Computational Modelling of Molecular Nanomagnets 3031310373, 9783031310379

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Challenges and Advances in Computational Chemistry and Physics 34 Series Editor: Jerzy Leszczynski

Gopalan Rajaraman Editor

Computational Modelling of Molecular Nanomagnets

Challenges and Advances in Computational Chemistry and Physics Volume 34

Series Editor Jerzy Leszczynski, Department of Chemistry and Biochemistry, Jackson State University, Jackson, MS, USA

This book series provides reviews on the most recent developments in computational chemistry and physics. It covers both the method developments and their applications. Each volume consists of chapters devoted to the one research area. The series highlights the most notable advances in applications of the computational methods. The volumes include nanotechnology, material sciences, molecular biology, structures and bonding in molecular complexes, and atmospheric chemistry. The authors are recruited from among the most prominent researchers in their research areas. As computational chemistry and physics is one of the most rapidly advancing scientific areas such timely overviews are desired by chemists, physicists, molecular biologists and material scientists. The books are intended for graduate students and researchers. All contributions to edited volumes should undergo standard peer review to ensure high scientific quality, while monographs should be reviewed by at least two experts in the field. Submitted manuscripts will be reviewed and decided by the series editor, Prof. Jerzy Leszczynski.

Gopalan Rajaraman Editor

Computational Modelling of Molecular Nanomagnets

Editor Gopalan Rajaraman Department of Chemistry Indian Institute of Technology Bombay Mumbai, India

ISSN 2542-4491 ISSN 2542-4483 (electronic) Challenges and Advances in Computational Chemistry and Physics ISBN 978-3-031-31037-9 ISBN 978-3-031-31038-6 (eBook) https://doi.org/10.1007/978-3-031-31038-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Chapter 6 is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/). For further details see license information in the chapter. This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Molecular Nano Magnets (MNMs) encompass a wide range of molecules exhibiting fascinating magnetic behaviour, from zero-dimensional isolated molecules to twodimensional layered compounds. The interest in these classes of molecules has grown exponentially in the last three decades due to their intriguing properties and potential applications in high-dense information storage devices, Qubits in quantum computing, molecular refrigerant and spintronics devices. Despite several breakthroughs in this area over the years, one of the formidable challenges in taking these molecules to end-user application lies in achieving control over their magnetic properties. Microscopic spin Hamiltonian (SH) parameters describe the magnetic properties of MNMs, and understanding the origin of these parameters and developing viable ways to regulate these SH parameters is highly desirable. Computational tools play a proactive role hand-in-hand with experiments to achieve this goal. While several books written so far in this area covered the synthesis and characterisation of MNMs, the theoretical tools that played a central role in their design and developments are not extensively covered, and we aim to achieve this here. This book attempts to give an overview of the computational modelling of MNMs by introducing the theoretical concepts and methods and applying these developed tools to compute/extract various SH parameters for transition metals/lanthanides/ actinide MNMs. Other contemporary applications of modelling of MNMs on surfaces or at the interfaces are also covered in this volume. This volume is divided into nine chapters. In Chap. 1, Liviu F. Chibotaru describes the application of ab initio approaches to the investigation of strongly anisotropic magnetic complexes with particular emphasis on first-principles calculations in extracting comprehensive information on magnetic interactions. This chapter highlights the flexibility of ab initio approaches allowing their use in combination with different levels of modelisation of interactions which still cannot be reliably treated from the first principles, such as exchange interactions in polynuclear complexes. Finally, the perspectives of applying ab initio methods for describing novel phenomena in molecular magnets are discussed. In Chap. 2, Rémi Maurice, Nicolas Suaud, and Nathalie Guihéry show analytical derivations to predict the nature and magnitude of both the zerofield-splitting and the anisotropic magnetic exchange in transition metal complexes v

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with a particular emphasis on Dzyaloshinskii-Moriya antisymmetric exchange interaction. A special focus of this chapter is devoted to describing the effect of spinorbit coupling (SOC) on the degenerate and non-degenerate ground states of metal complexes. Giang Truong Nguyen and Liviu Ungur describe the different computational approaches for estimating the magnetic exchange of polynuclear complexes in Chap. 3. The combination between accurate on-site eigenstates obtained in complete active space self-consistent field (CASSCF) calculations and theoretical modelling of the magnetic dipole-dipole and exchange interactions within the lines model and the recently developed kinetic exchange model is described alongside several recent examples. In Chap. 4, Jürgen Schnack describes methods to extract SH parameters from experimental magnetic data. It is imperative to understand various models and approximations involved in this approach as often such results are benchmarked to assess the quality of theoretical methods employed. Particularly in this chapter, the finite-temperature Lanczos method application to evaluate low-energy spectroscopic data of very large polynuclear complexes with a large Hilbert space dimension of about 100000 is described. In Chap. 5, Hélène Bolvin demonstrates the electronic structure and magnetic properties of open-shell actinide complexes, which are gaining attention in recent times. Since the crystal-field effects, electron-electron repulsion, and spin-orbit interactions are of the same order, ab initio calculations on these complexes are challenging. This chapter explains that Spin-Orbit Complete Active Space (SO-CAS)-based methods are still the quantum chemistry tool of choice since they include a balanced description of the three effects. The first-principles strategies applied to the problem of spin-phonon relaxation in magnetic molecules have recently gained unprecedented interest in designing potential Single Molecule Magnets (SMMs). In Chap. 6, Alessandro Lunghi presents a rigorous formalism of spin-phonon relaxation based on open quantum systems theory. Examples from the literature, including transition metals and lanthanides compounds, have been discussed to illustrate how Direct, Orbach and Raman relaxation mechanisms can affect spin dynamics for this class of compounds. In the past few years, we have witnessed the evolution of blocking temperature (TB ) and blocking barrier (Ueff ) of higher-order Dnh symmetry (n = 4, 5, 6 and 8) lanthanide SMMs and Ln encapsulated fullerenes. In Chap. 7, Sourav Dey, Tanu Sharma, Arup Sarkar and Gopalan Rajaraman showcase the strength of ab initio calculations in understanding the mechanism of magnetisation relaxation and the role of symmetry in dictating the desired magnetic characteristics. Beyond rationalising the observed magnetic properties, how these methods emerged over the years to predict molecules possessing desired magnetic properties are highlighted. Once successful MNMs are identified for a suitable application, the next step involves device fabrication which mandates anchoring them on the surfaces, and this brings in several challenges on its own. Computational tools play a pivotal role in this area which is the focal point of Chap. 8 by Matteo Briganti and Federico Totti. Here authors discuss various criteria to be considered for modelling molecules on the surfaces, followed by a discussion on the role of the surface and its effect on the geometry and magnetic properties of adsorbed MNMs. In Chap. 9, Eliseo

Preface

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Ruiz and Daniel Aravena demonstrate molecular-based spintronic devices and the current approaches for modelling electron and spin transport in magnetic systems, particularly emphasising systems based on SMMs and the emerging interest in the interplay between spin and chirality for the design of new spintronic devices. As molecular modelling is widely practised now by both my experimentalists and theoreticians, we hope this book will provide the essential background, choice of methods and limitations to set a stage for doctoral and postdoctoral students who aspire to work in the area of Molecular Nano Magnets. I would like to profusely thank all the authors for their fine contributions to this book. I would also take this opportunity to express my heartfelt gratitude to my students for their invaluable assistance. Additionally, I am deeply thankful to my family members, including Sishakha, Krishikha, M Amsaveni (late), G Bragathambal (late), and M Vijyalakshmi, for their unwavering support and encouragement throughout the daunting journey that spanned more than two years. Their presence and encouragement were instrumental in helping me complete this significant endeavor. Mumbai, India

Gopalan Rajaraman

Contents

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization Blocking in Metal Complexes . . . . . . . . . . . . . . . . . . Liviu F. Chibotaru

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2 Analytical Derivations for the Description of Magnetic Anisotropy in Transition Metal Complexes . . . . . . . . . . . . . . . . . . . . . . . Rémi Maurice, Nicolas Suaud, and Nathalie Guihéry

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3 Calculations of Magnetic Exchange in Multinuclear Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Giang Truong Nguyen and Liviu Ungur 4 Exact Diagonalization Techniques for Quantum Spin Systems . . . . . . 155 Jürgen Schnack 5 Modeling Magnetic Properties of Actinide Complexes . . . . . . . . . . . . . 179 Hélène Bolvin 6 Spin-Phonon Relaxation in Magnetic Molecules: Theory, Predictions and Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Alessandro Lunghi 7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where to from Here? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Sourav Dey, Tanu Sharma, Arup Sarkar, and Gopalan Rajaraman 8 Molecular Magnets on Surfaces: In Silico Recipes for a Successful Marriage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Matteo Briganti and Federico Totti 9 Theoretical Approaches for Electron Transport Through Magnetic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Eliseo Ruiz and Daniel Aravena Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 ix

Contributors

Daniel Aravena Department of Materials Chemistry, Universidad de Santiago de Chile, Santiago de Chile, Chile Hélène Bolvin Laboratoire de Chimie et de Physique Quantiques, CNRS, Université Toulouse 3, Toulouse, France Matteo Briganti Department of Chemistry “U. Schiff”, University of Florence, Florence, Italy Liviu F. Chibotaru Theory of Nanomaterials Group, KU Leuven, Leuven, Belgium Sourav Dey Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai, India Nathalie Guihéry Laboratoire de Chimie et Physique Quantiques, UMR5626, Université de Toulouse 3, Toulouse, France Alessandro Lunghi School of Physics, AMBER and CRANN Institute, Trinity College, Dublin, Ireland Rémi Maurice Univ Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes)—UMR 6226, Rennes, France Giang Truong Nguyen Department of Chemistry, National University of Singapore, Singapore, Singapore Gopalan Rajaraman Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai, India Eliseo Ruiz Department of Inorganic and Organic Chemistry, Institute of Theoretical and Computational Chemistry, Universitat de Barcelona, Barcelona, Spain Arup Sarkar Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai, India Jürgen Schnack Faculty of Physics, Bielefeld University, Bielefeld, Germany

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Tanu Sharma Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai, India Nicolas Suaud Laboratoire de Chimie et Physique Quantiques, UMR5626, Université de Toulouse 3, Toulouse, France Federico Totti Department of Chemistry “U. Schiff”, University of Florence, Florence, Italy Liviu Ungur Department of Chemistry, National University of Singapore, Singapore, Singapore

Chapter 1

Ab Initio Investigation of Anisotropic Magnetism and Magnetization Blocking in Metal Complexes Liviu F. Chibotaru

Abstract A thorough understanding of magnetic anisotropy and dynamical magnetic properties of molecular nanomagnets requires the application of refined experimental techniques and advanced theoretical tools. In this chapter, the application of ab initio approaches to the investigation of strongly anisotropic magnetic complexes is reviewed. The role of first-principles calculations in the extraction of comprehensive information on magnetic interactions, a large deal of which being unavailable from experiment, is emphasized. A special attention is drawn to the flexibility of ab initio approaches allowing their use in combination with different levels of modelization of interactions which still cannot be reliably treated from the first principles such as exchange interactions in polynuclear complexes. Finally, the perspectives of application of ab initio methods for the description of novel phenomena in molecular magnets are discussed. Keywords Ab initio calculations · Magnetic anisotropy · Metal complexes · Single-molecule magnets · Magnetization blocking · Exchange interaction

1.1 Introduction The discovery of magnetic bistability in the Mn12 ac complex [1], qualified nowadays as first single-molecule magnet (SMM) [2], has sparked much interest in the effects of magnetic anisotropy in metal complexes, whose investigation became a mainstream in magnetochemistry research [3–6]. Indeed, with the aid of magnetic anisotropy, the phenomenology of magnetic properties becomes much richer compared to isotropic (spin) magnetic complexes [7]. For example, a magnetic state in strongly anisotropic magnetic complexes is characterized not only by the total magnetic moment, like in

L. F. Chibotaru (B) Theory of Nanomaterials Group, KU Leuven, Leuven, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Rajaraman (ed.), Computational Modelling of Molecular Nanomagnets, Challenges and Advances in Computational Chemistry and Physics 34, https://doi.org/10.1007/978-3-031-31038-6_1

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isotropic systems [8], but also by local magnetic moments on the metal sites, especially, their relative directions, a feature playing a crucial role in the magnetism of such compounds [9]. Even more involved is the description of anisotropic magnetic properties. While the description of magnetism in isotropic complexes reduces to the knowledge of one single g-factor in the case of mononuclear compounds and one exchange parameter per pair of exchange-coupled magnetic centers in polynuclear compounds [8], the theoretical modeling of anisotropic magnetic complexes requires tens of parameters [7]. This makes the approaches for the investigation of magnetic properties in isotropic and (strongly) anisotropic metal complexes radically different. In the former case, the structure of spin multiplets responsible for magnetism can be derived from measured static magnetic properties, temperature-dependent Van Vleck susceptibility and field-dependent magnetization, by simulating these experiments with Heisenberg exchange models containing few fitting exchange parameters [8]. In the second case, these experimental data are by far insufficient to elucidate the magnetism, and new experiments like luminescence and far infrared spectroscopy [10, 11], inelastic neutron scattering [12], magnetic circular dichroism [13] and electron and nuclear paramagnetic resonances [14, 15] should be done in order to acquire additional information over the systems. The quest for additional data is especially stringent in the case of lanthanides, where recently performed single-crystal angular-dependent magnetic susceptibility proved to be very informative [16, 17]. The complexity of magnetic interactions in strongly anisotropic complexes prompted a wide use of various theoretical approaches for their description, ranging from traditional crystal-field models [18, 19] to new ab initio methodologies [20, 21]. The ultimate goal of these combined experimental and theoretical studies is the elucidation of the structure of the ground and low-lying magnetic states responsible for anisotropic magnetic properties of complexes, as well as their control via the geometry and atomic (ligand) structure of the compounds. In this review, we discuss the application of ab initio methods for the investigation of magnetic anisotropy and magnetization blocking in different metal complexes. Differences of their description in weakly and strongly anisotropic compounds, as well as between single- and polynuclear complexes, will be revealed.

1.2 Spin–Orbit Coupling Effects and Their Ab Initio Treatment The spin–orbit coupling at metal ions is usually much larger than on the other atoms in the complexes. As a result of spin–orbit entanglement of crystal-field orbitals with the spin components of electrons in the occupied orbitals at the metal sites, the value of induced magnetic moments will vary with the direction of the applied magnetic field. This gives rise to magnetic anisotropy which is generally understood as the dependence of magnetic properties of a system on the direction of applied magnetic

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field. In this chapter, we discuss the manifestations and the ab initio description of magnetic anisotropy in metal complexes.

1.2.1 Manifestations of Magnetic Anisotropy It is well known that the isolated (metal) ions are magnetically isotropic despite the presence of spin–orbit coupling. The ultimate reason for this isotropy is the 2J + 1 degeneracy of atomic multiplets |J, M⟩ after the projection of angular momentum J on a given axis, M = −J, −J +1, . . . J . Then, it is clear that lifting the degeneracy of angular momentum eigenstates, for instance, via the crystal field of the environment, will result in magnetic anisotropy. Consider as an example a V3+ ion with 3d2 electronic configuration. Its groundstate term corresponds to S = 1 and L = 3, while spin–orbit interaction couples these two momenta into the total J = L − S = 2. In an axial crystal field (CF), Fig. 1.1a, the J = 2 multiplet splits into two double-degenerate levels, called Ising doublets (ID), |2, ± 2⟩ and |2, ± 1⟩, and a nonmagnetic state |2, 0⟩ (Fig. 1.1b). Due to high rotational symmetry, the projection of total angular momentum on the main axis Z (Fig. 1.1a) has a definite value in all CF multiplets, the reason why the corresponding eigenfunctions are merely the |J, M⟩ states. According to well-known selection rules [22], the matrix elements of total angular momentum (and of the corresponding total magnetic moment) on these eigenfunctions are only nonzero for ΔM = 0, ± 1. Then, a relatively weak magnetic field, not mixing states from different CF multiplets, will induce only axial magnetization along Z in the two IDs (M = ± 1, ± 2). This means that both IDs are strongly anisotropic. The same physics explains the arising of magnetic anisotropy in complexes with odd number of electrons on the metal sites, the so-called Kramers complexes. The atomic multiplets in such complexes are characterized by half-integer values of J Fig. 1.1 Effect of lift of degeneracy on magnetic anisotropy [23]. a Example of ligand configuration leading to axial crystal field. b Splitting of the ground-state atomic multiplet J = 2 of V3+ ion into two doublets and one singlet in an axial crystal field

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and M. In an axial field as in Fig. 1.1a, the atomic multiplet J splits into Kramers doublets (KD), |J, ±M⟩. The smallest angular momenta S = 1/2 and J = 1/2 belong to a special case since the twofold degeneracy of their eigenstates cannot be removed in virtue of Kramers theorem [22]. Nevertheless, these KDs can become anisotropic too in complexes of low symmetry. The anisotropy arises in this case via spin–orbit (for S) and crystal-field (for J) admixture of excited states. The necessary condition that such admixture leads to anisotropy is the removal of degeneracy after angular momentum projection of the excited states. Thus, as in the cases discussed above, the lift of degeneracy is the ultimate reason for the observed anisotropy in this case too. However, now this lift of degeneracy is manifested in the second order of perturbation theory implying that the resulting anisotropy is not strong. Examples of this kind are Cu2+ complexes in orbitally non-degenerate ground state [22]. Compared to them, the anisotropy arising from the splitting of the ground manifold (Fig. 1.1) can be much stronger. In mononuclear complexes, there are three physical manifestations of magnetic anisotropy, which we will illustrate here on the example of spin complexes (Scomplexes). These are complexes characterized by orbitally non-degenerate ground state with a total spin S, in which spin–orbit coupling leads to the following effects. (1) 2S + 1—degenerate spin levels, corresponding to S > 1/2, become split into: • KDs for half-integer spin. • Non-degenerate levels for integer spin. This effect, called zero-field splitting (ZFS), is described by the following spin Hamiltonian: Σ (1.1) Hˆ ZFS = Sˆα Dαβ Sˆβ , α, β = x, y, z, αβ

where D is the ZFS tensor and the spin operators Sˆα , Sˆβ act on the spin-wave functions of the ground molecular term. (2) The Zeeman interaction becomes anisotropic, and the gyromagnetic factor becomes a tensor: g → gαβ , α, β = x, y, z.

