Physical analysis of the through-ligand long-distance magnetic coupling: Spin-polarization versus Anderson mechanism

HAL Id: hal-00864023 https://hal.science/hal-00864023 Submitted on 19 Mar 2018 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Physical analysis of the through-ligand long-distance magnetic coupling: spin-polarization versus Anderson mechanism Thibaut Térencio, Roland Bastardis, Nicolas Suaud, Daniel Maynau, Jacques Bonvoisin, Jean-Paul Malrieu, C. J. Calzado, Nathalie Guihéry To cite this version: Thibaut Térencio, Roland Bastardis, Nicolas Suaud, Daniel Maynau, Jacques Bonvoisin, et al.. Physical analysis of the through-ligand long-distance magnetic coupling: spin-polarization versus Anderson mechanism. Physical Chemistry Chemical Physics, 2011, 13 (26), pp.12314-12320. �10.1039/c1cp20179d�. �hal-00864023� 1 Physical analysis of the through-ligand long-distance magnetic coupling: spin-polarization versus Anderson mechanism Authors: T. Terencio(a) , R. Bastardis(b) , N. Suaud(a), D. Maynau(a), J. Bonvoisin(c), J.P. Malrieu(a), C. J. Calzado(d) and N. Guihéry(a) Affiliations: (a) Laboratoire de Chimie et Physique Quantiques, Université de Toulouse III, 118, route de Narbonne, 31062 Toulouse, France (b) Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, Via Domitia, 52 avenue Paul Alduy, 66860 Perpignan, France (c) CEMES/CNRS, NanoSciences Group, BP 94347, 29 rue Jeanne Marvig, 31055 Toulouse cedex 4, France (d) Departamento de Química Física, Universidad de Sevilla, Profesor García González s/n, 41012 Sevilla, Spain Abstract The physical factors governing the magnetic coupling between two magnetic sites are analyzed and quantified as functions of the length of the bridging conjugated ligand. Using wave-function-theory based ab initio calculations, it has been possible to separate and calculate the various contributions to the magnetic coupling, i.e. the direct exchange, the spin polarization and the kinetic exchange. It is shown on model systems that while the Anderson mechanism brings the leading contribution for short-length ligands, the spin polarization dominates the through-long-ligand couplings. Since the spin polarization decreases more slowly than the kinetic exchange, highly spin polarizable bridging ligand would generate a good magneto-communication between interacting magnetic units. 2 I Introduction The electronic communication between two magnetic sites connected through bridges has been a topic of large interest in molecular electronics[1-4] and spintronics[5,6]. Much effort from experimentalists has been invested in the elucidation of the role of extended bridges between two interacting units, both concerning the electron transfer in mixed valence compounds (electro-communication) and the magnetic coupling of unpaired electrons in pure magnetic compounds (magneto-communication). From a theoretical point of view, several works reviewed in references [7,8] have been devoted to the methodology required to accurately reproduce the magnitude and sign of magnetic couplings in bi-nuclear compounds involving short bridges. Beyond the contribution to quantum chemistry computational techniques, these papers [9-11] illustrate how the required methodology is intimately related to the physical factors governing these effective interactions. In the line of these previous works, the electronic mechanisms involved in the charge or spin communication through extended bridging ligands have to be understood. The decrease of J as a function of the conjugated-bridge length has been theoretically evidenced, using UDFT calculations in nytronil nitroxide diradicals[12] and dinitrenes[13]. In recent experimental and theoretical studies[14,15] performed on dinuclear copper(II) complexes connected through oligoacenebis(oxamate) or oligo-para- phenylenediamine bridging ligands, Unrestricted Density Fonctional Theory (UDFT) calculations support the occurrence of a spin polarization mechanism for the exchange interaction between the two distant unpaired electrons. This alternative channel for the propagation of the exchange coupling through extended ligand could exhibit a different decay with the inter-metallic distance than the kinetic exchange, and this could explain the previously commented discrepancy. The prevalence of the spin polarization mechanism was proposed on the basis of the appearance of non-negligible spin density on the bridge in the UDFT MS=0 solution. This feature can only be considered as an indication since there is no spin density in the real singlet state and it does not give any information on the relative contributions of spin polarization and kinetic exchange to the resulting magnetic coupling. Actually this work was also initially motivated by the unexpected behaviour found in a set of new phenylcyanamido-bridge dinuclear Ru complexes, recently synthesized and characterized[16] by one of us (JB). The mixed valent Ru(II)Ru(III) and homovalent paramagnetic Ru(III)Ru(III) forms of all the complexes were studied by ultra-violet-visible- 3 near-infrared (UV-vis-NIR) and electron paramagnetic resonance (EPR) spectroscopy. Electronic communication was quantified by the hopping integral VAB (between metals A and B) extracted from intervalence transition measurements in the near IR area, and magnetic communication was quantified in terms of the exchange coupling interaction J, accessible from the intensity of the EPR signal when