2 þ1 E þ 3 A1 þ 23 A2 þ 23 E þ 5 A2 þ 5 E:

ð25Þ

One can see the 48-dimensional space is blocked into the subspaces with the maximal dimension 2 that allowed (see [98,99]) to find the analytical solutions of the eigen-problem. The results show that the delocalization effect produce a motional averaging of the exchange parameters and influences essentially the zero-field splittings. A more complicated isostructural dodecanuclear systems [V12As8O40(HCO2)]3 (I) and [V12As8O40(HCO2)]5 (II) with partial electron delocalization have been considered in Refs. [100,101]. These compounds contain localized spins in the central V4þ 4 square sandwiched between two mixed-valence squares containing one (in I) or two (in II) delocalized electrons. The theoretical model includes Coulomb interaction in different localized configurations, delocalization effects and exchange interactions between localized and itinerant electrons. This rather complicated problem could be solved due to use of symmetry arguments (group-theoretical classification, symmetry-adapted basissets). An interesting feature of these kind of system is the effect of polarization of the localized square subunit by the moving electrons. 5. Site-symmetry approach for the delocalized electronic pair Delocalization of the electronic pair in MV compounds represents a concerted motion of two electrons over the network of the magnetic (spin) cores affected by Coulomb repulsion of the electrons, exchange interaction between electrons and spin sites and also vibronic interaction that determines the degree of localization. In this respect molecular polyoxometalates [102,103] are especially important because they provide a large variety of high-nuclearity spin-clusters possessing different topology, different kinds of exchange networks and possibility to accommodate metal ions. Two polyanion structures, Keggin and Wells-Dawson (Fig. 12) are especially important and well studied. These anions can be reversibly reduced by addition of specific numbers of electrons which prove to be delocalized over a network of the metal centers as schematically shown in Fig. 13a for the Keggin anion. The localized and mobile electrons can coexist and interact giving rise to many important features of these compounds that also are very suitable for testing different magnetic and vibronic models in the

area mixed valency. It was found that the reduced polyoxometalates containing delocalized electronic pair are strongly antiferromagnetic and this phenomenon cannot be explained with the model assuming coupling of electrons via multi-route superexchange. An explanation based on the concept of delocalization that can stabilize the spin-paired ground state without implying directly exchange interaction was worked out in Refs. [104–106]. As one can see (Fig. 13) the metal sites form four triangles (a1a2a3, etc.) that are perpendicular to four C3 axes of a regular tetrahedron abcd (Td-symmetry). A view of this network along one of the C3 axis allows to observe that the Keggin structure contains a hexagonal belt M6 in between two triangles M3; in one of them the MO6 sites are sharing edges while in the another one they are sharing corners. As one can see from Fig. 13 one can distinguish up to five types of localized states for the electronic pair with different distances between sites of localization (from 6.997 to 3.295 Å). For example the largest one denoted as I in Fig. 13b involves metals of the M6 belt separated by two other metals, the following types of pairs (from II to V) have larger distances. Each kind of the pair produces a set of delocalized states which is separated from the remaining ones by the energy gaps that are of the order of the Coulomb repulsion inside of the electronic pair. The pair I with the largest distance gives rise to a low lying set of the delocalized states and the energies of the groups of the levels increase with the decrease of the distance between the sites (from I to V). In general, two electrons are shared between 12 sites forming thus spin-triplets (Sp = 1) and spin-singlets (Sp = 0). For each Sp there are 66 different dispositions of the electronic pair which are different in the positions and distances that are indicated in [104]. Since the transfer operator does not mix different spin states, from the computational point of view this means that the we face 66  66 matrix of the full Hamiltonian (for each Sp). Let us illustrate how the sitesymmetry approach can simplify the problem. Site-symmetry or local symmetry of the atom in a molecule is determined by the site group Ga whose elements are of the full molecular symmetry group G that, when acting on the molecule, leave that atom in its initial position (for example, C3v in the site group of the atom B in AB4 molecule of Td symmetry). Site group Ga is always a subgroup of the full molecular group G. Usually site-symmetry approach is used for the classification of hybrid orbitals in molecules. A relevant definition of the site-symmetry of an electronic pair and application of this basic concept makes possible group-theoretical classification of the delocalized states for the electronic pair. As distinguished from the site-symmetry of an atom in Td molecule, we determine the site-symmetry of a pair as a site-symmetry of the side in the tetrahedron abcd. This