(1.2)

(3) The magnetization becomes anisotropic, and the magnetic susceptibility becomes a tensor: Σ χ (T , H ) → χαβ (T , H ), Mα = χαβ Hβ , α, β = x, y, z. (1.3) β

Note that we use the cgs electromagnetic system commonly employed in molecular magnetism [8]. In this system, the intensity of magnetic field H (measured in oersted), entering Eq. (1.3), is equal in vacuum to the magnetic induction B

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(measured in gauss), entering the expression for the Zeeman interaction. We will call both quantities “magnetic field” and will use oersted (or tesla, 1 T = 10,000 Oe) as a unit. In polynuclear complexes, the spins localized at different magnetic centers interact with each other via Heisenberg exchange interaction [8]. In the case of two centers, this interaction is described by the Heisenberg Hamiltonian: Hˆ ex = −J Sˆ 1 · Sˆ 2 ,

(1.4)

where S 1 and S 2 are ground-state spins of the two centers and J is the exchange coupling constant. The scalar-product form of (1.4) is dictated by the requirement of conservation of the total spin of the complex, S = S1 + S2 . This is a particular case of a general property: the total spin of a multielectronic system is conserved in the absence of spin–orbit coupling and externally applied magnetic field [24]. Indeed, one can check directly that Hˆ ex commutes with the square of the total spin, Sˆ 2 , and with any of its projections, e.g., Sˆ z . This means, in particular, that the eigenvalues of the exchange operator (1.4) are (2S + 1)-degenerate after the projection of total spin on arbitrary axis. Then again, as in the case of isolated atoms discussed above, we have full isotropy, now with respect to the direction (quantization axis) of the total spin. On this reason, the exchange interaction described by Eq. (1.4) is called isotropic. The spin–orbit coupling on the metal ions makes the spins of the corresponding magnetic centers anisotropic, as reflected in Eqs. (1.1)–(1.3). This anisotropy is imprinted also on the exchange interaction between the spins localized at the metal ions, which is not described anymore by a pure Heisenberg model (1.4). Actually, as shown by Moriya [7, 25], the exchange Hamiltonian between two spins is described by the operator: Hˆ ex = −J Sˆ 1 · Sˆ 2 + Sˆ 1 · D12 · Sˆ 2 + d12 · Sˆ 1 × Sˆ 2

(1.5)

which besides the Heisenberg term contains the symmetric anisotropic (second term) and the antisymmetric (last term) contributions. The symmetric anisotropic part is defined by the symmetric traceless tensor D12 , involving five independent parameters, while the antisymmetric part—by the vector d12 , involving three independent parameters. These two contributions to the exchange interaction do not conserve the total spin S of the pair, which can be checked straightforwardly by calculating their commutators with Sˆ 2 and Sˆ z . As a result, the eigenvalues of (1.5) will not be characterized anymore by (2S + 1)-fold degeneracy after the total spin projection. Accordingly, the corresponding eigenfunctions will not be |S M⟩ states, but rather their combinations, i.e., will manifest space and magnetic anisotropy. In the case when the spin–orbit coupling within an orbitally degenerate term is stronger than its CF splitting, a typical situation in lanthanides, the ground and lowlying excited states can be seen as arising from the CF splitting of the corresponding J-multiplet. The CF states of such J-complexes will manifest magnetic anisotropy as exemplified in Fig. 1.1. Polynuclear complexes involving J-ions will be characterized

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by strongly anisotropic exchange interactions, which are generally more complex than in S-complexes (Eq. 1.5).

1.2.2 Origin of Magnetization Blocking As was mentioned in the introduction, one of the most intriguing manifestations of magnetic anisotropy in complexes is the phenomenon of magnetization blocking or, generally, slowing down of magnetic relaxation in nanomolecules [3]. Magnetization blocking is ubiquitous in bulk magnetic materials, where it simply corresponds to ordering (freezing) of localized magnetic moments below some critical temperature [26]. In these ordered phases, the steady orientation of magnetic moments at the metal sites is caused by the presence of low-symmetry molecular field arising from exchange interaction with neighbor sites (Weiss field), which persists due to the macroscopic size of the material. For smaller pieces of material, containing a few metal sites, or in magnetic complexes, such a field does not develop and the magnetization relaxes quickly to its equilibrium value, i.e., to zero in the absence of an applied dc magnetic field. This occurs via fast quantum tunneling transitions between |S M⟩ states with neighboring values of momentum projection (M , = M ± 1), which remain quasi-degenerate in the absence of Weiss field. Due to these transitions, a paramagnetic nanoparticle prepared in a state |S|M⟩ will ⟩ lose its magnetization M via a thermodynamic equilibration with all states | S M , , M , = −S, . . . , S, within a relaxation time which can be as short as 10−9 s in concentrated isotropic systems [3]. Accordingly, a ZFS splitting of the ground-state spin S into doublets (|S M⟩, |S − M⟩), an often occurring situation in complexes with axial symmetry, will suppress strongly the tunneling transition between the two components of these doublets because the difference of their momentum projection, |ΔM| = 2M, significantly exceeds 1. Thus, the ground and excited doublets in Fig. 1.1b are characterized by |ΔM| = 4 and 2, respectively. On the other hand, the tunneling between different doublets, if they are not brought in resonance by an external bias magnetic field, is suppressed by their ZFS separation exceeding much the tunneling splitting within each doublet. One can state, therefore, that the axial magnetic anisotropy of doublets arising in complexes with axial ZFS is the reason for the suppression of tunneling transitions between |S M⟩ states. Such complexes (SMMs) exhibit slow relaxation of magnetization seen as magnetization blocking for a finite time interval in different magnetic measurements. Real complexes do not possess an ideal axial symmetry. Therefore, the rate of quantum tunneling of magnetization (QTM) in their ground ZFS doublet can exhibit largely differing values depending on the degree of deviation from an ideal symmetry. The necessary condition for a complex to display blocking of magnetization in a given temperature domain is the sufficiently low rate of tunneling flipping of magnetization M ↔ −M in the ground doublet state. This condition is satisfied when the corresponding tunneling matrix element is much smaller than the Zeeman spitting of the

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(M, −M) doublet induced by a relatively weak dc magnetic fields of the order of internal fields in the crystal. In other words, a weak magnetic field should be able to bring efficiently out of resonance of the two ground doublet components [3, 24]. Creating a bias between these levels with, e.g., external dc magnetic field gives rise to one-phonon (direct) relaxation processes [27, 28]. The energy separation between components of different doublets, which intrinsically present in most complexes, makes the one-phonon transitions between them a dominant relaxation mechanism at low temperature [22]. Rising the temperature, the Raman and the Orbach twophonon relaxation mechanisms become activated [22]. Raman processes subdivide into first-order and second-order types. Second-order Raman and Orbach mechanisms represent two-step processes involving an excited doublet component, e.g., − 1 → − 2 → 1 and − 1 → 2 → 1 (Fig. 1.2). However, while in the former the absorption of one phonon (transition − 1 → ± 2) and the emission of the other one (transition ± 2 → 1) are virtual processes, in the latter relaxation mechanism, they correspond to two direct (real) one-phonon transitions. Contrary to them, the first-order Raman is a one-step two-phonon real process (e.g., a direct two-phonon transition − 1 → − 2 in Fig. 1.2). In complexes without strong axiality (see Sect. 1.3.4) such as Ln(III) ions embedded in non-axial environment, these processes are the main relaxation mechanisms [29–31]. They become strongly suppressed in performant SMMs characterized by strong axial crystal field [32]. In such complexes also, the temperature-assisted quantum tunneling of magnetization (TA-QTM) [33] in the first excited doublet state becomes strongly quenched. Consequently, the relaxation proceeds from the first excited doublet, n = 2, via the second excited doublet, n = 3 (Fig. 1.2), involving the same processes as described above. Should these indirect relaxation mechanisms (Raman and Orbach) connecting the states −2 and 2 be suppressed too, and the relaxation proceeds via the third excited doublet, etc. In this way, the relaxation path will involve only direct spin–phonon transitions between neighbor multiplet states (bold arrows in Fig. 1.2). Climbing the stair of these states, each such transition requires an absorption of one phonon; therefore, the entire process is of activation type [22]. Accordingly, the relaxation rate is described by an Arrhenius low [3] with the highest involved doublet state (n = 3 in Fig. 1.2) corresponding to the magnetization blocking barrier U eff . In performant SMMs, the rate of over-barrier reversal of magnetization, C exp(−U eff /kT ), should be kept low at elevated temperature, which means that U eff is desired to be as high as possible. The latter implies involvement of as many doublet states as can be provided by the structure CF multiplets. To this end, the unwanted relaxation processes (all but thick arrows in Fig. 1.2) should be largely quenched. Besides direct and Raman transitions, at sufficiently high temperature, these include also the TA-QTM for excited doublets (yellow dashed arrows in Fig. 1.2). All described relaxation processes, including incoherent tunneling in the ground doublet and TA-QTM [33], are various versions of spin–phonon transitions between the connected multiplet states described by the Fermi golden rule [22]. The corresponding expressions involve a transition matrix element of an electronic operator and a temperature-dependent phonon factor. Therefore, designing efficient

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Fig. 1.2 Scheme of spin–orbit doublet states (± n) corresponding to ZFS components of the ground S = 5/2 term or to CF components of the ground atomic J = 5/2 multiplet. The two states of a doublet have opposite values of magnetization, |M| and −|M|, respectively. The arrows show the mechanisms of relaxation of magnetization (see the text). An ideal SMM corresponds to suppression of all relaxation channels except for ones shown by bold arrows. The wavy green arrows correspond to one-phonon (virtual) transitions for the second-order Raman processes and to two-phonon (real) transitions for the first-order Raman processes. Reprinted with the permission from [32]. Copyright 2016 American Chemical Society

SMMs requires, first of all, an engineering of the structure of low-lying multiplet states which determine the electronic matrix elements in different relaxation processes [32].

1.2.3 Theoretical Description Magnetic anisotropy in complexes has a long history of theoretical description by different phenomenological models. In the case of weak spin–orbit coupling effects, the model description is based on spin Hamiltonians as in Eqs. (1.1) and (1.5) [7, 8, 22] (but see the end of Sect. 1.3.1). In the case of moderate and strong spin– orbit coupling effects, the energies and wave functions of low-lying multiplets of the complex are described by effective spin (pseudospin) Hamiltonians [22, 34]. The pseudospin S˜ is not related to a physical angular momentum of the complex. Its size is defined by the dimension N of the manifold of states which is chosen for the model description: 2 S˜ + 1 = N. For example, in the case of doublet states arising from CF splitting of atomic J-multiplets (Fig. 1.2b), the corresponding pseudospin is S˜ = 1/2 (2 S˜ + 1 = 2). Despite its formal nature, the pseudospin vector ( S˜ x , S˜ y , S˜ z ) can be related to real-space coordinate system, i.e., completely defined, when it is close to a true spin (S-pseudospin), a true total angular momentum (J-pseudospin) or corresponds to a degenerate irreducible representation of high-symmetry group (⎡-pseudospin) [23]. Contrary to spin Hamiltonians used for the description of Zeeman, ZFS and exchange interactions in complexes with weak spin–orbit coupling effects, the

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

9

number of parameters entering the corresponding pseudospin Hamiltonians increases tremendously with the size of pseudospins. For the description of ionic anisotropy in mononuclear complexes and fragments also crystal (ligand) field models have been applied [35]. As in the case of pseudospin Hamiltonians, they can involve many parameters. For instance, to describe the CF spectrum arising only from the ground atomic multiplet of a Ln ion, one needs to use up to 27 crystal-field parameters if the complexes have low symmetry [22]. At the early stage, when the theory was not able to provide the parameters of pseudospin Hamiltonians with sufficient accuracy, the latter was efficiently constructed as invariants of given symmetry groups of the complexes [34]. Indeed, written in a coordinate system related to the symmetry axes of the complex, they involve a minimal number of parameters which can in principle be extracted from experiment. To this end, the eigenvalues of pseudospin Hamiltonians were fitted to the observed transition energies, while their eigenfunctions were used to simulate the experimental probabilities of transitions between different energy levels [22, 34]. Experimentally, the parameters of the spin Hamiltonians are most often extracted from EPR spectroscopy [7, 22, 34]. The intrinsic limitation of phenomenological models prompted the use of quantum chemistry methods, which have been first applied to the evaluation of spin Hamiltonians parameters, suitable for the description of most S-complexes. Thus, it became possible to determine the Zeeman-splitting g-tensor either perturbatively [36] or straightforwardly from DFT [37, 38] and Dirac–Fock (four-component) calculations [39, 40]. The parameters of ZFS D-tensor have been extracted from DFT calculations via a direct perturbative approach [41], by analyzing the second-order spin– orbit coupling contributions to the magnetic anisotropy energy [42, 43], via a DFTbased ligand field model [44], and from explicitly correlated ab initio calculations by analyzing the zero-field splitting of spin levels [38, 45–50]. The effect of weak spin–orbit coupling on the exchange interactions was also studied [51–53]. Recently, the increased power of explicitly correlated ab initio methods made the accurate determination of the parameters of pseudospin Hamiltonians for experimentally investigated complexes feasible. Contrary to spin Hamiltonians, the pseudospin description requires a non-perturbative treatment of spin–orbit coupling, i.e., the latter should be included in the quantum chemistry calculations from very beginning. The second important aspect is the essentially multiconfigurational character of the corresponding wave functions, which excludes the use of DFT methods for most complexes requiring a pseudospin description. The method of choice, which meets these requirements, proved to be the complete active space self-consistent field (CASSCF) approach [54], often followed by second-order perturbation treatment of dynamic electronic correlation (CASPT2) [55] as implemented, e.g., in the MOLCAS package [56]. In these calculations, relativistic effects are taken into account via the Douglas–Kroll Hamiltonian [57], first as scalar ones in the CASSCF/CASPT2 calculations and then via spin–orbit mixing of the obtained molecular terms within the spin–orbit restricted active space state interaction code (RASSI-SO) [58]. The methodology for the extraction of pseudospin magnetic Hamiltonians from ab initio calculations has been recently developed [21, 23].

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L. F. Chibotaru

Table 1.1 Ab initio calculated g-tensors of [Mo(CN)7 ]4− complexes in different geometries Ideal PBP (D5h )

Ideal CTP (C 2v )

K2 [Mn(H2 O)2 ]3 [Mo(CN)7 ]2 ·6H2 O fragment (C 1 )

g|| = 3.20 g⊥ ≈ 0.00

gy = 2.23 gx = 1.88 gz = 1.87

g1 = 2.53 g2 = 1.41 g3 = 1.39

ζ = 489 cm−1

ζ = 317 cm−1

ζ = 310 cm−1

k = 0.60

k = 0.39

k = 0.38

Reprinted with the permission from [59]. Copyright 2005 American Chemical Society

The efficiency of this ab initio approach for the description of the effects of strong magnetic anisotropy has been first demonstrated on the non-perturbative calculations of the g-tensor for the ground Kramers doublet of strong anisotropic [Mo(CN)7 ]4− complexes and fragments [59]. Table 1.1 shows that the obtained gtensor strongly depends on the geometry of the complex, i.e., whether the hepta-cyano environment forms an ideal pentagonal bipyramid (PBP), an ideal capped trigonal prisms (CTP), or has no symmetry at all being a fragment of an extended network K2 [Mn(H2 O)2 ]3 [Mo(CN)7 ]2 ·6H2 O [60]. Calculations of this type have been done for other transition-metal complexes with strong magnetic anisotropy [61–63], as well as for lanthanide [9, 16, 64–66] and actinide [67, 68] complexes. The implementation and application of ab initio methodology for the description of anisotropic magnetic properties of lanthanides (J-complexes) have been recently reviewed [69].

1.2.4 Ab Initio Methodology for Magnetic Properties A decent description of spin–orbit multiplets by multiconfigurational ab initio approaches provides reliable predictions for static magnetic properties of complexes with arbitrary strength of spin–orbit coupling. The basis of the approach is the use of the matrix elements of total magnetic moment, µ = −μB

Nel Σ

(ge si + li )

(1.6)

i=1

evaluated on the multiplet wave functions of the complex relevant for its magnetism. The expressions for all static magnetic properties in terms of these matrix elements and the multiplet energies have been derived [70]. These include magnetic susceptibility tensor and its powder-averaged value in function of temperature and applied field; the magnetization vector as function of the vector of applied magnetic field and its powder-averaged value; magnetic torque, magnetic entropy, the magnetization blocking barrier, etc. The same quantities are used for the unique definition of pseudospin corresponding to a chosen manifold of multiplet wave functions and

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

11

for the derivation on its basis of pseudospin magnetic (Zeeman) and ZFS Hamiltonians [21, 23]. The latter includes besides conventional first-rank Zeeman (described by the g-tensor) and second-rank ZFS (described by D-tensor) contributions also higher-rank tensors (Sect. 1.3.2). The derivation of these effective Hamiltonians is non-perturbative, i.e., essentially exact. There is an important methodological difference in the CASSCF/CASPT2 description of single-ion (mononuclear) and (exchange-coupled) polynuclear complexes. While the multiplet structure of the former is reliably calculated by these ab initio methods, a similar description of the latter cannot be achieved with sufficient accuracy. Accordingly, the ab initio methodology for the description of magnetism of mononuclear complexes is based on CASSCF/CASPT2/RASSI-SO calculation of their low-lying multiplet structure, as described above. It is implemented in the SINGLE_ANISO program [69, 71]. For polynuclear complexes, the ab initio calculation of the multiplet structure at individual metal ions in the complex is performed as a prerequisite. In these calculations, all metal ions except the chosen one are replaced by close nonmagnetic ions (e.g., Cu2+ by Zn2+ , Dy3+ by Lu3+ , etc.). In a following step, the anisotropic exchange interaction between metal ions is simulated within the Lines model using the multiplet structure at individual metal ions [72]. This is done via the calculation of the matrix of a model isotropic exchange interaction with adjustable parameters (the Lines exchange parameters) on the multiplicative basis of a chosen group of low-lying multiplets at the metal centers. Such an approach allows to use one single exchange parameter for each pair of exchangecoupled metal ions, which are determined from the fitting of available magnetic data of the complex [9, 63]. The obtained spectrum of spin–orbit exchange multiplets is used for the calculation of magnetic properties and magnetic Hamiltonians (for a chosen group of exchange multiplets) in a similar way as it is done for single-ion complexes. This methodology is implemented in the POLY_ANISO program [9, 63, 69]. The implemented approach is especially efficient for the evaluation of exchange interaction in complexes with strong axial anisotropy on the metal ions. The structure of the output of the two codes is shown in Scheme 1.1. Besides the quantities similarly derived in SINGLE_ and POLY_ANISO, the former code also allows for the derivation in a unique way of CF (Stevens) parameters for each metal center while the latter gives the parameters of anisotropic exchange Hamiltonian and magnetic dipolar interaction in terms of pseudospin operators at the metal sites. Both codes are able to take into account the intermolecular exchange interaction in a mean-field fashion [8] and also the shape anisotropy arising in samples of low symmetry [73]. The SINGLE_ANISO and the POLY_ANISO programs have been implemented as modules in several versions of MOLCAS [74], in the OpenMOLCAS [75] and ORCA [76] quantum chemistry packages. One should stress that the description of magnetism and the extraction of effective Hamiltonians is similar for all kinds of metal complexes, involving lanthanides, actinides, transition metals and radicals (in mixed polynuclear compounds). Moreover, the procedure itself is essentially

12

L. F. Chibotaru

Scheme 1.1 Physical quantities calculated by the SINGLE_ANISO and POLY_ANISO software

exact, and all approximations are being entailed in the quantum chemistry calculation of multiplet wave functions. In the following, the application of this ab initio methodology is exemplified on a number of complexes.