varying temperature. VAB and J versus intermetallic distances exhibit an exponential decay with two different laws. The slopes  and ’ of the ln(|VAB|) and ln(J) lines respectively are such that ’/2. At the first glance, this relationship between the two parameters is quite surprising. Indeed, the superexchange magnetic interaction is usually described as resulting from the Anderson mechanism (see Section II) in which J scales as t2/U (kinetic exchange) where t and U are characteristics of the magnetic compound; t is the inter-site hopping integral and U the relative energy of the ionic form A-B+ resulting from a charge transfer between the metals. Neglecting the dependence of U to the inter-metallic distance and making the assumption that the hopping integrals t (in the homovalent magnetic system) and VAB (in the mixed-valent system) have the same exponential decay would lead to the relation ’=. While the elucidation of this unexpected behavior which would require the study of mixed valent systems is out of the scope of the present study, we will analyze the relative importance and decay laws of both the kinetic exchange and spin polarization contributions to the magnetic coupling. This work takes benefit of the possibility offered by wave function based calculations to distinguish the effect of the configurations involved in each mechanism in order to evaluate their separate contributions and to analyze their decay with the inter-magnetic-site distance. Furthermore, the separation of both mechanisms enables us to quantify all effective interactions involved in the magnetic coupling, i.e. direct exchange, spin polarization, hopping integral t and on-site electron repulsion U. In section II, the various physical factors governing the magnetic interaction are briefly summarized and the methodology required to accurately reproduce them is recalled. Section III is devoted to an ab initio study of the different contributions to the magnetic coupling as functions of the distance in model compounds. Conclusion and perspectives are presented in section IV. 4 II Theory and computation The magnetic coupling is the leading effective interaction of spin Hamiltonians. For a binuclear system of spin S=1/2, the model space of this Hamiltonian is constituted of the two neutral valence bond determinants in which each magnetic sites has one unpaired electron. As shown in several papers, the microscopic origins of the magnetic coupling are multiple. This section recalls briefly the main physical factors that govern the magnitude and sign of this interaction [9-11,18]. II.a Physical factors governing the magnetic coupling: recalls For a centro-symmetric binuclear magnetic system, the local magnetic molecular orbitals (MOs) a on site A and b on site B are obtained from the gerade g and ungerade u MOs: 2 ug b 2 ug a     (1) It is important to recall that the quantum chemical optimization of the g and u orbitals results in delocalizations between the magnetic site and the ligand, i.e. the magnetic orbitals have tails on the ligand and reciprocally the ligand orbitals have tails on the magnetic sites. This feature has important consequences on the magnitude of the various contributions to the coupling described here-after. Two main factors govern the magnetic coupling between these unpaired electrons: i) The direct exchange K defined as: barabK1 12   (2) This integral is always positive and hence favors a ferromagnetic coupling. It is large when the two a and b MOs present an important “differential overlap”, i.e. when, despite their orthogonality, they both have significant amplitudes in some regions of space. This integral is expected to decrease rapidly with the distance between the two magnetic sites. Nevertheless the delocalization of the magnetic orbitals on the bridging ligand may lead to non-negligible values of this contribution. 5 ii) The kinetic exchange. This contribution is brought by the delocalization of the electrons between the two magnetic MOs, i.e. the electron on site A may jump from a to b provided that the hopping integral t between a and b is non-zero. This integral represents the interaction between the neutral valence-bond (VB) forms ba and ab (A.B.) and the ionic VB forms aa (A-B+) and bb (A+B-). It is non-zero except for symmetry reasons. The on-site repulsion U is the energy difference between the neutral ungerade triplet and the ionic gerade singlet function. In the basis of the following neutral and ionic functions:     bbaa 2 1 abba 2 1 I N   (3) the Hamiltonian matrix obtained for the two-magnetic-electron case is:         Ut2 t2K2 where the triplet state energy is taken as zero of energy. The corresponding electronic mechanism, known as the Anderson mechanism [19] (see scheme 1), introduces ionic VB components in the wave functions of the lowest singlet state: )bbacore.(aC)abbcore.(aCucore.uμgcore.gλΨ INg 1 (4) where core describes the product of the closed shell MOs and CN and CI are normalized coefficients, with |CN| > |CI|. 