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B. Tsukerblat / Inorganica Chimica Acta 361 (2008) 3746–3760 Table 4 Terms SC of the delocalized electronic pair in the Keggin structure (numbers of positions are indicated in parentheses) Type of pair

Terms SC

I(6)

II(24)

III(12)

IV(12)

V(12)

3

3

23T1,23T2 1 A1,1A2,21E,1T1,1T2

3

3

T1,3T2 1 A1,1E,1T1

A1,3A2,23E,33T1,33T2 1 A1,1A2,21E,31T1,31T2

polyhedron is the simplest one that retains full Td symmetry of the Keggin unit. Site-symmetry of each site is thus C2v and one can assign six local coordinates to six faces of the cube defining thus instant positions of the pair that can be specified by the number of the face. The wave-functions of the pair I belong to the irreps of C2v and can be expressed in terms of Slater determinants (for a2b2 position) as /I ð½a2 b2 ; Sp ¼ 1; M ¼ 1Þ ¼ ja2 b2 j; 1  2 j þ ja 2 b2 jÞ: /I ð½a2 b2 ; Sp ¼ 0Þ ¼ pffiffiffi ðja2 b 2

ð26Þ

It can be easily seen that these functions belong to the irrep B1 and A1 of C2v and consequently give terms 3B1 and 1A1. Each of four pairs II does not belong to the C2v site group, but the part of the full Hamiltonian involving two triangles that are associated with the ab side is invariant under the transformation of C2v. This allows to construct the basis functions of C2v from four localized functions of the pair II. This procedure can be extended (see details in [104]) and all irreps C0 for the localized electronic pairs can be obtained. For example, for the pairs I, II associated with the ab side one finds: pair I: 3B1, 1 A1; pair II: 1A1, 3A2, 3B1, 3B2, 1A1, 1A2,1B1. Applying symmetry operations of the full symmetry Td group to each basis function belonging to a certain term SC0 of the site group C2v one can generate the reducible six-dimensional irreps Cf of Td. By decomposing Cf into irreps one can classify the delocalized states of the pairs according to the irreps C of group symmetry of the system (Table 4). One can see that the full matrix (66  66) can be blocked into submatrices of smaller sizes accordingly to the numbers of repeated irreps. For the spin-triplet and spin-singlet terms we get DðSp ¼1Þ ð66Þ¼3 A1 þ 33 A2 þ 43 E þ 103 T 1 þ 83 T 2 ; DðSp ¼0Þ ð66Þ ¼ 51 A1 þ 21 A2 þ 71 E þ 71 T 1 þ 81 T 2 :

ð27Þ

Therefore the maximum size of the energy matrix for spin-triplet terms is 10 while for spin-singlet this size is 8. The computational problem can be further simplified when the Coulomb interaction is strong enough to separate the sublevels belonging to different pairs, the mixing can be neglected and the low lying levels can be

A2,3E,23T1,3T2 1 A1,1E,1T1,21T2

A2,3E,23T1,3T2 A1,1E,1T1,21T2

1

found Fig. 14. For example, for the pair I each irrep enter once and the evaluation of the energy levels can be done directly if the mixing of configurations is neglected. As it was demonstrated for Keggin [104] and Wells-Dawson [105] structures the site-symmetry approach can be employed to easily construct the basis functions belonging to the definite irreps. One can see that site-symmetry approach dealing with the both spin and point symmetries is an efficient tool for the consideration of the high-nuclearity MV compounds with delocalized electronic pairs. 6. Symmetry adapted bases for the vibronic problems in mixedvalence clusters The vibronic Jahn–Teller (JT) and pseudo JT interaction [67,68] is inherently related to the problem of mixed valency [75,76,107–110]. In fact, in mixed-valence systems the ‘‘extra” electron produces significant deformation of the surrounding that in particular, caused the so-called charge transfer bands of the light absorption with significant spectral half-width. In general, especially in the case of intermediate vibronic coupling (class II in Robin–Day classification scheme) the semiclassical approximation based on the concept of slowly moving ions is inapplicable for the description of the energy levels (they cannot be identified with the adiabatic surfaces) and the quantum-mechanical approach is desired, especially for the adequate description of the light absorption. This means that the full Hamiltonian involving electron transfer, free nuclear vibrations and vibronic interaction has to be diagonalized. In this case we are dealing with the infinite basis of the unperturbed electron-vibrational states so that even the truncated basis possess high dimensionality due to high dimensions of the excited vibrational levels of the degenerate vibrations. This situation is schematically shown in Fig. 15 for the case of an unique electron delocalized over three equivalent sites giving rise to an electronic singlet and a doublet separated by the gap 3jtj (t is the transfer integral). The electronic wave-functions A1 and E types can be easily constructed from the localized functions /A, /B, /C by applying projection operator: 0 1 0 pffiffiffi pffiffiffi pffiffiffi 1 0 1 /A1 /A 1= 3 1= 3 1= 3 p ffiffiffi p ffiffiffi B/ C B C B C ð28Þ @ Ex A ¼ @ 1= 2 1= 2 A  @ /B A: 0 pffiffiffi pffiffiffi pffiffiffi /Ey /C 1= 6 1= 6 2= 6 In the absence of the electron-vibrational interaction the two levels A1 and E are accompanied by the sets of the vibrational (harmonic oscillator) levels forming a multilevel system. The number of