1.3 Magnetization Blocking in Mononuclear Complexes The interest for the effects of magnetic anisotropy first arose in connection with SMM behavior of Mn12 ac [1], while their study in mononuclear complexes started a decade later, after the discovery of SMM behavior in [Tb(Pc)2 ]− [18]. This is because the magnetization blocking is easier to achieve in polynuclear than in mononuclear complexes. On the other hand, the anisotropic magnetic interactions are obviously less complex in the latter.

1.3.1 Weak Spin–Orbit Coupling Effects In transition-metal complexes, the effects of spin–orbit coupling are usually weak. Indeed, a typical situation in these complexes is an orbitally non-degenerate electronic ground term, characterized by the total spin S and well separated from excited states, so that the effect of spin–orbit coupling can be considered as a perturbation. This is the case in most of the first-row transition-metal complexes involving ions like Cr3+ , Mn2+ , Fe3+ , Ni2+ , etc., where the spin–orbit coupling is of the order of

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

13

several hundred wavenumbers, while the CF splitting is of the order of several thousand wavenumbers, so that the perturbation theory works rather well [77]. Due to the orbital non-degeneracy of the ground term, the effect of spin–orbit coupling arises only in the second order of perturbation theory, so the ZFS Hamiltonian is always bilinear in S, having the form given by Eq. (1.1). The Zeeman interaction in the lowest, first-order perturbation theory after spin–orbit coupling is always linear in S and is described by the g-tensor (1.2) [22]: Hˆ Zee = μB

Σ

Hα gαβ Sˆβ ,

α, β = x, y, z.

(1.7)

α,β

The D- and g-tensors entering magnetic Hamiltonians (1.1) and (1.7), respectively, were straightforwardly derived for various levels of description of electronic terms of complexes. The simplest description is provided by the CF theory [22, 34, 35, 77]. Thus, in the case of weak and intermediate CF, when the lowest terms of the complexes can be thought as arising from the ground atomic LS term of the corresponding metal ion, the spin–orbit coupling acquires a simple form: ξ Hˆ so (L S) = λL · S, λ = ± κ , 2S

(1.8)

where ξ is the spin–orbit coupling constant of the metal ion [22], κ is the orbital reduction factor, taking into account the effect of metal–ligand covalency [22, 78], and λ is the term-projected spin–orbit coupling constant for a given spin S [22, 24, 78]. The two signs in the expression for λ correspond to less than half-filled and more than half-filled shell of magnetic electrons on the metal site, respectively. Perturbation theory applied to the operator (1.8) leads to spin Hamiltonians in Eqs. (1.1) and (1.7) with the following parameters: g = ge 1 + λΛ, D = λ2 Λ, Σ ⟨Ψ S M | Lˆ α |Ψν S M ⟩⟨Ψν S M | Lˆ β |Ψ S M ⟩ Λαβ = , α, β = x, y, z, E S − Eν S ν

(1.9)

where Lˆ α , Lˆ β are Cartesian components of the total orbital momentum, Ψ S M and E S are the wave function and energy of the ground term, while Ψν S M and E ν S —of the νth excited term of the same spin S. The fact that no terms of other spin admix in Eq. (1.9) is the consequence of the form (1.8) of the spin–orbit coupling, i.e., of the LS approximation [22]. A general spin–orbit coupling operator [24, 79] will admix also terms with S , = S ± 1 [41, 80, 81]. By proper choice of the directions of coordinate axes (X, Y, Z), one can bring the tensors g and D to diagonal form. These coordinate systems define the main magnetic axes for g and the main anisotropy axes for D. The eigenvalues are called main values of the g-tensor (gX , gY , gZ ), or simply g-factors, and main values of D-tensor (DX ,

14

L. F. Chibotaru

DY , DZ ). Since the latter defines the splitting of the S multiplet, choosing the zero of energy in the center of gravity of the split levels makes D traceless, i.e., DX + DY + DZ = 0. Then, the ZFS Hamiltonian, written in the coordinate system of main anisotropy axes, is defined by only two independent parameters [8]: ) ( ) ( Hˆ ZFS = D Sˆ Z2 − S(S + 1)/3 + E Sˆ X2 − SˆY2 , D = 3D Z /2,

E = (D X − DY )/2.

(1.10)

D is called parameter of axial magnetic anisotropy because it defines the ZFS splitting of S in complexes with axial symmetry (possessing main rotational axis of the order higher than two, e.g., C3 or S4 ); E is the parameter of rhombic magnetic anisotropy because it is only nonzero for non-axial symmetries, e.g., D2 . In the LS approximation, both g and D-tensors are expressed via a common tensor Λ. Therefore, the main axes of these tensors coincide for any symmetry of the complex since in both cases they are the main axes of the tensor Λ (1 in Eq. (1.9) is a unity matrix). This is an artifact of the simplified description of spin–orbit coupling (1.8) having a perfect spherical symmetry, i.e., being unrelated to the actual symmetry of the complex. Lowest-order perturbation theory based on true wave functions of the electronic terms and realistic spin–orbit coupling give “quantum chemistry” expressions for g and D which are free of this drawback [23]. As an example, Fig. 1.3 shows the main magnetic and main anisotropy axes of a Ni(II) complex [Ni-(HIM2 -py)2 NO3 ]NO3 [82] obtained with SINGLE_ANISO module on the basis of CASSCF /CASPT2/RASSI-SO calculations [21]. We can see that despite the fact that this complex is in a weak spin–orbit coupling limit, the directions of its main magnetic and main anisotropy axes differ significantly. Note that the two sets of axes will coincide in complexes with symmetry not lower than orthorhombic [22, 23]. One should stress that the obtained magnetic Hamiltonians, Eqs. (1.1) and (1.7), are represented by true-spin operators Sˆα (acting on the spin-wave functions of the ground molecular term) in both LS and “quantum chemistry” perturbative derivations. The important difference is that in the former case, the magnetic tensors, Eq. (1.9), are always symmetric, while in the latter, g is generally a non-symmetric matrix (unless the complex possesses sufficiently high symmetry) albeit D remains symmetric. Such a situation is not acceptable since a non-symmetric matrix cannot be diagonalized in a usual way, i.e., the g-tensors cannot be brought to main magnetic axes by a proper rotation of Cartesian axes; therefore, its main values (g-factors) cannot be determined. To overcome this complication, an artificial symmetrization of the g matrix has been applied [41, 80] which, however, yielded a Zeeman Hamiltonian predicting the Zeeman splitting and other magnetic properties diverging from the predictions of the exact (quantum chemistry) treatment. The paradox is resolved if we remember that in the presence of spin–orbit coupling (even a weak one), the spin is not a good quantum number anymore and the magnetic and other effective Hamiltonians should be expressed in terms of corresponding pseudospins (Sect. 1.2.3). Passing to them results in pseudospin Hamiltonians described by symmetric magnetic tensors

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

15

Fig. 1.3 Molecular structure of [Ni-(HIM2 -py)2 NO3 ]NO3 [82]. Orientation of the main magnetic axes (X m , Y m , Z m ) (green arrows) and the main anisotropy axes (X a , Y a , Z a ) (purple arrows) calculated ab initio. Color scheme: turquoise Ni; red O; blue N; gray C; white H. Reproduced with the permission from [21]. Copyright 2012 AIP Publishing

and giving equivalent predictions of magnetic properties with the original true-spin Hamiltonians (see [23] for derivations and discussion).

1.3.2 Strong Spin–Orbit Coupling Effects This is the case of complexes containing lanthanides (except Gd3+ ), actinides, most of 4d and 5d and some of 3d transition metals (e.g., Co2+ , Fe2+ , etc., in some geometries). In these complexes, the low-lying terms are often closely spaced, so that their spin–orbit mixing cannot be treated as a perturbation. As a result, the spin of the ground term cannot be considered as a good quantum number even approximately. Therefore, contrary to the case of weak spin–orbit coupling effects, the Zeeman and ZFS Hamiltonians cannot be expressed via the ground term’s spin but should be formulated instead via the pseudospin (see Sect. 1.2.3). The main difference from ˜ does not have the meaning of a physthe spin description is that the pseudospin ( S) ical angular momentum of the complex and its size is defined by the number N of low-lying multiplets involved in the model description (2 S˜ + 1 = N). The magnetic Hamiltonians expressed in terms of pseudospin operators formally look similarly to spin Hamiltonians but contain much more terms and independent parameters. Thus, the spin ZFS Hamiltonian in Eqs. (1.1) and (1.10) contains only terms bilinear in spin operators and five independent parameters (three angular parameters defining the main anisotropy axes, X a , Y a and Z a , and two ZFS parameters, E and D). The spin Zeeman Hamiltonian contains only linear terms in S α and six parameters (three angles defining the directions of main magnetic axes, X m , Y m and

16

L. F. Chibotaru

Z m , and three main values of the g-tensor: gX , gY and gZ ). In the case of pseudospin description, these Hamiltonians keep their form (with Sˆα replaced by Sˆ˜α ) for S˜ ≤ 1 in the case of Zeeman and for S˜ ≤ 3/2 in the case of ZFS interaction, respectively [21, 23]. For larger pseudospins, terms of higher order in Sˆ˜α should be added. In order to construct possible independent (irreducible) combination of polynomials in Sˆ˜α , the technique of irreducible tensor operators (ITOs) is conveniently applied for the construction of pseudospin magnetic Hamiltonians [23]. It can be shown that independent combinations of polynomial terms of a rank l can be represented by 2l + 1 familiar spherical harmonics Y lm (θ, ϕ), m = −l, −l + 1, … l, in which the spherical angular coordinates defining the direction of an electron in space, r/r, ˆ˜ S. ˜ These can be replaced by more simple are replaced by pseudospin unit vector S/ ) ( ) ( ˆ˜ S˜ , or their Sˆ˜ ∝ Yˆ S/ expressions, the Stevens operators [22, 34, 83], Oˆ n,m

n,m

Hermitian combinations (m > 0): ) ) 1( i(ˆ m ˆ ˆm O . = − 1) Oˆ nm = O (− 1)m Oˆ n,m + Oˆ n,−m , Ω (− n,−m n,m n 2 2

(1.11)

ˆ defining the Zeeman interaction, Since the operator of magnetic moment, µ, ˆ · H, is a time-odd operator which changes its sign under time inversion Hˆ Zee = −µ [22], its ITO decomposition will involve only irreducible operators of odd rank: ˆ =µ ˆ1 +µ ˆ3 +µ ˆ 5 + · · · , μˆ nα = µ

n ( Σ

) α ˆm α ˆm Ωn , α = x, y, z. bnm On + cnm

m=0

(1.12) On the other hand, the ZFS Hamiltonian is a time-even operator; therefore, it will contain only even-rank irreducible tensors in its decomposition: 2 4 6 + Hˆ ZFS + HZFS + ··· , Hˆ ZFS = Hˆ ZFS

n Hˆ ZFS =

n ( ) Σ α ˆm α ˆm Ωn . (1.13) enm On + f nm m=0

The highest rank of ITO entering the decomposition of Eqs. (1.12) and (1.13) is ˜ which gives the structure of Zeeman and ZFS defined by the condition n max ≤ 2 S, Hamiltonians as function of the size of pseudospin S˜ shown in Table 1.2. We can see α that the number of independent parameters describing the Zeeman interaction (bnm α α α and cnm in Eq. (1.12)) and the ZFS interaction (enm and f nm in Eq. (1.13)) increases quickly with the size of pseudospin in the case of low symmetry of complexes. However, in high-symmetry complexes, their number can be strongly reduced. For example, the Zeeman Hamiltonian for pseudospin S˜ = 3/2 will include in a general case six parameters in the first-rank part and 21 parameters in the third-rank part (see Table 1.2), while only one parameter for contributions of each rank in the case of cubic symmetry of the complex [22, 23].

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization … Table 1.2 ITO decomposition of Zeeman and ZFS pseudospin Hamiltonians

S˜

ˆ ·H Hˆ Zee = −µ

Hˆ ZFS

1/2

ˆ1 µ

1

ˆ1 µ

– Hˆ 2

3/2

ˆ1 +µ ˆ3 µ

2 Hˆ ZFS

2

ˆ1 +µ ˆ3 µ

5/2

17

ZFS

2 +H ˆ4 Hˆ ZFS ZFS

1

3

5

2 +H ˆ4 Hˆ ZFS ZFS

1

3

5

2 +H ˆ 4 + Hˆ 6 Hˆ ZFS ZFS ZFS

ˆ +µ ˆ +µ ˆ µ

3

ˆ +µ ˆ +µ ˆ µ

7/2

ˆ1 +µ ˆ3 +µ ˆ5 +µ ˆ7 µ

2 +H ˆ 4 + Hˆ 6 Hˆ ZFS ZFS ZFS

4

ˆ1 +µ ˆ3 +µ ˆ5 +µ ˆ7 µ

2 +H ˆ 4 + Hˆ 6 + Hˆ 8 Hˆ ZFS ZFS ZFS ZFS

Modified with permission from [21]. Copyright 2012 John Wiley and Sons

A useful tool for theoretical investigation of magnetic anisotropy, especially, in lanthanide complexes is the CF analysis of the lowest multiplets [18, 19, 22]. This theory describes the splitting of an nl shell of the metal ion in the field of surrounding ligands’ atoms. Given the spherical symmetry of the |nlm⟩ orbitals on the metal ion, the expansion of CF potential in spherical harmonics Y km (θ, ϕ), m = −k, −k + 1, … k, will involve only terms with even k ≤ 2l, which in the case of lanthanides and actinides (l = 3) reduce to k = 2, 4, 6 [22]. For the description of magnetic properties of lanthanide complexes, including its anisotropy, it is sufficient to describe the CF splitting of the ground atomic J-multiplet of the corresponding Ln3+ ion. This ) ( is achieved by the projection of electronic operators Y km (θ, ϕ) onto the ITO Oˆ k,m Jˆ ) ( and their Hermitian combinations (1.11), where Jˆ = Jˆx , Jˆy , Jˆz is the operator of total angular momentum. Thus, the CF Hamiltonian acting in the space of 2J + 1 wave functions |J M⟩ of the ground atomic J-multiplet has the form [22]: J Hˆ CF

=

Σ n=2,4,6

(

n Σ

m=0

c ˆm Bnm On

+

n Σ

) s ˆm Ωn Bnm

.

(1.14)

m=1

Recently, an ab initio methodology for first-principles derivation of parameters of J-projected CF Hamiltonian for lanthanides has been proposed and implemented in the SINGLE_ANISO code [84]. Such J-projected Hamiltonians are less justified in the case of actinides, where important mixing of excited terms takes place due to a much stronger metal–ligand covalency and the CF theory should be formulated in terms of J˜-pseudospin states [85]. In the absence of point group symmetry, which is a typical situation in lanthanides, the Hamiltonian (1.14) involves 27 independent parameters, whose correct determination poses the main problem in the application of CF theory to lanthanides. In such a situation, the unique derivation of all CF parameters provided by the ab initio methodology represents a major advantage over other approaches employed so far.

18

L. F. Chibotaru

The traditional approach to CF description is based on the extraction of these parameters from available experiments. This approach becomes feasible when the number of CF parameters is reduced due to a high symmetry of the complex [18, 19, 86]. For example, in the case of complexes with square antiprism geometry, such as [LnPh2 ]− with Ph = phtalocyanine [18] and polyoxometalates [19], having a D4d symmetry c c c , B40 and B60 in the absence of distortions (ideal geometry), only the parameters B20 enter Eq. (1.14) and can be determined from magnetic data. However, the ab initio CF analysis of the [TbPh2 ]− complex has shown that despite its close geometry to the D4d symmetry, many non-axial CF parameters are far from being negligible [84]. One should emphasize that the extracted set of J-multiplet CF parameters within the ITO approach [84] is not only unique but also exact in the sense that diagonalization of Eq. (1.14) reproduces exactly the corresponding energies and wave functions of the ab initio calculation from which this set was derived. This is a key difference from approaches based on the fitting of phenomenological ligand field models to the ab initio calculations [87]. The latter employs a set of one-electron parameters modeling an effective CF potential, one effective spin–orbit coupling constant and three effective Racah parameters. While such a CF description can give some insight, especially when analyzing a series of isostructural complexes [88], the fitted parameters may not be always physically sound because of the oversimplification of the employed models. Thus, due to the covalent admixture of ligand orbitals to the 4f ones, the actual interaction between electrons in the seven molecular orbitals of 4f type is described by hundreds of independent bielectronic parameters, contrary to few Racah parameters used in phenomenological ligand field models [89]. This can become an additional important source of CF splitting, as was demonstrated by the ligand field analysis of [Mo(CN)7 ]4− complexes (Table 1.1), where the variation of isotropic Racah among various 4d orbitals was found to reach several thousand wave numbers [59]. Another important manifestation of strong spin–orbit coupling is the possibility of negative g-factors in some multiplets [90]. This was confirmed in resonance experiments with circularly polarized radiation performed on cubic actinide complexes [91–93]. The quantity extracted in this measurement is the sign of the product of three g-factors of the investigated KD, whose ab initio calculation has been recently implemented in SINGLE_ANISO and applied to various complexes [23, 94]. The ab initio methodology allowed, in particular, to achieve a unique definition of the g-tensors of the ground KDs in several actinide complexes [91–93], including cases of their deviation from cubic symmetry, and to confirm the experimental findings concerning their g-factors [95]. An analysis of the conditions for the occurring of negative g-factors has shown that the dominant CF components in Eq. (1.14) of a rank higher than second are indispensable [94]. This explains immediately why negative g-factors cannot appear in KDs of S-complexes, even in those of them which are of Ising type as in Mn12 ac [3]: the ZFS interaction responsible for their formation is always of second rank (see Eq. 1.10). The same is true for other complexes where second-rank CF contribution is dominant such as Ln complexes in strong axial environment discussed below. On the contrary, if the geometry of the complex is far from

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

19

being axial, negative g-factors do appear as demonstrated recently on the example of Er3+ [94], Yb3+ [67] and even Cu2+ [94] complexes.