6 Scheme 1: Neutral and ionic forms involved in the Anderson mechanism. The matrix elements t between the neutral and ionic determinants is indicated in red. On the contrary, the triplet state cannot be stabilized by the inter-site electron delocalization and its wave-function is only constituted of neutral VB forms. For instance, the Ms=1 spin component wave function is: core.ab u gu.core 3 (5) The kinetic exchange always brings an antiferromagnetic contribution to the magnetic coupling and therefore favors the lowest spin state. Working only in the space constituted of neutral and ionic VB forms bbandaa,ab,ba , it is possible to show that the perturbative second-order analytical expression of the magnetic coupling is: U t KEJ 2 ST 42EE TS  (6) When the inter-site delocalization proceeds through the tails of the magnetic MOs on the bridging ligand, this magnetic interaction is called super-exchange and the electronic mechanism is known as the Anderson generalized mechanism which is represented in scheme 2. The electrons jump from one site to the other through the ligand orbitals. 7 Scheme 2: Configurations involved in the Anderson generalized mechanism. Only one of the two ionic forms is represented. The matrix elements (not indicated) are hopping integrals between the magnetic centers and the ligands. Different pathways coupling the neutral forms can be proposed, depending on the involved intermediate configurations either ligand to metal charge transfer (LMCT) and/or metal to ligand charge transfer (MLCT). One may note that the Anderson generalized mechanism also involves ionic forms as intermediate steps. Electron correlation deeply affects the above-mentioned interactions. When using correlated wave function based calculations, several types of excitations contribute to the differential stabilizations of the valence bond determinants (ionic versus neutral) and their interactions. Several papers [9-11] have been devoted to the detailed analysis of these correlation effects. The correlated wave function is expanded on a very large number (millions) of determinants resulting from excitations from the closed shell MOs to the unoccupied ones. As it will be shown below, it is possible to concentrate this information in the sub-space S0 defined by the neutral and ionic VB forms, using the theory of effective Hamiltonians [20]. It consists in defining effective interactions between the neutral and ionic VB determinants such that i) the eigenvalues of the effective Hamiltonian matrix are those of the all-electron exact electronic Hamiltonian, i.e. they are equal to the correlated ab initio energies and ii) the eigenvectors of the effective matrix are the projections of the ab initio wave functions onto the sub-space S0. The resulting effective interactions, noted here-after Keff, teff and Ueff, take into account the effect of electron correlation. 8 Let us distinguish two classes of correlation effects: the first one affects the K exchange integral while the second one modifies the kinetic contribution to the magnetic coupling. i) Correlation effect on the exchange integral K: the spin polarization of the doubly occupied MOs. The spin polarization mechanism is a real physical phenomenon. It is for instance responsible for the appearance of local spin densities () which are opposite to the dominant one () in Ms=1/2 radicals. Its first manifestation, evidenced in the EPR spectroscopy of the CH3 radical, was the non intuitive appearance of a contact term between the spin of the unpaired electron of the carbon and the protons spins. Since the protons are located in the (XY) plane which is a nodal plane for the singly occupied 2pz orbital bearing the unpaired electron, a zero contact term was expected. Its appearance comes from the spin polarization of the  and  spin electrons of the  bonds. Due to a weaker repulsion between electrons having the same spin, the  spins of the  bonds go closer to the carbon while the  spins go closer to the protons, generating the contact term between the nuclear spin and the induced electronic  spin density. As shown in previous papers [9], the spin polarization mechanism in diradicals goes through excited configurations involving a triplet state on the bridge and a triplet state in the magnetic orbitals as shown in scheme 3. Since this mechanism never involves any ionic VB configuration, its effect on the magnetic coupling results in a modification of the interaction between the neutral determinants, i.e. a modification of the exchange integral K such that:   r,i ir ba eff iKrrKi 2KK (7) where i and r are respectively occupied and vacant MOs of the ligand and the integral rKi a is the interaction )1(a)2(rr)2(a)1(i 1 12  . One should notice that while K is always positive, eff K can be either positive or negative. A detailed analysis of the physical effects on the spin polarization phenomenon of doubly excited determinants which do not involve any ionic forms is out of the scope of the present study. Let us however mention that their contribution to eff K is non-negligible and will be taken into account in the following ab inito calculations. 15 therefore expect that long ligands highly spin polarizable, such as polyacene[30] would induce a good magneto-communication between magnetic units. As reference [16], the present work questions the prevalence of the Anderson mechanism in the magnetic coupling. Nevertheless, while the present study rationalizes the appearance of a ratio slightly different from 2 between the slopes of ln(|teff|) and ln(J), it does not provide any explanation for the obtained ratio ~0.5 from experimental measurements of reference [16], since the overall ratio slope of the magnetic coupling is 1.89. A remaining question concerns the possible difference between the values of t and VAB. 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