Fig. 14. Energy pattern of the pair I as function of the transfer parameter (arbitrary scale).

Fig. 15. Illustration for the electron-vibrational sets of the levels unperturbed by the vibronic coupling.

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the states, i.e. the dimension of the basis for the matrix to be diagonalized depend on the strength of the vibronic coupling. The difficulties are multiply increased even for the relatively simple systems in which several sets of degenerate vibrations are active, for example tetrameric MV clusters. Here we show how to design the symmetry adopted set of the electron-vibrational (or spin–electronvibrational) states. Such kind of the basis set is expected to provide an essential reduction for the vibronic matrices at the expense of some simple and easily programmable algebraic operations. 6.1. Trinuclear mixed-valence systems Let us refer to a relatively simple case of a trinuclear mixed-valence system containing one electron shared among three-spinless sites A, B and C (point symmetry C3v). Delocalization processes lead to a singlet A1 and a doublet E levels separated by the gap 3jtj with t being the transfer integral. One can see that the delocalized electron is coupled to the double degenerate vibrational mode E that results in a combined JT problem within the doublet E and pseudo JT mixing of E and A1 levels. Before constructing of the symmetry adopted electron-vibrational functions let us note that the wave-functions an one-dimensional harmonic oscillator can be denoted as jni,n = 0, 1, 2, . . . An excited state jni can be obtained by successive applications of the bosonic creation operator b+: 1 þ jni ¼ pffiffiffiffiffi ðb Þn j0i: n!

ð29Þ

For the two-dimensional (vibrations of E symmetry) harmonic oscillator jnxnyi, nx, ny = 0, 1, 2, . . . with the orthogonal and normalized state vectors, hnx ny jmx my i ¼ dnx mx dny my , one obtains 1 þ þ jnx ny i ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ðbx Þnx ðby Þny j00i: nx !ny !

ð30Þ

Each level n = nx + ny is g = (n + 1)-fold degenerate, for example the firs excited level n = 1 is double degenerate (nx = 1, ny = 0 and nx = 0, ny = 1), etc. This shows that the dimension of the vibrational P space is pv ¼ Nn¼0 ðn þ 1Þ ¼ 12 ðN þ 1ÞðN þ 2Þ where N is the number of the vibrational levels included in the basis. The two functions j01i, j10i form irrep E (basis x and y in C3v). The excited state behave like polynomials xnx yny accordingly to Eq. (30) and belong thus to the reducible sets. Creation operators þ þ bx and by can be related to the irrep E, the wave-function of the ground state j00i (vacuum) belongs to A1. The basis functions belonging to the definite irreps can be obtained for each n by the application of symmetry adopted polynomials constructed from þ þ operators ðbx Þnx ðby Þny to the ground state. For example applying þ þ simply bx and by one creates E basis j10i x and j01i y. Symmetry adapted operator polynomials (let say, of type Cc, symbol c enumerates the basis functions) are applied to the ground state j00i create thus the vibrational functions jnCci with a given n belonging to an irrep Cc. This can be done by the use of the conventional coupling procedure that allows to construct a symmetry adapted basis set wCc from the direct products of two irreducible basis sets uC1 c1 and uC2 c2 [111]: X uC1 c1 tC2 c2 hC1 c1 C2 c2 jCci; ð31Þ wCc ¼ c1 c2

where hC1c1C2c2jCci are the coupling (Clebsch–Gordan) coefficients for the point groups (they are given by Koster et al. [112]). The stepwise procedure can be illustrated by evaluation for several low lying vibronic levels n: (1) n = 0, g = 1, basis j0, A1i, þ þ (2) n = 1, g = 2, direct product Eðbx ; by Þ  A1 ðh00jÞ ¼ E, so the basis with n = 1 form irrep E:

þ

bx j00i ¼ j10i;

þ

by j00i ¼ j01i;

ð32Þ

Finally, for the first excited level the basis of E type is simply: j1; Exi ¼ j10i;

ð33Þ

j1; Eyi ¼ j01i:

(3) n = 2, g = 3, the next functions can be obtained by the applying the creation operators to the n = 1 basis. The direct prodþ þ uct Eðbx ; by Þ  Eðh10 j; h01jÞ ¼ A1 þ A2 þ E, so one can detect an excessive degeneracy. Using coupling procedure, Eq. (31) one can obtain three non-vanishing functions A1 + E, while A2 state disappears. Normalized functions j2,SCi are the following: 1 j2; A1 i ¼ pffiffiffi ðj20i þ j02iÞ; 2 ) j2; Exi ¼ j11i : 1 j2; Eyi ¼ pffiffi ðj20i  j02iÞ

ð34Þ

2

(4) n = 3, g = 4, direct product E(bx,by)  (n = 2  basis)  E(bx, by)  (A1 + E) = A1 + A2 + 2E.One can see that we have obtained six-dimensional space instead of required fourdimensional. Orthogonal four-dimensional space includes A1 + A2 + E: 1 pffiffiffi ð 3j21i  j03iÞ; 2 pffiffiffi 1 j3; A2 i ¼ ðj30i  3j12iÞ; 2 pffiffiffi ) j3; Exi ¼ 12 ð 3j30i þ j12iÞ : pffiffiffi j3; Eyi ¼ 12 ðj21i þ 3j03iÞ j3; A1 i ¼

ð35Þ

(5) n = 4, g = 5, direct product E(bx, by)  (A1 + A2 + E) = A1 + A2 + 3E, but the five-dimensional orthogonal space includes A1 + 2E: pffiffiffi pffiffiffi pffiffiffi j4; A1 i ¼ 2p1 ffiffi2 ð 3j40i þ 2j22i þ 3j04iÞ; ð3j31i  j13iÞ j4; aExi ¼ p1ffiffiffiffi 10

9 =

; pffiffiffi pffiffiffi ; j4; aEyi ¼ p1ffiffi5 ð 3j22i  2j04iÞ ðj31i þ 3j13iÞ j4; bExi ¼  p1ffiffiffiffi 10 pffiffiffi ð 6j22i  5j40i þ 3j04iÞ j4; bEyi ¼ 2p1ffiffiffiffi 10

ð36Þ 9 = : ;

The last case illustrates what one can expect at the subsequent steps of this procedure, namely, several identical irreps resulting in an excessive dimension of the space, so that each step should be supplemented by the procedure of orthogonalization within repeated irreps (that at the same time eliminates excessive functions) by so-called Gram–Schmidt procedure that takes an arbitrary basis and generates an orthonormal one (included for example, in the Standard Packages of Wolfram’s ‘‘Mathematica”). It does this by sequentially processing the list of vectors and generating a vector perpendicular to the previous vectors in the list. Finally, symmetry adapted vibrational functions are to be coupled to the electronic ones (or spin–orbital, etc.), this should not bring any complications. The procedure so far discussed assumes only symmetry, so it is applicable to the linear, quadratic, etc., vibronic interactions. The calculation can be facilitated by the preliminary assignation of the vibrational multiplets. The group-theoretical assignation includes the following steps that will be illustrated for the E vibrations in C3v. Each basis set with a given n, jnxny, ni should be put in correspondence to a set jl, mi, l being the quantum number of

B. Tsukerblat / Inorganica Chimica Acta 361 (2008) 3746–3760 Table 5 Group-theoretical classification of the vibrational multiplets Vibrational number n Irreps D(l) Irreps of C3v group Assignation n ? C