1.3.3 Success of Ab Initio Description of Ln Complexes The calculation of temperature-dependent molecular susceptibility and fielddependent magnetization of lanthanide complexes within ab initio-based methodology gives a good agreement with experiment, sometimes with experimental accuracy. These results have been reviewed elsewhere [69]. One may conclude that at least the low-lying CF multiplets are well described by the CASSCF/CASPT2/SORASSI ab initio calculations. It turns out, however, that the entire spectrum of CF multiplets originating from the ground atomic J-multiplet of the Ln ion can be well described by this ab initio methodology. As an example, let us consider the Er-trensal complex (H3 trensal = 2,2, ,2,, tris (salicylideneimido)trimethylamine) (Fig. 1.4a) whose CF spectrum was firmly established from luminescence [96] and INS [28] studies. The results of ab initio calculations of the eight KDs arising from the CF splitting of the ground atomic multiplet 4 I 15/2 of Er3+ are shown in Fig. 1.4b for different levels of approximations [84]. In this plot, black, blue and green levels were calculated within CAS (11 in 7) (A), CAS (11 in 14) (B) and CAS (17 in 25) (C) active spaces, respectively. Er, O, N and C atoms close to ER were described by ANO-RCC-VDZP basis sets, and the Madelung potential from surrounding molecules in the crystals has been simulated with Mulliken atomic charges of five layers of surrounding unit cells (see [84] for the details). We can see from Fig. 1.4b that all CASSCF calculations underestimate the energies of the higher four KDs. At the same time, the four low-lying KDs are well described for all active spaces, explaining why the magnetism is well reproduced even at a low level of ab initio theory, a conclusion valid for most of Ln complexes. The CASPT2 step in the smallest active space (A) pushes up all multiplet energies, which come closer to the experimental ones for larger active spaces. Note that the B and C active spaces, besides describing well the entire CF spectrum, also reproduce the three g-factors of the ground KD close to experiment. The ab initio analysis also allows to assess the contribution of electrostatic field from electronic distribution at the ligand to the CF splitting of the 4 I 15/2 of Er3+ . First, the effect of atomic point charges (Mulliken and LoProp [97]) obtained in DFT and SCF/MP2 calculations of the trensal ligand (charged − 3) on the CF spectrum has been elucidated within CASSCF (A) approach. Figure 1.4c shows that the resulting overall CF splitting is more than six times smaller than the experimental CF splitting. In the second approach, the true space distribution of the electronic density in the ligand was considered explicitly in the electrostatic potential. The electronic density distribution (Fig. 1.4a) was found self-consistently from concomitant Hartree–Fock calculation of the (trensal)3− ligand and CASSCF (A) calculation of Er3+ . A similar calculation has been done employing fragment ab initio embedding model potential (FAIEMP) [98], which besides true electrostatic potential includes also the exchange

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Fig. 1.4 Ab initio CF in Er-trensal complex. a Distribution of electronic density of the (trensal)3− ligand surrounding Er3+ . b Comparison between various ab initio approaches for the calculation of CF spectrum with data extracted from luminescence [96, 97]. All calculations have been done for the experimental geometry [28] and included the Madelung potential. c Comparison of predictions of various electrostatic models and the minimal CASSCF calculation of the CF spectrum. Reprinted with the permission from [84]. Copyright 2017 John Wiley and Sons

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and the Pauli’s repulsion contributions. Figure 1.4c shows that both approaches give close CF spectrum implying that the electrostatic potential yields the major contribution to the total non-covalent embedding potential of the metal ion. The obtained CF splitting is much larger than the one obtained from point charges’ calculations but still twice smaller than the overall splitting given by conventional CASSCF (A) calculation and amounts to ca 40% from the experimental CF splitting. This analysis points to a major role played by metal–ligand covalency for the CF in lanthanides, a conclusion yet not widely recognized [99–102]. It also proves that the magnetic anisotropy of individual KDs of the complex cannot be treated adequately when the effect of electrostatic potential from the ligand is only taken into account [84]. In particular, the g-factors for the ground KD strongly disagree with experimental data when extracted from electrostatic CF calculations listed in Fig. 1.4c. Moreover, calculations based on electrostatic potential of true charge distribution (Fig. 1.4a) and FAIEMP predict easy-plane-type magnetic anisotropy of the ground-state g-tensor in major disagreement with experiment showing its pronounced easy-axis (axial) anisotropy. On the contrary, CASSCF /CASPT2 calculations based on active spaces of B and C types, fully including the effects of metal–ligand covalency and the Madelung potential, are in close agreement with experiment. In view of these results, as well as of CF analysis of other Ln complexes [84], the claim that the lowest CF multiplets in Ln complexes are dominated by electrostatic effects from the ligands [103, 104] appears ungrounded. One should stress that Er-trensal is a typical Ln3+ complex whose multiplets basically arise from a 4f11 configuration and are well described within both a minimal CAS of seven 4f orbitals (A) and a CAS of 14 orbitals, 4f11 + 5f0 (B), allowing to take into account the double-shell effect [74] which is the basic ab initio approach to lanthanides. Both the CF spectrum (Fig. 1.4b) and the g-factors of the ground KD are obtained close to experiment already for calculations with B active space when the CASPT2 step is performed. Compared to it, a similar calculation with 26 active orbitals, additionally including 5p6 + 5d0 + 6p0 orbitals as a restricted active space (C), does not improve the results, neither for CF spectrum (Fig. 1.4a) nor for the ground-state g-factors [84]. On the contrary, in the Ln2+ complexes, the additional electron may reside outside the 4f shell; hence, some of atomic orbitals which were empty in Ln3+ (5d, 6s, 6p) now must be included in the CAS. To deal with this more complicated situation, the multiplet structure of the simplest, diatomic Ln2+ complexes [LnO] has been investigated at a high calculational level including the RASPT2 step for the enlarged active space [105]. The calculations have shown that the main contribution to the low-lying terms in [LnO] comes from the electronic configuration 4fN 6s1 , where N = 8, 9 and 10 for Ln = Tb, Dy and Ho, respectively. The effects of spin–orbit coupling in [LnO] were considered within RASSI calculations. The resulting multiplet spectrum radically differs from trivalent lanthanide complexes. In Ln3+ compounds, the molecular multiplets originate from the atomic 2S+1 L J ones, merely corresponding to relatively small CF splitting of the latter. For instance, in [DyO]+ , the spectrum of lower multiplets arise from the atomic multiplets 6 H 15/2 and 6 H 13/2 , respectively (Fig. 1.6). In divalent

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L. F. Chibotaru

lanthanides, the multiplet spectrum differs significantly. The ground atomic multiplet 2S+1 L J arises from the Hund’s LS term in which the spin S corresponds to the addition of the spin 1/2 of one electron in the outer 6s orbital to the core spin S−1/2 of the Hund’s term of the 4fN shell. On the other hand, L is the same as in the Hund’s term of the configuration 4fN of Ln3+ due to the lack of orbital momentum for the added electron in the 6s orbital. Contrary to trivalent lanthanides, the next J-multiplet is not of LS type anymore but originates from a non-Hund LS−1 term in which the core spin S−1/2 of the 4fN shell couples antiparallel to the spin of the added 6s electron. The resulting multiplet is 2S−1 L J−1 , i.e., corresponding to a reduction of the ground-state J by unity. The second excited atomic multiplet corresponds to the first excited multiplet in trivalent lanthanides, 2S+1 L J−1 , i.e., derives from the ground LS term. The modified order of the lowest two multiplets compared to Ln3+ compounds can be attributed to relatively weak intra-atomic exchange interaction between 6s and 4f orbitals. The reason for the latter is a large radius of the 6s orbitals yielding a Hund splitting of the atomic terms of ca 2000 cm−1 [105], a much smaller value than the separation of spin–orbit J-multiplets. The calculated multiplet spectrum for [TbO] is shown in Fig. 1.5a where the levels are arranged according to the total momentum projection m on the anisotropy axis (the Tb–O axis), which is conserved due to a perfect axial symmetry of the complex. As one can see, in the ground doublet state, the magnetic moment increases by 1 μB in [LnO] compared to [LnO]+ compound containing a Ln3+ ion. Should the additional electron be accommodated not in a 6s but in a (quasi) degenerate 5d orbital, the rising of magnetic moment would be smaller due to an opposite orientation of unquenched orbital momentum to the spin of the electron required by spin–orbit coupling in the 5d shell. Such a reduced increase of magnetic moment was considered an evidence that the extra electron occupies a 5d orbital in divalent lanthanides [106, 107]. One should not forget, however, that the reduction of the overall magnetic moment can also occur due to partial CF quenching of the 4f orbital momenta in complexes [78, 108]; in 5d orbitals, they can be quenched completely by strong axial CF [109], in this sense, there would not be a difference for the extra electron to reside in the 6s or a 5d orbital. Finally, we should note the excellent agreement between the calculated three branches of multiplets in Fig. 1.5a with the positions and the magnetic moments of the low-lying multiplets of [TbO] extracted with laser excitation spectroscopy with selective fluorescence detection (Fig. 1.5b) [110]. Indeed, the three groups of levels at the bottom of spectroscopic diagram in Fig. 1.5b practically superimpose the calculated respective branches of multiplets (the levels with m = ± 1/2 in the lowest and with m = ± 1/2, ± 3/2 in the middle branch are absent in the diagram, presumably being not detected in these measurements). One may conclude that the employed ab initio approach is the method of choice for the accurate investigation of the multiplet spectrum and magnetism of lanthanide complexes. Its success steams from weak hybridization of the metal (4f) orbitals with the ligand orbitals, a situation well dealt with by post Hartree–Fock methods such as CASSCF. Compared to lanthanides, this hybridization is much stronger in actinide (5f) and TM (nd) complexes, which together with strong spin–orbit coupling makes the

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Fig. 1.5 a Multiplet spectrum and the magnetization blocking barrier of [TbO]. Each doublet state ± m arising from a given atomic multiplet 2S+1 L J is placed according to its saturation magnetic moment (horizontal axis). The intensity of the pink lines indicates the amplitude of the average transition magnetic dipole moment in μB between the connected states (the legend in the righthand side), the square of which roughly scales with the rate of spin–phonon transition between them [32]. The low-lying red lines outline the magnetization blocking barrier. Reprinted with the permission from Ref. [105]. Copyright 2020 John Wiley and Sons. b Electronic transitions (0–0 bands) of [TbO]obtained using laser excitation spectroscopy with selective fluorescence detection (see [110] for further details). The number at each horizontal solid line corresponds to the value of angular momentum projection m of the corresponding doublet state. Reprinted with the permission from [110]. Copyright 1984 Canadian Science Publishing

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L. F. Chibotaru

Fig. 1.6 Multiplet spectrum and the blocking barrier of [DyO]+ . Each doublet state ± M J arising from a given atomic multiplet 2S+1 L J is placed according to its saturation magnetic moment (horizontal axis). The intensity of the pink lines indicates the amplitude of the average transition magnetic dipole moment in μB between the connected states (legend for color and lines similar to Fig. 1.5a). Reprinted with the permission from [105]. Copyright 2020 John Wiley and Sons

accurate description of their magnetism within the same ab initio approach rather problematic. Thus, calculations of magnetic susceptibility of [NpIV (COT)2 ] and [OsIV Cl4 (κN1 -Hind)2 ], where Hind = 2H-indazole, show that it differs from the experiment by ca 20% [111, 112]. Even more dramatically looks the discrepancy for the calculated averaged orbital angular momentum ⟨l⟩ and the effective spin– orbit coupling λSO at the heavy transition-metal sites in various compounds. The extraction of these quantities for double perovskites Ba2 LiOsO6 , Ba2 NaOsO6 and Ba2 YMoO6 from fragment ab initio calculation looks exaggerated by ca 40% with respect to values obtained from experiment [113, 114], while being reasonably well reproduced by DFT calculations [115]. At the same time, in heavy-metal complexes with weaker metal–ligand covalency, the magnetic properties are described satisfactorily. Thus, the parameters of Zeeman interaction (g and G-factors) in the ground ⎡8 multiplet of Np4+ impurity substituting Zr4+ in one of the octahedral units [ZrCl6 ]2− of the cubic molecular crystal Cs2 ZrCl6 are in a fairly good agreement (including the signs) with experiment [116]. One may conclude that the current post Hartree–Fock calculation methodology is ill-defined for the treatment of magnetism in complexes exhibiting concomitantly strong covalency between the magnetic orbitals of the metal with the ligands’ orbitals and strong spin–orbit coupling on the metal site (5f, 4d and 5d metal complexes). A solution would be the straightforward use of spin– orbitals in the active space which, however, would increase twice its size making such multiconfigurational calculations nowadays impractical.

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1.3.4 Role of Axiality in Magnetization Blocking The axiality of spin–orbit multiplets of magnetic complexes characterizes the difference of magnetic properties along the main magnetic/anisotropy axes in these multiplets [117]. Strong axiality is achieved when the magnetism along one of these axes (principal axis Z) differs significantly from the other two (transversal axes X and Y ). Axial multiplets always appear as spin–orbit doublets. In complexes with odd numbers of electrons, these are KDs whose degeneracy in the absence of applied magnetic field is guaranteed by the Kramers theorem [22]. The axiality of a KD is measured by the ratio of the principal g-factor, gZ , to the transversal ones, gX and gY . The highest possible (perfect) axiality corresponds to vanishing values of the latter, gX = gY = 0. This situation is realized in the case of pure axial crystal field, c , where n = 2, 4, 6 in the case of lanthanides containing in Eq. (1.14) only terms ~ Bn0 and actinides and n = 2, 4 in the case of transition-metal complexes [22]. In this case, the KDs are characterized by definite projections M of the total angular momentum J on the axis Z (we take a lanthanide as an example). The corresponding wave functions are |J M⟩ and |J − M⟩, i.e., are described by equal projections |M| of opposite sign which is a requirement imposed by time reversal symmetry [22]. As explained in Sect. 1.2.2, the angular momentum eigenfunctions |J M1 ⟩ and |J M2 ⟩ with the difference |M1 − M2 | > 1 will have zero matrix elements of angular momentum projection operators, ⟨J M1 | Jˆα |J M2 ⟩ = 0. This means that the KDs |J ± M⟩ for M > 1 will have zero matrix elements of transverse components of the angular momentum resulting in gX = gY = 0, i.e., will be characterized by a perfect axiality. As an example, Fig. 1.6 shows the ab initio calculated [105] spectrum of low-lying KDs in the axial diatomic complex [DyO]+ . The crystal field in this complex stabilizes the KDs with large values [ ] of M, the main contribution in (1.14) coming from the c 2 ˆ term B20 3 J Z − J ( J + 1) , where J = 15/2. This term is equivalent to the axial ZFS term for S-complexes, Eq. (1.10), where S, Sˆ Z are to be replaced by J, JˆZ . However, c , in the present case, we have also non-negligible contributions from CF terms ~ B40 c B60 containing even powers of JˆZ up to fourth and sixth orders, respectively. In the case of complexes with even number of electrons, the multiplets in strongly axial CF form Ising doublets. An example of the latter is the CF eigenstates |2 ± 1⟩ and |2 ± 2⟩ in Fig. 1.1. A perfect axiality in these complexes corresponds to zero energy gap (tunneling gap) between the two components of the doublet. This is particularly the case of the IDs in Fig. 1.1. Strong axiality of the ground doublet is the necessary condition for the observation of magnetization blocking (SMM behavior) in a complex [32]. Indeed, the smallest rate of reversal of magnetization (magnetization relaxation rate) is achieved at low temperature, when only the ground doublet state is populated. In this case, the relaxation of magnetization takes place via the tunneling of magnetic moment into the opposite direction, M ↔ −M [3]. In KDs, the tunneling splitting (Δtun ) can arise only due to Zeeman interaction of transversal magnetic moments with external magnetic field (from surrounding nuclear spins and other magnetic complexes),

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( ) ⊥ = μB g X sˆ˜ X H X + gY sˆ˜Y HY , where sˆ˜ X and sˆ˜Y are operators of transversal Hˆ Zee components of pseudospin s˜ = 1/2 (2˜s + 1 = 2). Strong axiality is characterized by vanishing values of gX and gY , and therefore, Δtun ≈ 0, which leads to quenching of QTM in the ground doublet. At higher temperature, the spin–lattice relaxation via excited KDs becomes possible (see Sect. 1.2.2); however, this will not proceed via the tunneling of magnetization in the excited doublets if they are strongly axial too. In this case, the relaxation process will go via consecutive transition between |J M⟩ states with close values of magnetic moment projection M thus outlining an activation barrier of reversal of magnetization (Fig. 1.2). In the case of Ising doublets, realized in complexes with even number of electrons, Δtun is an intrinsic gap, always present if the symmetry is not high enough. High axiality of the complex means in this case disappearance of this tunneling gap, i.e., again the quenching of QTM. Diatomic compounds like [DyO]+ represent a limit for SMM performance of 3+ Ln -based complexes because their blocking barriers involve all states arising from the ground atomic J-multiplet of the corresponding ion [117], i.e., all eight low-lying KDs in the case of Dy3+ (Fig. 1.6). All performant single-Ln SMMs only approach this ideal limit to some extent. Thus, on individual Dy3+ sites in Dy5 and K2 Dy4 alkoxide cage complexes [118], as well in the NCN-pincer ligand dysprosium singleion complex [119], the relaxation goes via the second excited KD, in the pentagonal bipyramidal Dy(III)-containing complex [Dy(bbpen)X] [120] via the third excited KD, and in a hypothetical near-linear bis(amide) Dy(III) complex [121] via the fourth excited KD. Recently, synthesized dysprosium metallocene complexes, Cp–Dy–Cp, displaying the strongest blocking of magnetization among molecular complexes to date (ca 1200 cm−1 ), possess barriers corresponding to relaxation via fourth/fifth excited KD [122–125]. The much higher barrier obtained in [DyO]+ is due to a much shorter Dy–O bond (Fig. 1.6) compared to usual Dy3+ complexes and the accompanying more strong covalency effects to the CF splitting. Note that complexes of Ln2+ ions exhibit similar SMM performance as those of three-valent lanthanides; in particular, the magnetization blocking barrier of [DyO] [105] practically coincides with the one in [DyO]+ (Fig. 1.6). In view of potential applications, it is important that SMMs deposited on various surfaces retain their ability for magnetization blocking. To investigate this aspect, we considered a hexagonal boron–nitride (h-BN) substrate with a neutral [DyO] deposited on it. Recently, the magnetization blocking of Co ions adsorbed on such surface was investigated showing promising results [126]. The DFT-optimized structure of the DyO@h-BN complex is shown in Fig. 1.7a. The axis of adsorbed DyO is almost parallel to one of the B–N bonds, and the latter is being pushed above the plane of the substrate. The calculations predict an adsorption energy of 53.5 kcal/mol (2.3 eV). The calculated barrier for blocking of magnetization on the Dy ion is shown in Fig. 1.7b. The calculated height of this barrier is still larger than 2000 cm−1 , thus matching the lanthanide diatomics in the gas phase [105]. This is remarkable given a strong non-axial contribution present in the CF of Dy(II) ion due to the alignment of the DyO to the surface of DyO@h-BN. This finding is important because it proves via

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Fig. 1.7 a DFT-predicted structure of the deposited [DyO] on h-BN surface. b Calculated magnetization blocking barrier for the optimized structure (legend for color and lines similar to Fig. 1.5a). Reprinted with the permission from [105]. Copyright 2020 John Wiley and Sons

state-of-the-art calculations the existence of practical possibility to design magnetization blocking barriers which could overcome twice those in the best reported SMMs. Moreover, this is achievable with structurally robust diatomic units having a large adsorption energy to the surface. One should stress two implications of this finding. First, it allows to keep increasing the magnetization blocking barrier in future SMMs. Indeed, as the current studies show [122–125], the nowadays SMMs based on Cp–Dy–Cp units have approached the limit for the blocking barrier (ca 1200 cm−1 ) due to intrinsic difficulties in the synthesis of low-coordination lanthanide complexes. In such a situation, the [LnO] diataomics offer an alternate route due to their short Ln–O bond resulting in strong covalent contribution to the CF and ultimately to a very large axial CF component. Secondly, the SMMs’ applications in molecular electronics require their function on various surfaces. In this respect, [LnO] are again units of choice due to their structural and magnetic robustness upon surface deposition.

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1.4 Magnetization Blocking in Polynuclear Complexes In polynuclear compounds, the magnetic properties are determined by the common effect of ionic anisotropy on the metal sites and the anisotropic exchange interaction between them. The description of strongly anisotropic magnetic complexes can require a large number of parameters making their extraction from experiment a rather complicated task.

1.4.1 Complexes in Strong Exchange Limit In function of the relative strength of ionic anisotropy on the metal sites and of the exchange interactions, the polynuclear complexes belong to one of the two groups. The first one corresponds to the case of weak ZFS on the metal sites compared to the spread of exchange spectrum. This is the case of strong exchange limit [7], involving most of transition-metal complexes. As was discussed in Sect. 1.3, the situation of weak ZFS, Eqs. (1.1) and (1.9), is achieved in the case when the CF splitting of the lowest states exceeds considerably the strength of spin–orbit interaction. In this case, the exchange interaction between arbitrary metal sites is described by the generic exchange Hamiltonian (1.5). In this Hamiltonian, the isotropic exchange part is by far the main term, while the antisymmetric and the symmetric anisotropic contributions depend on the extent of spin–orbit admixture of excited terms on the metal sites [27, 127] and are, therefore, much smaller. In many complexes of this limit, the exchange multiplets arising from isotropic Heisenberg interaction (1.4) and characterized by the total spin S of the complex are separated by energy gaps exceeding the anisotropic contributions to the exchange interaction [last two terms in (1.5)] and the ZFS on the metal sites. Then, the spectrum of spin–orbit multiplets of such complexes merely corresponds to the splitting of isotropic exchange terms S by anisotropic contributions. This splitting is described in a good approximation by the ZFS Hamiltonian, Eq. (1.10), in which the spin projection operators refer to the ground exchange term S of the complex [1, 3]. An example of spectrum of multiplets resulting from the ZFS splitting of the ground exchange term is shown in Fig. 1.8 for the complex [Fe8 O2 (OH)12 (tacn)6 ]8+ , where tacn = 1,4,7-triazcyclononane [3, 128]. One of the distinctive features of complexes in strong exchange limit is an almost collinear arrangement of local magnetic moments on the metal sites in all multiplets. Figure 1.8a shows that the magnetic moments on the Fe(III) centers are all aligned along the main anisotropy axis of the ground S = 10 term of the Fe8 complex. This is the result of much stronger isotropic exchange interaction in this compound, stabilizing collinear arrangement of magnetic moments, compared to anisotropic contributions which might induce their non-collinear arrangement. Magnetization blocking in complexes in strong exchange limit is determined entirely by the character of ZFS splitting of its ground exchange term. Best SMMs are obtained in the case when only the axial component of ZFS Hamiltonian for

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Fig. 1.8 a Molecular structure of the [Fe8 O2 (OH)12 (tacn)6 ]8+ complex [3] and the ordering of local magnetic moments on the Fe(III) sites in the ground exchange doublet. b The structure of ZFS levels of the ground exchange term (S = 10) in function of their saturated magnetic moment along the main anisotropy axis of the complex (Z in Eq. (1.10))

the S term, D Sˆ Z2 in Eq. (1.10), is present, which is guaranteed if the polynuclear complex possesses an axial symmetry. This was precisely the case for the first discovered SMM, [Mn12 O12 (CH3 COO)16 (H2 O)4 ] (or Mn12 ac) [1–3], which possesses an S4 symmetry axis. A second important feature is the negative sign of axial ZFS constant (D < 0) which ensures the stabilization of a doublet state with maximal spin projection on the anisotropy axis (|10 ± 10⟩ in the case of Mn12 ac) [3]. Since the axial ZFS provides a high degree of axiality of the ground and the low-lying spin–orbit doublets of the complex, the reversal of magnetization will proceed via relaxation steps involving multiplets with close values of spin (momentum) projection, as described in Sect. 1.2.2. In the case of very strong axiality of all doublets, this chain of relaxation transitions can accede the state with minimal spin projection (M = 0 for integer S and M = ± 1/2 for half-integer S), which will be the highest in energy for D < 0. In this case, the reversal of magnetization between the states |SS⟩ and |S − S⟩ of the ground doublet will require overcoming a barrier of a height equal to the ZFS splitting of the term S. The amplitude of ZFS splitting of the term S depends on the way the individual ionic and exchange anistropic contributions are projected into the ZFS Hamiltonian of this term [7]. For example, in the case of ferromagnetic ground exchange term, the contribution of ZFS at individual metal ions to the negative D will be maximal when the local anisotropy parameters on the metal sites will be all negative (Di < 0), while the corresponding local anisotropy axes (Z i ) will be all parallel to the main anisotropy axis of the complex. In the case of Mn12 ac, such contribution from ionic anisotropy to D, as well as to fourth- and sixth-rank ZFS tensors (which are operative for S = 10, see Table 1.2), is expected to

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be predominant [129]. We note that the Ising doublets |10 ± M⟩ in Mn12 ac are not ideally axial (Sect. 1.3.4) even in the case of a perfect D2d symmetry of the complex. Then, the fourth- and sixth-rank ZFS contributions and the interaction with transverse magnetic field can lead to small Δtun in these doublets [3]. However, the opening of the tunneling gaps occurs via several perturbation steps, their number being proportional to M. Therefore, the large value of S is necessary to have suppressed QTM in the low-lying multiplets. The tunneling splitting in the multiplets |S ± M⟩ increases quickly with the diminishing M from its maximal value (S), leading sometimes to the shortcutting of the relaxation path at some intermediate value M 0 < M min , thus effectively reducing the height of the blocking barrier from its highest possible value [33]. This is because in higher-lying exchange doublets, corresponding to M < S, the opening of the tunneling gap arises via a lesser number of perturbation steps than in the ground exchange doublet (M = S). On the other hand, the arguments based on perturbation theory and the large ground-state spin explain why the lowlying doublets of Fe8 complex (S = 10) [3] remain axial despite a relatively strong rhombic contribution (|E/D| = 0.19) to the ZFS interaction (1.10) [128]. This is also confirmed by Fig. 1.8b showing an almost parabolic momentum dependence of the energy of low-lying multiplets, similarly to Mn12 ac [3], suggesting predominant effect from axial anisotropy for these levels.

1.4.2 Complexes in Weak Exchange Limit When the spin–orbit coupling effects on the metal sites become stronger, i.e., the separation of the low-lying local terms becomes of the order or smaller than their spin–orbit coupling, the magnetic interaction on the metal sites and the exchange interaction between them are described in terms of pseudospins (see Sect. 1.3.2). The reason why the total spin of the complex is not a good quantum number anymore is the strong mixing of several spin terms, which cannot be described via perturbation theory with respect to spin–orbit coupling. One of the consequences of this situation was a large number (compared to the case of weak spin–orbit coupling effects) of parameters of ZFS and Zeeman pseudospin Hamiltonians describing the magnetic interactions on individual magnetic centers, which a priori are of comparable order of magnitude. As the corresponding Eqs. (1.12) and (1.13) show, the number of independent parameters, equal to the number of allowed tensorial components, increases ˜ The situation becomes even more dramatic quickly with the size of pseudospin S. in the case of exchange interaction between two metal ions with strong spin–orbit coupling effects. Similar to magnetic pseudospin Hamiltonians of individual metal sites, the exchange interaction will include a priori all allowed tensorial forms of two pseudospin operators. The only constraints imposed on these forms are (i) the overall even tensorial ranks of all contributions to the exchange interaction, which is the consequence of invariance of the exchange Hamiltonian with respect to time inversion, and (ii) the order of polynomials of pseudospin operators ( Sˆ˜α ) should not

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˜ which is the consequence of a basic property of spin and pseudospin exceed 2 S, operators. Then, the general form (in the absence of symmetry) of exchange interaction between magnetic centers A and B, described by the pseudospins S˜A and S˜B , respectively, is the following: Hˆ ex =

≤2 S˜A Σ k

S˜B Σ k ≤2 l [ Σ Σ p=0

l

p q cc Jkp,lq Oˆ k (S˜ A ) Oˆ l (S˜ B )

q=0

sc ˆ kp (S˜ A ) Oˆ lq (S˜ B ) + Jkp,lq Ω ] ss ˆ lq (S˜ B ) , ˆ kp (S˜ A )Ω + Jkp,lq Ω

+

p cs ˆ lq (S˜ B ) Jkp,lq Oˆ k (S˜ A )Ω

(1.15)

where the ranks of the ITOs in each term can be simultaneously odd, k, l = 1, 3, …, or simultaneously even, k, l = 2, 4, … and Ω0k = Ωl0 = 0. We can see that the ab in this exchange Hamiltonian can amount number of independent parameters Jkp,lq to many tens, even in the case of pseudospins of moderate size. On the contrary, the exchange interaction between metal ions with weak spin–orbit coupling effects, Eq. (1.5), involves one isotropic (J), three antisymmetric (Dzyaloshinskii–Moriya vector d) and five symmetric anisotropic (Dαβ of zero trace) exchange parameters, i.e., nine independent parameters at most for any size of interacting spins. These contributions correspond to first-rank terms (k = l = 1) in Eq. (1.15). In the case of very strong spin–orbit coupling effects, the ZFS on the metal sites exceeds the exchange interaction between them. The corresponding complexes belong to the so-called weak exchange limit [7]. This is the case of almost all lanthanide complexes (except Gd3+ ) and also actinides and transition-metal complexes in special geometries allowing for orbital (quasi)degeneracy which leads to unquenched orbital momenta on the metal sites [78, 108]. Hence, the hierarchy of interactions is opposite to the case of complexes in strong exchange limit. Figure 1.9 shows the spectrum of low-lying exchange multiplets of a trinuclear Dy3 complex. The levels denoted by Dy1 –Dy3 in the left-hand side correspond to single-ion CF excitations on the corresponding Dy sites (Fig. 1.9a). The exchange interaction between metal centers “broadens” the CF levels into “bands” (right-hand side of Fig. 1.9b). However, the width of these exchange bands is much smaller than the separation between the CF levels. The latter usually amounts to several tens cm−1 , while the exchange splitting for a pair of two lanthanide ions is of the order of a few cm−1 . As a result, the spectrum of levels arising from exchange interaction of ground doublets on Ln sites is well separated from the group of exchange multiplets corresponding to excited local doublets (Fig. 1.9b). Then, the low-lying exchange spectrum can be described as exchange interaction between ground doublets on Ln ions, i.e., between corresponding pseudospins s˜ = 1/2. According to general rules described above, only terms of first rank will be retained in the corresponding exchange Hamiltonian (1.15), i.e., will have the generic form (1.5) in which Sˆ 1 and Sˆ 2 are to be replaced by sˆ˜ 1 and sˆ˜ 2 , respectively. As was discussed earlier in Sect. 1.3.4 [69, 117], the ground doublets on Ln sites in complexes are usually strongly axial. This allows to simplify

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L. F. Chibotaru

a)

b)

Fig. 1.9 a Molecular structure of a Dy3 complex (Murugesu et al., unpublished) and the arrangement of local magnetic moments on the Dy(III) sites along the corresponding principal magnetic axes (dashed lines) in the ground exchange KD. b The ab initio calculated energies of low-lying crystal-field multiplets (left) and their splitting by the exchange and dipolar interaction (right)

the description of interaction in these systems further. Given the high interest for lanthanide complexes, below we discuss in detail this exchange interaction and its effect on magnetization blocking.

1.4.3 Pure Lanthanide Complexes The high axiality on lanthanide sites in combination with the weak exchange limit in the corresponding complexes leads to a situation where, at least in the ground and lowlying exchange multiplets, the local magnetic moments on Ln sites will be directed along the corresponding principal magnetic axes. The latter is defined as having the largest g-factor among the three main magnetic axes of the corresponding doublet on the metal site, denoted by gZ in Sect. 1.3.4. Figure 1.9a shows the arrangement of local magnetic moments in the ground exchange KD of a Dy3 complex. We can see a strong non-collinearity of magnetic moments on Dy sites, which is a fingerprint of complexes in weak exchange limit. Within the Lines model, the exchange interaction between strong axial doublet states of two Ln ions, described by pseudospins s˜1 = s˜2 = 1/2, reduces to Ising exchange interaction between the projections of pseudspins on the corresponding principal magnetic axes, z1 and z2 of the two doublets (Fig. 1.10a). When these axes are not parallel to each other, the exchange is of non-collinear Ising type, which is the case in most polynuclear lanthanide complexes. This conclusion is supported by straightforward derivation of the exchange interaction between doublets of ideal axiality (|J, ±M⟩) straightforwardly on the basis of microscopic models of electronic structure of interacting metal pairs [130]. Therefore, if the directions of

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

a)

Fig. 1.10 Exchange interaction between two strongly axial doublets S˜1 and S˜2 (a) and between an axial doublet S˜1 and an isotropic spin S 2 (b). Dashed lines show principal magnetic axes (zi ) of the corresponding axial doublets i

33

b)

principal magnetic axes are known, either from ab initio calculations [69, 131] or from magnetic measurements [16, 17], then the simulation of exchange interaction between the corresponding strongly axial Ln ions will require knowledge of one single exchange parameter (J ex in Fig. 1.10a), i.e., will be as simple as the Heisenberg exchange interaction for two spins (1.4). This is a great simplification for the theory which allowed to rationalize a large number of polynuclear lanthanide complexes [9, 69, 118]. The exchange interaction between lanthanide doublets amounts to few wavenumbers in most cases. At the same time, the magnetic moments in the ground doublet states at Ln ions can reach values up to 10 μB . In this situation, the dipolar interaction between local magnetic moments on Ln sites is of the order of or larger than the net exchange interaction and should be taken into account along with the latter in the description of magnetic properties of lanthanide complexes. The dipolar interaction has the form: )( ) ( ˆ 2 · n12 ˆ2 −3 µ ˆ 1 · n12 µ ˆ1 ·µ µ , (1.16) Hˆ dip = − 3 r12 where r 12 is the distance between the magnetic moments and n12 is the unit vector along the axis connecting them. Substituting for magnetic moments their expressions in terms of pseudospins (˜si = 1/2): µi = −μB g (iZ i) sˆ˜i zi e Z i , i = 1, 2, we obtain again an expression for non-collinear Ising interaction of pseudospins (Fig. 1.10a) with the coefficient which should be identified as the parameter of dipolar magnetic interaction (J dip ). Then, the total magnetic interaction between two strongly axial Ln doublets reduces to ) ( Hˆ int = − Jex + Jdip sˆ˜1z1 sˆ˜2z2 ,

(1.17)

with (2) Jdip = μ2B g (1) Z1 gZ2

cos θ12 − 3 cos θ1n cos θ2n , 3 r12

(1.18)