n=0

n=1

n=2

n=3

g=4

ð0Þ

D(1/2) E E

Dð1Þ  A1 + E A1 + E

D(3/2) A1 þ A2 þ E A1 + A2 + E

Dþ A1 + 2E A1 + 2E

Dþ A1 A1

ð2Þ

n=5 D(5/2) A1 þ A2 þ 2E A1 + A2 + 2E

angular momentum. This quantum number is defined as l = n/2 and the corresponding parity p for the even l is p = (1)l  (1)n/2. Then using the conventional rules [30] the reducible representation D(l) should be decomposed in C3v group. The D(l) with the half-integer gives double-valued irreps (noted by bar). To get final assignation the double-valued irreps should be replaced by the single-valued ones: A1 ! A1 ; A2 ! A2 ; E ! E (we omit the details).The results are given in Table 5 for n = 1,2, . . . 5. One can see that the total degeneracy of five vibrational levels is pv = 21, meanwhile this space is blocked into 5A1 + 2A2 + 7E and the symmetry adapted vibrational functions are already found. To get full results for the electronvibrational problem the direct products of electronic and vibrational irreps should be multiplied. This can be symbolized as: CðelÞ  CðvibÞ ¼ ðA1 þ EÞ  ð5A1 þ 2A2 þ 7EÞ ¼ 12A1 þ 9A2 þ 21E:

ð37Þ

The labels so far obtained are related to the full electron-vibrational problem and therefore one can see that the total Hilbert space of the dimension 63 is blocked, maximal size of the vibronic sub-matrix being 21. The simplification of the vibronic problem is essentially helpful when we are dealing with several vibronic modes that occurs in tetrameric and in systems with higher nuclearity. 7. Problem of orbital degeneracy in the exchange and double exchange This problem requires a special review that is why here we give only some general comments and references. If the constituent ions are placed in a high-symmetric (octahedral) surrounding the ground states can be orbitally degenerate so that the electronic shells carry not only spin but also the orbital angular momentum contributions. Under this condition the conventionally accepted HDVV Hamiltonian is inapplicable even as a rough approximation. A general form of the effective Hamiltonian for the systems containing ions with the unquenched orbital angular momenta has been deduced in Refs. [113–117]. This requires the use along with the ITO of the rotation group also ITO of the point groups [52]. The exchange interaction in clusters comprising ions with unquenched orbital angular momenta includes HDVV contribution and orbitally-dependent terms of the effective exchange Hamiltonian that are, in general, of the same significance as spin–spin coupling. These terms (along with all accompanying interactions, like spin–orbit coupling, low symmetry crystal field and Zeeman interaction) can also be expressed in terms of ITO that facilitates evaluation of the energy pattern and allows to analyze the magnetic anisotropy in general terms (the detains are given in Ref. [116]). The orbitally-dependent contributions was shown to result in the anomalously strong magnetic anisotropy. As an example the binuclear face-shared Ti(III) clusters in Cs3Ti2Cl9 crystals and pentanuclear Mn5 cyano-bridged clusters containing orbitally degenerate Mn(III) ions [118,119] exhibiting strong magnetic anisotropy can be mentioned.

8. Conclusion In summary, the following main results of the quantummechanical applications of the symmetry concepts in molecular

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magnetism are reviewed: (1) spin-symmetry that is taken into account through application of the ITO approach. The use of ITO allows to essentially reduce the matrices of HDVV Hamiltonian and facilitate evaluation of the thermodynamic and spectroscopic properties of high-nuclearity metal clusters. Application of ITO approach proved to be of crucial importance for the study of the magnetic anisotropy in clusters containing orbitally degenerate metal ions; (2) group-theoretical classification (assignment) of the exchange multiplets dealing with both spin-symmetry and point symmetry establishes one-to-one correspondence between spin quantum numbers and irreps of the point group to which the system belongs. This theoretical tool allows to analyze the non-Heisenberg forms of the exchange interaction and magnetic anisotropy in general terms. Knowledge of the irreps of the appropriate point group of symmetry establishes selection rules for magnetic resonance transitions and facilitates computation of spin levels. This approach allows also to reveal the selection rules for the active Jahn–Teller coupling and to clear understand the interrelation between spin frustration and structural instabilities; (3) group-theoretical classification of the delocalized electronic and electron-vibrational states of mixed-valence compounds in terms of spin and point symmetries (including delocalization of the electronic pair) that essentially reduces the time of calculations and provides direct assess to the selection rules for different kinds of transitions. This becomes crucially important in the dynamical vibronic problems inherently related to mixed-valency even for the truncated basis sets when the calculations become hardly executable not only in the case of strong vibronic coupling but even provided that the vibronic coupling is moderate. The proposed approach that includes the design of the symmetry adapted vibronic basis enormously extends computational abilities in the dynamical problem of mixed-valency. The applications so far discussed clearly demonstrate that the symmetry concepts are vitally important for the contemporary molecular magnetism.

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