34

L. F. Chibotaru

where θ 12 is the angle between the principal anisotropy axes on the two Ln sites and θ in , i = 1, 2, is the angle between the principal magnetic axis on the center i and the vector n12 connecting the two Ln ions. Note that knowledge of principal magnetic axes (e Z i ) and principal values of the g-tensors (g iZ i ), e.g., from ab initio calculations of monolanthanide fragments, allows to calculate the parameter J dip in Eq. (1.18) from the first principles. Thus, the only quantity which requires extraction from other data is the exchange parameter J ex . The latter is expressed via the Lines exchange parameter (see Sect. 1.2.4) which at its turn is either extracted from the fitting of the experimental temperature-dependent magnetic susceptibility and field-dependent magnetization of the complex [9, 63, 105, 132] or by means of broken-symmetry (BS) DFT calculations [133]. The derivation of the Lines exchange parameter in the latter case is done in two steps. First, the Heisenberg exchange parameter of the isostructural Gd compound (obtained by substitution of all Ln ions by Gd in the original complex) is calculated by applying a known BS-DFT methodology [134]. Its value is next rescaled by the coefficient (S Gd /S Ln1 ) * (S Gd /S Ln2 ), where S Ln1 and S Ln2 are the spins of the ground atomic multiplets at lanthanide ions Ln1 and Ln2 involved in the exchange interaction. This corresponds to the Lines exchange parameter for the Ln1 –Ln2 pair [133]. It can be subsequently slightly modified to simulate better the experimental magnetic data, in which case the calculated values represent a reliable starting point for the fitting. Such BS-DFT estimations of the Lines exchange parameters are especially useful for polynuclear complexes containing many pairs of exchange-coupled magnetic centers, notably in mixed Ln–TM complexes (see the next section). Consider as an example an asymmetric binuclear Dy2 complex, [Dy2 ovph2 Cl2 (MeOH)3 ]·MeCN, where H2 ovph = pyridine-2-carbolxylic acid [(2-hydroxy-3-methoxyphenyl)methylene] hydrazide, which is one of the first reported binuclear lanthanide SMMs [135]. Figure 1.11a shows a different environment for two Dy ions in this complex, which leads to different spectra of local KDs at the corresponding metal sites. In particular, the ab initio calculations give different orientations of principal magnetic axes on two Dy sites (red dashed lines in Fig. 1.11a), although both are being directed closely to the Dy–Dy bond. Then, according to Eq. (1.18), the dipolar interaction between these KDs is ferromagnetic (J dip > 0). Ab initio calculation finds very strong axiality of ground-state KDs on both Dy sites implying the applicability of non-collinear Ising model (1.17) for the description of their exchange interaction. From ab initio results, we obtain J dip = 5.36 cm−1 , while simulation of magnetic susceptibility (Fig. 1.11b) gave for the exchange contribution in (1.17) J ex = 0.52 cm−1 . One consequence of the almost ferromagnetic arrangement of two Dy magnetic moments in the ground exchange doublet (coupled into a total magnetic moment of ca 20 μB ) is the very strong dipolar magnetic interaction between neighbor complexes, implying an unusually large effective exchange parameter between ground-state exchange doublets on neighbor complexes (zJ in Fig. 1.11b). The corresponding stray field H bias is calculated as 0.11 T and coincides with the measured one [135]. The almost net Ising exchange interaction between Dy sites, together with strong axiality of their KDs, is the reason

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35

Fig. 1.11 a Molecular structure of [Dy2 ovph2 Cl2 (MeOH)3 ]·MeCN. Color legend: Dy purple, C gray, N blue, O red, Cl green. Dashed lines show principal magnetic axes on Dy sites and arrows show the local magnetic moment in the ground exchange doublet. b Molar magnetic susceptibility multiplied by temperature versus T for the Dy2 complex. The inset is a plot of the reduced magnetization versus H/T. The solid lines are guides for the eye. Reprinted with the permission from [135]. Copyright 2011 American Chemical Society

for a very strong axiality of the resulting two exchange Ising doublets, displaying a negligible tunneling splitting Δ. The latter is the reason for the quenching of QTM in this complex which leads to the observed hysteresis loops of magnetization and long relaxation times at low temperatures [135]. The reversal of magnetization in the low-temperature domain goes via over-barrier relaxation mechanism. At temperatures considerably exceeding the separation between two lowest exchange doublets (≈ 3 cm−1 ), the magnetic relaxation occurs via excited KDs on individual Dy sites. The fact that two relaxation times are observed in this temperature region [135] is an additional evidence for non-equivalent Dy sites having, in particular, different energies of the first excited KDs and, therefore, different activation energies for on-site reversal of magnetization. Another example where the ab initio description was highly successful is the detection for the first time of a molecular toroidal magnetic moment in dysprosium triangles [9], one of the three structures [Dy3 (µ3 -OH)2 L3 Cl(H2 O)5 ]Cl3 ·4H2 O shown in Fig. 1.12a [136]. The dashed lines in the figure show calculated local main magnetic axes in the strongly axial ground KDs at three Dy sites, forming the sides of an almost perfect equilateral triangle. Then, the dipolar contribution (1.18) to the interaction of local KDs (1.17) and the negative sign of the Lines exchange parameters (resulting in J ex > 0 [9]) leads to an almost toroidal arrangement of the magnetic moments at Dy sites (arrows in Fig. 1.12). They sum up into a total magnetic moment < 0.5 μB (calculated local magnetic moments on Dy sites are ≈ 10 μB ), in full accord with the experimental slope of M(H) and χ T at low temperature [136]. An additional evidence for the accurate prediction by ab initio calculations of the directions of the local main magnetic (anisotropy) axes at the metal sites comes from their comparison with the experimentally extracted ones. Figure 1.12b shows that the calculations predict small

36

L. F. Chibotaru

deviations of the directions of the local magnetic moments at Dy sites from perfect tangential directions (φ i ) amounting to 8.7–9.5°. These quantities have been also extracted from measured single-crystal magnetic susceptibility via its simulations with a symmetric non-collinear Ising model (one common φ for all three Dy sites) [137]. These simulations gave, however, six different arrangements of local magnetic moments, corresponding to six different values of φ, all being compatible with the measured susceptibility [137]. One of these fitted values corresponds to φ = 13° which compares very well with ab initio values for φ i . We may conclude, therefore, that even sophisticated magnetic measurements are not sufficient to decide over the arrangement of local magnetic moments in non-collinear magnetic structures, and the ab initio analysis is being indispensable for their assessment. Another conclusion is that the employed ab initio methodology represents a reliable tool for the accurate determination of main magnetic (anisotropy) axes at the metal sites in complexes.

1.4.4 Mixed Ln–TM Complexes In the case of mixed Ln–TM complexes, according to the Lines model, the high axiality (if present) of the ground-state doublet of the lanthanide ion will lead again to the Ising exchange interaction. However, contrary to the previous case, this interaction involves now the projection of Ln pseudospin s˜1 = 1/2 on the corresponding principal magnetic axis (Z 1 ) and the projection of the true spin S 2 of the transitionmetal ion on the same axis Z 1 (Fig. 1.10b), i.e., it is of collinear Ising type. An analysis based on Anderson’s superexchange model [138] extended to treat interacting anisotropic metal pairs [139] shows that this is indeed the form of exchange interaction when the strongly axial Ln doublet is close to have a maximal projection of the total moment J on the principal magnetic axis, i.e., to be ≈ |J, ±J ⟩. This is, in particular, the situation in most lanthanide ions with low site symmetry in the complex. For doublets characterized by lower projections of the total moment, e.g., ≈ |J, ±( J − 1)⟩, their exchange interaction with TMs involves also transversal components of S which are, however, relatively small [139] and can be treated within a non-collinear Ising model [130]. Another difference from the previous case is that the magnetic dipolar interaction between a strongly axial Ln doublet and an isotropic TM spin is not reduced to an Ising form as in Eq. (1.17). Indeed, substituting in Eq. (1.16) the expression for µ1 corresponding to a strongly axial Ln doublet (Sect. 1.4.3) and taking µ2 = −μB g (2) S2 , we obtain the following contribution: ( ) Hˆ dip = −Jdip s˜ˆ1z1 Sˆ2z1 − 3 cos θ1n sˆ˜1z1 Sˆ2n ,

(1.19)

(2) 3 Jdip = μ2B g (1) Z 1 g /r 12 ,

(1.20)

with

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

37

Fig. 1.12 a Molecular structure of [Dy3 (µ3 -OH)2 L3 Cl(H2 O)5 ]Cl3 ·4H2 O [136], containing two chlorides coordinated to Dy(3) sites, which was used in ab initio calculations. Color scheme: blue, Dy; red, O; green, Cl; dark gray, C; white, H. Dashed lines show the calculated anisotropy axes on the dysprosium fragments and the arrows show the ordering of local magnetizations in the ground state of the complex. b The two components of the ground KD of the complex. The angle φ i measures the deviation of the corresponding local anisotropy axes from the tangential direction. Reproduced with the permission from [9]. Copyright 2008 John Wiley and Sons

where S 2n is the projection of the TM spin on the axis n12 connecting the two metal ions and the angle θ 1n is the same as in Eq. (1.18). Compared to Ln–Ln pairs, the Ln–TM exchange interaction is usually one order of magnitude larger due to more diffuse 3d orbitals of transition metals compared to 4f orbitals of lanthanides. Then, the description of the Ln–TM exchange interaction involving one single (ground) doublet on Ln site is only justified when the resulting exchange splitting is significantly smaller than the energy of the first excited doublet at the lanthanide ion. If this condition is not fulfilled, more doublets on the Ln sites should be involved in the description of exchange interaction, which taken together will correspond to larger pseudospins [23] and will lead to a more complex exchange interaction according to Eq. (1.15). On the contrary, the dipolar magnetic interaction within Ln–TM pairs is expected to be significantly smaller than in Ln–Ln pairs due

38

L. F. Chibotaru

to a smaller magnetic dipole moments of TM ions compared to lanthanide ions. Then, the dominant collinear Ising Ln–TM interaction (Fig. 1.10b) will impose a collinear alignment of all local magnetic moments to the principal magnetic axes of Ln ions if the latter is parallel to each other. At the same time, the effect of other contributions to the exchange and dipolar magnetic interactions, favoring a noncollinear arrangement of local magnetic moments, will be effectively suppressed by the collinear Ising contribution. An example of such situation is the tetranuclear complex [CrIII 2 DyIII 2 (OMe)2 (O2 CPh)4 (mdea)2 (NO3 )2 ] shown in Fig. 1.13a [133]. This complex has an inversion center which connects the pairs of dysprosium and chromium atoms. Ab initio calculations have shown that the ground KD at Dy sites is very axial. The principal magnetic axes (Z) on the two Dy ions, indicated in Fig. 1.13a by dashed lines, are parallel to each other because of the inversion symmetry of the complex. The Lines exchange parameters have been calculated by BS-DFT method as described in the previous section. The only difference is that now the rescaling coefficient contains one single factor, S Gd /S Ln , corresponding to one single lanthanide ion in the exchangecoupled pair Ln-S. The calculated exchange parameters by broken-symmetry DFT methods are shown in Table 1.3 (they refer to s˜Dy = 1/2, S Cr = 3/2, see Ref. [133] for the details). We can see that for Cr-Dy pairs, J ex is several times larger than J dip , calculated with Eq. (1.20) on the basis of ab initio results. Varying the values of exchange parameters in order to fit the experimental temperature-dependent susceptibility (Fig. 1.13b) does not modify them drastically, as can be seen in Table 1.3. The large value of obtained J ex is the reason for an almost collinear arrangements of magnetic moments on the metal sites in the ground (Fig. 1.17a) and the lowlying exchange doublets. A similar behavior has been found in a series of Ln–TM complexes with almost isotropic TM sites [140]. On this reason, the general Hamiltonian of magnetic interaction between low-lying states on the metal centers in the CrIII 2 DyIII 2 complex Hˆ =

4 ( 4 Σ Σ

ij ij Hˆ ex + Hˆ dip

) (1.21)

i=1 j>i

reduces to a collinear Ising one ( ) Hˆ = −JDy1−Dy1, sˆ˜Dy1,z sˆ˜Dy1, ,z − JDy1−Cr1 sˆ˜Dy1,z SˆCr1,z + sˆ˜Dy1, ,z SˆCr1, ,z ( ) − JCr1−Cr1, SˆCr1,z SˆCr1, ,z − JDy1−Cr1, sˆ˜Dy1,z SˆCr1, ,z + sˆ˜Dy1, ,z SˆCr1,z

(1.22)

with the parameters which include the effect of exchange and dipolar magnetic interaction between the four metal ions (last column in Table 1.3). The resulting spectrum of low-lying exchange doublets is shown in Fig. 1.14 together with the structure of blocking barrier for magnetization. We can see that the high axiality of low-lying exchange doublets, blocking the QTM relaxation in these excited states, is the reason for relatively high value of blocking barrier, amounting to ca 50 cm−1 . This

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

39

Fig. 1.13 a Molecular structure of [CrIII 2 DyIII 2 (OMe)2 (O2 CPh)4 (mdea)2 (NO3 )2 ]. Color legend: Dy purple, Cr yellow, C gray, N blue, O red. Dashed lines show principal magnetic axes on Dy sites and arrows show the local magnetic moment in the ground exchange doublet. b Molar magnetic susceptibility multiplied by temperature versus T: experiment (dashed line) and theory (solid line). Modified with the permission from [133]. Copyright 2013 John Wiley and Sons Table 1.3 Magnetic coupling parameters (cm−1 ) for different metal pairs of Cr2 Dy2 complex (Fig. 1.13) Metal pairs M1 –M2

Calculated

Fitted

J dip

J ex (BS-DFT)

J ex

J M1–M2 in Eq. (1.22)

Dy1–Dy1,

2.5

1.00

1.00

− 1.5

Cr1–Cr1,

0.34

0.12

0.10

0.12

Dy1–Cr1

5.2

− 26.0

− 20.5

− 20.3

Dy1–Cr1,

5.2

− 32.5

− 17.0

− 16.7

Modified with the permission from [133]. Copyright 2013 John Wiley and Sons

value matches nicely the activation energy for relaxation of magnetization extracted from ac susceptibility [133]. We would like to emphasize that the barrier shown in Fig. 1.14 is of exchange type since it is built on exchange multiplets originating from ground-state KDs on Dy sites. The many states involved in the relaxation path make this barrier opaque at low temperatures (significantly lower than the height of the barrier), which explains the very good blocking properties of the complex [133]. On the other hand, the activation energy describing the intra-ionic relaxation of magnetization, which becomes operative at higher temperatures, corresponds to the second excited KD on each Dy site which lies much higher in energy [133]. If the Ln ions in a mixed Ln–TM complex are characterized by non-collinear principal magnetic axes, then the arrangement of magnetic moments on the metal sites will also be non-collinear. This is because the isotropic spin of each TM site tends to align as parallel as possible to the principal magnetic axes of neighbor Ln ions and often ends up by choosing an intermediate orientation. For example, in the octanuclear CrIII 4 DyIII 4 complex of wheel type having an approximate fourfold symmetry axis [141], the principal magnetic axes on Dy sites lie in mutually orthogonal planes,

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L. F. Chibotaru

Fig. 1.14 Structure of magnetization blocking barriers in the Cr2 Dy2 complex (Fig. 1.13). The thick black lines represent the exchange levels as a function of their magnetic moment along the principal magnetic axis of the complex. The external red arrows connecting the neighbor levels outline the relaxation path. The numbers at red and green arrows are transition magnetic moment matric elements (in μB ) between the corresponding states; Δtun at each blue arrow is the intrinsic tunneling gap of the corresponding exchange doublet. The horizontal dashed line approximately corresponds to the height of the calculated blocking barrier. Reprinted with the permission from [133]. Copyright 2013 John Wiley and Sons

while the spin on each Cr lies in the bisecting plane between two neighbor dysprosium ions. Among the mixed complexes, the Ln–R compounds where R is a radical are viewed nowadays as the most perspective ones for the design of efficient SMMs. The main advantage of these complexes is their ability to exhibit a very strong Ln– R exchange interaction, which can overcome the Ln–TM exchange by an order of magnitude. This has been recently demonstrated on the example of a N2 3− radicalbridged lanthanide complexes [142], which have shown a record for the height of blocking barrier of exchange type and a highest to date blocking temperature among polynuclear SMMs [143]. Finally, one notices that SMM complexes combining strongly anisotropic and isotropic magnetic sites should not necessarily include lanthanide ions. As an example of a “non-lanthanide” mixed complex, Fig. 1.15a shows a trinuclear compound from the series [LCoII LnCoII L]+ [144], where L = N,N, ,N,, -tris(2-hydroxy-3-methoxybenzilidene)-2-(aminomethyl)-2-methyl-1,3propanediamine, for Ln = Gd. In this series, owing to the structure of L, the complexes possess an almost ideal trigonal axis connecting the three metal ions. The trigonal symmetry of the complex gives rise to a twofold orbital degeneracy of the two Co(II)

1 Ab Initio Investigation of Anisotropic Magnetism and Magnetization …

41

centers allowing for unquenched projection of orbital momentum, L z ≈ ± 2, on the common symmetry axis (z) [145]. Spin–orbit coupling between L z and the S z projection of the ground spin S = 3/2 on cobalt leads to four equidistant KDs on each Co site, separated by ≈ 2ζ /3, where ζ is the spin–orbit coupling constant of Co(II) ion. The ab initio calculation has confirmed this structure of low-lying KDs on the Co sites, giving for the energy separation between them an average value of 280 cm−1 [145]. This allows to describe the magnetic properties of the complex in terms of exchange interaction between the ground KDs on each Co center (˜si = 1/2, i = 1, 2) and the isotropic spin S = 7/2 of the Gd ion. The important feature is the obtained relatively strong axiality (gZ ≈ 9.3, gX , gY ≈ 0.3) of the ground KD on Co(II) sites. This is the result of predominant contribution of the projections of the total angular momentum M J = 7/2 and M J = − 7/2, respectively, to the two wave functions of the KD. The deviation from a perfect axiality (see Sect. 1.3.4) is due to the fact that other M J components can admix in trigonal symmetry. Nevertheless, the axiality is sufficiently high (gX,Y /gZ = 0.02–0.04) in order to make the exchange interaction between Co and Gd mainly of collinear Ising type (Fig. 1.10b). This is confirmed by the spectrum of exchange multiplets (Fig. 1.15b) calculated with the interaction Hamiltonian involving exchange parameters extracted from the fitting of magnetic data (in analogy with Cr2 Dy2 complex discussed above). We can see that the lowest four exchange multiplets are equidistant and correspond consecutively to the spin projections M = 7/2, 5/2, 3/2 and 1/2 on Gd site, respectively, under unchanged directions of local magnetization on Co sites. This gradual decrease of M with increasing the energy of the exchange doublet resembles much the structure of blocking barrier in the conventional SMMs for complexes in the strong exchange limit, like Mn12 ac.1,3 The main difference is that in the later the splitting of the ground spin manifold follows a parabolic dependence, E M = |D|M 2 , while in the present case, it is almost equidistant. The analysis of the structure of the blocking barrier (Fig. 1.16) shows that its top corresponds to ca 12 cm−1 [145], which is in good agreement with the value extracted from experiment [144]. The high SMM performance of this complex [144] is explained by a multilevel structure of the exchange barrier, as was also the case for the Cr2 Dy2 complex considered above.

1.5 Further Developments and Perspectives As emphasized in this review, the relevance of ab initio approach for the investigation of anisotropic magnetic complexes stems from two major factors, (i) the complexity of anisotropic magnetic properties requiring knowledge of many parameters for their description and (ii) the high performance of state-of-the-art ab initio methods allowing for the description of multiplet structure of mononuclear complexes and fragments with (sometimes) experimental accuracy. Two new domains where the ab initio methodology is expected to be equally successful are the first-principles description of anisotropic exchange interaction and of magnetization dynamics which are shortly outlined below.

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Fig. 1.15 a Molecular structure of [LCoGdCoL]+ [144]. Color legend: Gd yellow, Co green, C gray, N blue, O red. Dashed lines show principal magnetic axes on Co sites and arrows show the local magnetic moments in the ground exchange doublets. b Spectrum of exchange levels originating from the ground KDs on Co(II) sites (left) and the value/orientation of local magnetic moments on metal sites in the correspond states. Modified with the permission from [145]. Copyright 2013 American Chemical Society

Fig. 1.16 Magnetization blocking barrier in [LCoGdCoL]+ . The exchange states are arranged according to the values of their magnetic moments. The external red arrows connecting the neighbor levels outline the relaxation path. The meaning of the internal arrows is the same as in Fig. 1.14 except for the numbers at horizontal arrows denoting now transition magnetic moment matrix elements between the two components of the corresponding exchange KD. Reprinted with the permission from [145]. Copyright 2013 American Chemical Society

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Fig. 1.17 Plots of the exchange parameters (J fd )kp,lq between the ⎡ 8 multiplets of Nd3+ (in meV) for the nearest (a) and the next nearest (b) neighbors in NdN crystal. J covers all four groups of parameters with the corresponding indices in Eq. (1.15). The red and green show the ferromagnetic and antiferromagnetic contributions. Reprinted with the permission from [152]. Copyright 2022 American Physical Society

1.5.1 First-Principles Calculation of Anisotropic Exchange Interaction The Lines model is proved to be highly efficient for the treatment of exchange interaction in polynuclear complexes involving strongly axial magnetic ions (Sects. 1.4.3 and 1.4.4). Its applicability to most polynuclear SMMs with strongly anisotropic magnetic ions (lanthanides) is a blessing for the rationalization of their magnetization blocking behavior. However, this model appears oversimplified for the treatment of complexes involving metal sites with either weak spin–orbit coupling effects (Sect. 1.4.1) or even strong ones (e.g., resulting in unquenched orbital momentum) but exhibiting weak/moderate axiality in the ground CF multiplet or group of CF multiplets. In the latter situation, the exchange interaction is described by the full

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exchange Hamiltonian (1.15), while it involves only its first-rank terms (k, l = 1), reducing to Eq. (1.5), in the case of weak spin–orbit coupling effects on the metal sites. It is easily seen from Eq. (1.15) that the number of exchange parameters in such Hamiltonian amounts several dozens when at least one of the pseudospins exceeds the minimal value S˜ = 1/2. Obviously, these parameters cannot be extracted from any magnetic or INS measurements, so their derivation from ab initio calculations becomes indispensable. Current ab initio methods are not able to describe accurately the exchange interaction in complexes of experimental interest. This was the reason for the development of the semi-ab initio approach based on the Lines approximation. A decent derivation of general exchange Hamiltonian (1.15) requires knowledge of detailed electronic interaction between the multiplets of different magnetic centers involved in the exchange interaction. Given the localized character of on-site multiplets in complexes and magnetic insulators, the parameters of the exchange Hamiltonian are straightforwardly derived as perturbative expressions with respect to intercenter electronic interaction in the spirit of Anderson’s theory of superexchange [138]. For f metals, such expressions have been derived for a Hubbard model [146] and later for a complete electronic Hamiltonian for magnetic electrons [147]. Although the microscopic exchange theory for the f metals looks more involved than for the d metals due to a larger number of involved magnetic orbitals, its first-principles application has two important advantages compared to the latter. First, the group of molecular orbitals of f character (bands in the case of magnetic insulators) are well separated, so that the derivation of the parameters of intercenter one-electron interaction (the transfer parameters) can be done via straightforward application of Wannier transformation to these bands. Second, due to a very strong localization of magnetic f orbitals at the lanthanide or actinide sites, the exchange interaction between them is adequately described within the second order of the perturbation theory with respect to intercenter electron transfer. On the contrary, in transitionmetal compounds, usually there are no pronounced molecular orbitals (bands) of d type, so that the values of the derived parameters of the intercenter transfer Hamiltonian will depend on the number of molecular orbitals/bands involved in the Wannier transformation [148]. Also, the second-order perturbation theory with respect to intercenter electron transfer proved to be insufficient for a decent description of exchange interaction in these systems, sometimes even qualitatively [148]. The microscopic theory of Ref. [147] has been applied to first-principles description of exchange interaction and magnetization blocking in N2 3− radicalbridged lanthanide complexes [149], synthesized and investigated earlier [142, 143]. The giant exchange interaction in these radical-bridged binuclear compounds is contributed almost exclusively by the antiferromagnetic kinetic exchange interaction in each of the two Ln–N2 3− pairs, which is adequately described within a theoretical frame considering explicitly only the magnetic orbitals on the two sites [147]. The multiplet structure of Ln3+ centers has been calculated ab initio as described in Sect. 1.2.4. Besides, also the multiplet structure of reduced metal ions (Ln2+ ) has been calculated (for unchanged geometry) in order to assess the amplitude of kinetic delocalization of radical electron over the Ln3+ center [149]. The only free parameter

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in this treatment is the electron promotion energy N2 3− → Ln3+ into the ground state of Ln2+ center, which is extracted from the fitting of temperature-dependent magnetic susceptibility for each complex in this series. The extracted Ln–N2 3− exchange interaction exhibits strong anisotropy, with non-negligible contributions in Eq. (1.15) up to k, l = 7. Besides reproducing the magnetism, the calculated exchange spectrum enabled the understanding of magnetic relaxation in these complexes as proceeding via the first excited exchange doublet whose height agrees well with experimentally extracted blocking barrier for all investigated compounds, with Ln = Tb, Dy, Ho, Er [142, 143]. A key finding was that the very strong exchange interaction in these complexes intermixes efficiently the CF states on Ln sites, so that the axiality of the ground and excited exchange doublets is diminished dramatically and the blocking barriers do not exceed the energy of the first excited exchange KD. By quenching the exchange admixture of excited CF doublets to the ground ones, the activated relaxation proceeds via higher exchange doublet, thereby doubling the height of the blocking barriers [149]. This prediction was confirmed by subsequent synthesis of Tb–N2 3− –Tb complexes with more axial ligands surrounding the terbium sites compared to the previous complex [142] which indeed exhibited the doubling of the blocking barrier [150]. In many lanthanide materials, however, different exchange contributions are expected to be of comparable in strengths and should, therefore, be taken into account on equal footing with the kinetic exchange interaction between the magnetic electrons. For lanthanides (and possibly actinides), the most important contribution to be added is the kinetic delocalization of f electrons over empty atomic orbitals centered on neighbor magnetic sites, i.e., the Goodenough’s exchange mechanism [151]. To this end, the microscopic theory in Ref. [147] has been extended to include the microscopic description of Goodenough’s mechanism [152]. The first-principles implementation of this theory requires knowledge of much more local multiplets on Ln ions and electron transfer parameters due to the additional orbitals of 5d, 6p and other types at the Ln center involved in the description, which are extracted from ab initio and DFT calculations as previously [147]. This approach was applied to the analysis of exchange interaction and magnetic properties in both ordered and paramagnetic phases of neodymium nitride NdN, a member of a vast family of lanthanide nitrides exhibiting ferromagnetism with high critical (Curie) temperature of a few tens K [153]. These materials have a rock salt structure with each Ln3+ surrounded by a perfect octahedron of six N3− . In such an environment, the ground atomic multiplet J = 9/2 of Nd3+ splits as 2⎡ 8 + ⎡ 6 , the ground ⎡ 8 quadruplet being stabilized by ca 150 cm−1 . The latter corresponds to the pseudospin S˜ = 3/2; therefore, the exchange interaction in this compounds is described by terms with the rank 1 ≤ k, l ≤ 3 in Eq. (1.15). With the derived microscopic parameters, the kinetic superexchange and the magnetic dipolar interaction between 4f electrons was found to be less important than the Goodenough mechanism involving a 4f–5d electron delocalization between neighbor Nd sites. The only free parameter in its description was the electron promotion energy, whose reasonable value was obtained from the fitting of Curie temperature. Because the Nd nodes form a face-centered cubic (fcc) lattice, one should consider the exchange interaction in two types of Nd–Nd pairs,

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the nearest neighbor pairs, e.g., along (110) direction, and the next-nearest-neighbor one, e.g., along (100) direction. Figure 1.17 shows the magnitude of the calculated exchange parameters. The largest ones correspond to the highest rank 3, whereas parameters of other ranks (k, l = 1, 2) are non-negligible too, a reason why the exchange interaction is qualified as multipolar. With this interaction and extracted CF operator at Nd sites, the magnetic order of NdN has been further investigated by mean-field approach in combination with spin-wave theory [154]. The most stable magnetic order was found to be the ferromagnetic one with all magnetic moments on each site aligned along one of the crystal axes, e.g., c, in full agreement with the neutron diffraction data [155]. The obtained ferromagnetic phase is characterized by non-negligible high-order multipole moments. It nucleates in the form of two order parameters, the primary ϕT1u and the secondary ϕ Eg (T 1u and E g are irreps of Oh group), whose temperature dependence is shown in Fig. 1.18. ϕT1u includes magnetic moments (in order of importance) of ranks 7, 9 and 1, the last describing the usual (first-order) magnetic moment. On the contrary, ϕ Eg is nonmagnetic, including components of ranks 8, 4, 6 and 2 (in decreasing order). Clearly, such insight into the multipolar order cannot be extracted from conventional measurements, for which most of its components will be “hidden”. In a next step, the dispersion of spin-wave excitations and, finally, the magnetic and thermodynamic properties in both ordered and paramagnetic phases have been calculated for the entire temperature domain [152]. This study demonstrates that the ab initio approach to the derivation of low-energy Hamiltonians represents a powerful tool for the study of lanthanide and actinide materials with complex magnetic order. In particular, it allows also to elucidate the origin of ferromagnetism of NdN (and the entire family of lanthanide nitrides) as arising due to a number of contributions to the Goodenough mechanism [152], the earlier model by Kasuya and Li highlighting only few of them [156]. On the basis of these results, we can conjecture that also the strong ferromagnetic exchange interaction between Dy sites connected by a nitrogen bridge in endohedral fullerenes Dyn Sc3−n N@C80 (n = 2, 3) [157, 158] is due to the Goodenough mechanism. The existence of strong ferromagnetism in these complexes is also supported by BS-DFT calculations of the isostructural Gd compounds [159] showing strong ferromagnetic Fig. 1.18 Temperature dependence of the primary (T 1u ) and secondary (E g ) order parameters in NdN. Reprinted with the permission from [152]. Copyright 2022 American Physical Society

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exchange interaction between Gd sites, which after proper rescaling to Dy ions (see Sect. 1.4.3) gives values which fit perfectly the experiment [160].

1.5.2 First-Principles Description of Magnetization Dynamics Magnetic relaxation in metal complexes and substitutional impurities, notably lanthanides, in diamagnetic lattices started to be investigated since 1960s [22]. The first investigated systems were characterized by low axiality such as, e.g., Ln3+ in ethyl sulfate [29], in which the ground multiplet was either intrinsically nondegenerate (Ln = Tb) or with KD components brought out of resonance by external magnetic field. Accordingly, the magnetic relaxation involved the electron–phonon mechanisms described above (Sect. 1.2.2 and Fig. 1.2) without QTM relaxation. In the simplest form, the temperature dependence of the relaxation rate (inverse of the relaxation time τ ) is given by Eq. (1.23a) without the second term. On the contrary, the measurements of magnetization dynamics in SMMs involve the QTM relaxation [161], so that τ −1 includes all four terms corresponding to the direct, QTM, Raman and Orbach (activated) process, respectively. Under constant applied magnetic field, this equation involves six parameters determined from the fitting of experimental τ −1 (T ), extracted at its turn either from measured ac magnetic susceptibility or magnetization hysteresis loops [3]. τ −1 has a smooth monotonic temperature dependence (Fig. 1.19), so that even the simplified model (1.23a) often appears overparametrized. In the last years, a discussion has arisen regarding appropriate ways to fit magnetic relaxation processes in lanthanide SMMs, motivated by the claims from several authors that it is incorrect to model the temperature dependence of the Raman relaxation rate with power functions T n temperature containing arbitrary exponents n. These claims where based on early theoretical estimations made for the acoustic phonons, according to which exponents ranging between 2 and 7 do not appear [22, 162]. Theoretically was shown, however, that these exponents can take values n = 2–4 for non-Kramers complexes and n = 2–6 for the Kramers ones when optical and acoustic phonons are simultaneously involved in the Raman relaxation [163]. Moreover, it has also been found that the temperature dependence of the Orbach process acquires a power-low prefactor when two-phonon transitions to the intermediate electronic state are included along with conventional single-phonon transitions [164]. Recently, it was firmly demonstrated [165] that the relaxation of Ho adatom on a MgO surface proceeds via the Mills–Huang mechanism of Raman transition on a localized vibration [166–168], characterized by an activated relaxation rate with the activation energy corresponding to the frequency of this vibration. Gu and Wu [169, 170] admitted that this relaxation mechanism can become dominant in some SMMs. Concomitantly, Lunghi et al. [171, 172] suggested that anharmonicity effects on localized vibrational modes can give rise to direct (single-phonon) relaxation of activated character with the activation energy corresponding to half of the frequency

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Fig. 1.19 Fitting of the temperature dependence of the relaxation time in [Dy2 (phsq)4 (NO3 )2 (MeOH)2 ]·2MeOH [173] with three models based on Eqs. (1.23a)–(1.23c)

of this vibration. The models incorporating the localized two-phonon relaxation [169, 170] and the anharmonic one-phonon relaxation [172] are given in the simplest form in Eqs. (1.23b) and (1.23c), respectively. U

τ (T )−1 = AT + B + C T n + τ0−1 e kB T , Ua

(1.23a)

Ub

−1 kB T −1 kB T τ (T )−1 = AT + B + τ0a e + τ0b e ,

(1.23b)

w

−1

τ (T )

e kB T

U

−1 = AT + B + V ( w )2 + τ0 e kB T . kB T e −1

(1.23c)

In these equations, the first two terms are the rates for the direct (single-phonon) and the QTM relaxation processes within the ground KD and the last term is the Orbach relaxation rate through the first excited KD. The third term corresponds to the conventional Raman mechanism in Eq. (1.23a), to the Raman relaxation on the strongly localized vibration in Eq. (1.23b) and to the anharmonic one-phonon relaxation in Eq. (1.23c). We use the above equations to fit the τ −1 (T ) data for the description of temperature dependence of the relaxation time in the recently

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synthesized binuclear complexes [DyIII 2 (phsq)4 (NO3 )2 (MeOH)2 ]·2MeOH [173]. A binuclear complex was chosen because of an efficient quenching of QTM in such systems, caused by the need to reverse two magnetic moments during the tunneling [159, 174]. This allows to drop the second term (B) in Eqs. (1.23a)–(1.23c) thus reducing the number of fitting parameters. The results of three simulations are shown in Fig. 1.19. Fitting of the extracted τ values versus temperature by the conventional model (Eq. 1.23a) leads to a Raman exponent of n = 5.8 which is lower than exponents corresponding to acoustic phonons. Despite an apparently better agreement with experiment displayed by the last two models, the activation energy is obtained closer to the results of ab initio calculations only for the first (conventional) model. Note that w in the last model corresponds to a half of the energy of vibrational mode, and the latter is approaching the value of the Orbach barrier. Concerning other relaxation mechanisms involved in Eqs. (1.23a)–(1.23c), their presence in the complex cannot be firmly confirmed by the fitting procedure in Fig. 1.19. Thus, we can see that very different mechanisms involved in models (1.23b) and (1.23c) fit the relaxation time equally well (the corresponding τ −1 (T ) curves are indistinguishable in Fig. 1.19). At the same time, we emphasize that the models themselves are oversimplified in that they do not include contributions which a priori can be considered non-negligible. For instance, the Mills–Huang mechanism [166–168] was incorporated in the model (1.23b) in its extreme version, neglecting the Raman contribution from delocalized acoustic and optical phonons. This is only justified when the relevant local vibration is sufficiently localized (corresponds to a narrow energy distribution). At present, the realization of such a scenario has been firmly confirmed only for Ho adatom on a MgO surface, [165] where the energy width of Ho-type local vibration is of the order of 20 cm−1 . In bulk molecular crystals, the width of optical bands is usually significantly larger, so that the use of the Mills–Huang mechanism in its extreme version, as it is suggested in [169, 170], is not a priori justified and requires the assessment of many other relaxation mechanisms mentioned above. Concerning the better fitting of τ (T ) displayed by the last two models (1.23b, 1.23c) compared to the conventional model (1.23a), it is merely due to the involvement of two independent exponential functions in the former. This analysis proves that relaxation models involving several parameters, including the recently proposed [169, 170, 172], although still oversimplified, cannot be firmly extracted from the sole fitting of the temperature dependence of the relaxation time. Ab initio extraction of the parameters defining various relaxation mechanisms seems to be indispensable. The first-principles treatment of electron–phonon relaxation in magnetic molecules requires the knowledge of electron-vibrational coupling between magnetic sublevels for all phonons in the crystals, as well as phonon dispersion and polarization vectors (phonon eigenvectors) for all reciprocal space vectors k from the first Brillouin zone of the corresponding lattice [3, 22]. The latter subdivides into 3N nuc -3 optical phonon branches (number of phonons for each value of k) and three acoustical phonon branches with vanishing frequencies at k → 0. In early studied lanthanide compounds, due to a weakly axial ligand environment of Ln ions, the separation between the low-lying magnetic levels (CF multiplets) was sufficiently small to fit the frequencies of acoustic phonons. The latter was described in most

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applications within the Debye model, reducing the full phonon spectrum to three acoustic branches in an enlarged Brillouin zone [175]. The description of relaxation via acoustic phonons is further simplified by applying the “rotational” approximation to the electron–phonon coupling [176], which highlights the importance of librational modes, i.e., rotations of the molecule as a whole (the corresponding coupling vanishes in the case of a free molecule) [177]. The rationale behind this approximation is that the direct and Raman processes scale with the 5th and 10th inverse powers of velocity of sound, which is the smallest for the transversal acoustic branches, while the latter contributes to electron–phonon coupling mostly via the librational distortions of the unit cell [176]. Given the kinematic nature of this “rotational” contribution, the corresponding expression for the transition rate between two magnetic levels is solely determined by the ZFS Hamiltonian in the case of S-complexes (e.g., first-row TM) [176, 178] and the CF Hamiltonian in the case of J-complexes (e.g., Ln) [177, 179]. Note that the “rotational” model also underlies the simple method of construction of magnetization blocking barriers (see Sect. 1.2.2) on the basis of ab initio calculated matrix elements of magnetic moments between the wave functions of different CF multiplets (average transition moments) [32]. The “rotational” model in a first-principles form was successfully applied to the description of relaxation in the Mn12 acac SMM [33, 180], where the acoustic phonons play a dominant role [3, 181]. It was further extended for the description of onephonon and Raman relaxation in arbitrary Kramers and non-Kramers doublets [179]. It was shown that in the cases when these doublets have a clear genealogy from an S or J (pseudo) spin, the relaxation rates can be expressed solely through their gfactors and the intrinsic gap, respectively, i.e., quantities derivable straightforwardly from ab initio calculations [179] (Sect. 1.2.4). An example of application of this approach is the calculation of electron–phonon relaxation rate in [Co(acac)2 (H2 O)2 ] (acac = acetylacetonate) [182], a representative CoII complex showing slow relaxation [183]. Figure 1.20a shows the temperature dependence of the electron–phonon relaxation rate calculated within “rotational” approximation [179] in which the averaged sound velocity of transversal phonons (the only free parameter of the theory) was taken close to the value of the average speed of sound extracted from calorimetric measurements of this compound [183]. The agreement with experiment is already quite good for T > 4 K, whereas in the lower temperature domain, it improves significantly when an additional internal field is included. Thus, the data corresponding to an applied external field of 175 and 350 mT (red and blue symbols in Fig. 1.20a, respectively) are perfectly fit in the entire temperature domain for H int = 115 mT [179]. A similar behavior is displayed by the field-dependent relaxation rate at low temperature (1.8 K), showing a much better agreement with experiment when the internal field is taken into account. One should note, however, that a value of 115 mT for the latter is certainly exaggerated which points to the role in the relaxation of other; first of all, intramolecular phonon modes are not included in this calculation. Moreover, the rationalization of these data is also possible within a mechanism involving the nuclear spin–phonon interaction albeit for its unrealistically high strength [183]. Therefore, to get full insight into the relaxation mechanisms of this complex, a theory

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Fig. 1.20 a Calculated electron–phonon contribution (lines) versus experimental relaxation rate [183] (symbols) for [Co(acac)2 (H2 O)2 ] (acac = acetylacetonate) at three values of applied magnetic field. b Calculated electron–phonon contribution (solid lines) versus experimental relaxation time [184] (dashed line) for [Co(L)2 ]2− where H2 L = 1,2-bis(methanesulfonamido)benzene. Δ is Lorenzian width corresponding to the anharmonic enlargement of the relevant local vibration. Reprinted with the permission from [179]. Copyright 2022 American Physical Society

using first-principles electron–phonon and nuclear spin–phonon coupling constants for all phonon branches should be applied. Recently, several attempts of first-principles description of electron–phonon relaxation in magnetic complexes have been made [123, 125, 171, 172, 185–188]. The approach was based on the calculation of derivatives of CF parameters for Ln complexes or ZFS parameters for TM complexes with respect to nuclear displacements of the ligand atoms surrounding the metal ion. In the case of lanthanides, this was achieved via several ab initio calculations of CF parameters [84] for geometries of the complex differing by the amplitude of a given nuclear distortion Qi , from which the derivatives of all CF parameters in Eq. (1.14) with respect to these distorc s /∂ Q i , ∂ Bnm /∂ Q i , were calculated numerically. Note that the electrontions, ∂ Bnm vibrational operator looks precisely as the CF Hamiltonian (1.14), in which the CF parameters are replaced by these derivatives [32]. The TM complexes are treated similarly with the only difference that instead of CF Hamiltonian (1.14), the ZFS operator (1.1) is considered (the derivatives are respectively taken from the ZFS parameters Dαβ ). Given many dozens of nuclear distortions to be considered for a typical Ln complex, thousands of ab initio calculations are needed to calculate the electron-vibrational coupling parameters, thus requiring a large amount of computation which is usually done on a supercomputer [123]. With the knowledge of electron-vibrational coupling parameters, the electron–phonon coupling parameters can be obtained straightforwardly by decomposing the nuclear displacements around the metal ion into phonon eigenmodes. The latter can in principle be obtained from DFT calculations of the crystal [189]. This step, however, was never undertaken for the investigation of magnetic relaxation in complexes, although its feasibility was demonstrated in other studies involving full calculation of electron–phonon coupling in molecular crystals with large unit cells [190]. Instead, vibrational eigenmodes of one single complex were considered, while the phonon dispersion was simulated

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by assigning a width to each molecular vibrational level. The vibrational spectrum of the complex was simulated either for an isolated free molecule [123, 186] or as phonons in the ⎡ point of the Brillouin zone [171, 187]. In both cases, however, no acoustic vibrations can be simulated. The latter, however, plays an important role in magnetization reversal transitions and has been included within the “rotational” approximation [176, 179] in simulations of relaxation of magnetization in disprosocenium [186] and other complexes [191]. To achieve quantitative agreement with the relaxation data, three fitting parameters have been introduced for the treatment of electron–phonon relaxation. Additional fitting parameters were employed for the phenomenological description of the field dependence of Raman relaxation [192, 193] and the QTM in the low-T tunneling regime [194], which depends at its turn on the phenomenological (Gaussian) distribution of internal fields [195]. On the other hand, the simulation of the phonon dispersion by Lorenzian enlargement of the effective vibrations of an individual molecule in the crystal, including the three modes with ω = 0 corresponding to three acoustic modes at the ⎡ point, [188] apparently leads to serious overestimation of the electron–phonon relaxation rates. Thus, Fig. 1.20b shows that the calculated relaxation times for direct (Orbach) and second-order Raman mechanisms alone are shorter by several orders of magnitude (respectively, the relaxation rates larger by several orders of magnitudes) than the experimental relaxation time (relaxation rate) [187]. On the other hand, other relaxation mechanisms not considered in this calculation can be equally important, such as first-order Raman relaxation found dominant for Dy complexes in a broad temperature domain [186]. The ultimate goal of ab initio treatment of magnetization dynamics is the quantitative assessment of various mechanisms contributing to the observed relaxation of magnetic moment and decoherence of ground doublet state (qubit). Given the multitude of interactions present in complexes embedded in various hosts and the impossibility of their reliable extraction from experiment, this goal can only be achieved via accurate ab initio determination of all quantities defining these interactions. Besides the electron–phonon coupling discussed here, a first-principles description should be extended over hyperfine-phonon coupling [3, 183], recently found to give rise to qualitatively new relaxation behavior [196]; Zeeman-phonon coupling [185] argued to be important when the Zeeman splitting exceeds ZFS in the S-multiplets of TM complexes [197]; incoherent QTM and TA-QTM [3, 33, 180]; etc. A reliable description of CF splitting in lanthanides (Sect. 1.3.3) gives confidence to the ab initio calculations of electron–phonon coupling constants, albeit at high computational cost, especially, for high-order coupling constants. The intermolecular magnetic interactions, underlying the QTM in Kramers ions [198, 199] and the distribution of internal fields, can also be calculated without problems once the intramolecular magnetic anisotropy (g-tensors of individual doublets) is known from ab initio calculations. The real bottleneck, however, for the first-principles description of magnetization dynamics precluding its predictive power is the still unreliable description of phonons in molecular crystals by the current DFT calculations. Two aspects make the calculation of phonon spectrum critical. First of all, the already mentioned extremely high sensitivity of basic electron–phonon relaxation

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mechanisms to the slope of dispersion curve (velocity of sound) of three acoustic branches, while the low-frequency phonons play a major role in the reversal of magnetic moments in most SMMs. Second, the relative error in the calculation of the frequency of these phonons is higher than for the optical phonons, and this error certainly exceeds [197, 200–202] the limit above which the first-principles description of magnetic relaxation becomes unreliable. This aspect is illustrated in Fig. 1.21 showing the calculated phonon spectrum of archetypal molecular crystals, the cubic A3 C60 fullerides (A = K, Rb, Cs), for which the low-frequency phonons have been decently calculated for the first time by employing new ionized pseudopotentials for alkali atoms [190]. A remarkable feature of this material is the high-frequency intrafullerene vibrations, making the corresponding bands well separated (> 250 cm−1 ) from intermolecular ones (< 150 cm−1 ). We can see from the figure that the two used DFT exchange–correlation functionals, PBE and LDA GGA, give very close predictions for the intrafullerene optical phonons, which in addition found in good agreement with experimental Raman data [203, 204]. At the same time, the two approaches show large discrepancies in the description of intermolecular phonon bands. This is remarkably given these molecular crystals which are basically ionic solids for which a DFT description should not pose problems. The situation is more complicated in SMM materials, mostly representing Van der Waals solids putting limitations on their DFT description and requiring specially adapted exchange–correlation functionals [205–207]. Due to these complications, the quantitative description of electron–phonon relaxation in most SMMs is not possible by the current DFT means, sometimes even not by the order of magnitude. With rare exceptions such as Ho addatom on a MgO, for which the dominance of Mills–Huang twophonon relaxation mechanisms was firmly established, the current first-principles descriptions using several fitting parameters cannot discriminate between different relaxation mechanisms as emphasized above.

1.5.3 Perspectives Concerning prospective developments of ab initio methods related to magnetism, which look feasible nowadays, one should mention the following: (1) Analytical derivatives method for coupling constants. Although the accuracy of calculated electron–phonon coupling constants is not as critical as for the lowfrequency phonon spectrum (the former enter relaxation rates as second power), their full evaluation by numerical methods used so far can be intractable for large complexes, especially, for the second-order coupling constants. Therefore, their calculation via analytical derivatives of the ab initio energy matrix requiring one single full ab initio calculation at equilibrium geometry would allow treatment of arbitrary complexes with moderate computer resources. The expressions for analytical derivatives between different wave functions evaluated at stateaveraged CASSCF level of theory are currently available [208, 209] and already

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Fig. 1.21 Phonon dispersion for A3 C60 (A = K, Rb, Cs) crystals. The A15 corresponds bodycentered cubic (bcc) structure including two CsC60 in a unit cell due to a merohedral order of C60 molecules. Red and black lines correspond to PBE and LDA GGA functionals, respectively. Reprinted with the permission from [190]. Copyright 2022 American Physical Society

implemented in the MOLCAS package [75]. The only development needed for application to magnetic relaxation problems is to re-derive and implement them in the basis of the spin–orbit multiplets calculated by RASSI. Another feasible implementation would be the analytical second-order derivatives of the energy matrix (the Hessian) for the calculation of second-order electron-vibrational coupling constants. (2) Accurate evaluation of magnetic and electric dipolar momenta. It is well known that the CASSCF wave functions are built on the Hartree–Fock-type molecular orbitals and, therefore, underestimate the covalency because of unscreened Coulomb and exchange interaction in the frontier orbitals. This leads to systematic errors in the calculation of different physical properties such as, e.g., exaggerated deviations of g-tensors of S-complexes (e.g., of Cu2+ ) from the isotropic value ge = 2 [21]. This drawback cannot be remediated by a simple enlargement of the active space as it was recently demonstrated in a series of density matrix renormalization group (DMRG) CASSCF calculations [210]. On the other hand, the inclusion of dynamical correlation effects through CASPT2 (MS-CASPT2) calculations of wave functions (not only of energy shifts of CASSCF solutions as it is done nowadays) is expected to cure this drawback. Such calculations will improve the predicted anisotropic magnetic properties in two respects. First, the calculation of orbital contributions to magnetic moments will be much more realistic. The current calculations for lanthanide complexes look satisfactorily, less so for actinide compounds (even when the CASSCF wave functions are

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corrected by the MS-CASPT2 step) and usually unsatisfactorily for the calculation of heavy transition-metal complexes. For the latter, the DFT calculated orbital contribution to the magnetic moment [115] appears to be closer to experiment (see Sect. 1.3.3). Second, the off-diagonal matrix elements of magnetic moment defining, in particular, the rate of QTM relaxation at low temperature are expected to be modified significantly compared to their current very low values predicted by CASSCF calculations in complexes with strong axiality. Calculations with CASPT2 wave functions will improve also the ab initio calculation of electro-dipole matrix elements and, as a result, will contribute to a better description of the intensity of spectroscopic lines arising from electric dipolar transitions. (3) Investigation of multiferroic materials. These are materials displaying several forms of ferroic order (ferromagnetism, ferroelectricity, ferroelasticity and ferrotoroidicity) [211] or ferroic response to various external stimuli [212]. Typical multiferroics are materials with magnetoelectric response such as, e.g. BiFeO3 , in which the magnetic order can be induced by applied electric field and vice versa [213]. Such a complex response makes the multiferroics’ candidates of choice for the design of multifunctional materials and electronic devices [211, 214]. In magnetic molecules, the magnetoelectric coupling was the most investigated multiferroic property [215–218], due to the opportunity of electric control of spin qubits. A prospective application of magnetoelectric coupling is magnetic skyrmions representing nanometer-sized spin textures (similar to polynuclear complexes) of topological origin exhibiting, in particular, currentdriven motion [219]. Due to long lifetime, they are regarded nowadays as potential candidates for molecular memory units in skyrmion-based spintronic devices [6, 219]. So far, first-principles investigations of magnetoelectric coupling have only been done by DFT methods for both crystalline materials [220] and individual molecules [221]. Given the involvement of magnetic metal ions in most multiferroic materials, the decent description of various multiferroic interactions should be done by explicitly correlated ab initio methods similar to the anisotropic magnetic properties described here. Such ab initio study has been undertaken for another multiferroic property, the interaction of toroidal magnetic moment with external magnetic field in heterometallic CuII /DyIII 1D chiral polymers, which has shown good agreement with magnetic data [222]. Similar ab initio studies can be extended to many other multiferroic materials.

1.6 Concluding Remarks The understanding of mechanisms of magnetic anisotropy in molecular nanomagnets became a task of primary importance in the last decade. The description of the effects of magnetic anisotropy is relatively simple in complexes with weak spin–orbit coupling effects on the metal sites, corresponding to strong exchange limit in polynuclear compounds. The model parameters for such complexes can usually be extracted

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without problems from magnetic measurements. On the contrary, in complexes with strong spin–orbit coupling effects, leading to weak exchange limit in polynuclear compounds, the details of anisotropic magnetic interactions can be hardly revealed from experiment alone. The problem is in the large number of parameters describing the anisotropic magnetic interaction in this case. For such complexes, the theoretical description of magnetic anisotropy based on multiconfigurational ab initio calculations with non-perturbative treatment of spin–orbit coupling becomes indispensable. We presented several examples of such an approach allowing to extract details of anisotropic magnetic interaction which have not been available in the past. We also analyzed in detail the mechanisms leading to strong magnetic axiality and magnetization blocking in anisotropic magnetic complexes as being the necessary condition for the design of efficient SIMs and SMMs and emphasized the role of lanthanide-based complexes. In most cases, the ab initio methods cannot be applied straightforwardly, notably in the case of polynuclear complexes, but rather for the description of parts of the material in combination with a realistic modelization. Examples of such semi-ab initio approaches are the application of Lines model for the treatment of anisotropic exchange interaction and the transition magnetic moments method for the construction of magnetization blocking barriers. Both these approaches work well for strongly anisotropic complexes with pronounced axial anisotropy on the metal sites, which is just the favorable situation for magnetization blocking, the reason why they were successfully applied for rationalization of SMM behavior in many complexes. The full ab initio treatment of magnetization dynamics in complexes requires, however, knowledge not only of magnetic but also of several other interactions in the extended material making stringent the demands for their highly accurate determination from first-principles calculations. Answering these demands is a major task for future application of ab initio methods to novel materials.

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Chapter 2

Analytical Derivations for the Description of Magnetic Anisotropy in Transition Metal Complexes Rémi Maurice, Nicolas Suaud, and Nathalie Guihéry

Abstract This chapter is dedicated to the rationalization of magnetic anisotropy in metal complexes. Analytical derivations allow one to predict the nature and magnitude of both the zero-field-splitting and the anisotropies of magnetic exchange. The first section is devoted to mononuclear complexes. It addresses the effect of spin–orbit coupling (SOC) in two different cases: (i) when the ground state is non-degenerate and a second-order SOC applies. The effect of the SOC can then be modeled by an energy splitting of the M S components of the ground spin state. Illustrations of the power of these analytical derivations for the rationalization of the ZFS of various complexes are presented; (ii) when the ground state is (almost) degenerate, a firstorder SOC applies. A more sophisticated model is here derived which rationalizes the obtaining of a giant value of the ZFS in a Ni(II) complex. The second section is devoted to the derivation of multi-spin models for binuclear complexes. We will determine the physical content of both the symmetric and the antisymmetric exchange tensors in the case of two centers with spin S = 1/2. A peculiar derivation concerns the Dzyaloshinskii–Moriya (antisymmetric exchange) interaction in case of a local degeneracy of the orbitals and shows how the first-order SOC can generate giant values of this anisotropy of exchange. In the last subsection, we will show that the usual multi-spin model for spin S = 1 centers is not valid and derive an appropriate model involving a four-rank exchange tensor. Keywords Magnetic anisotropy · Model Hamiltonian derivation · Dzyaloshinskii–Moriya interaction · Correlated relativistic ab initio calculations · First-order spin–orbit coupling R. Maurice Univ Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes)—UMR 6226, 35000 Rennes, France N. Suaud · N. Guihéry (B) Laboratoire de Chimie et Physique Quantiques, UMR5626, Université de Toulouse 3, Paul Sabatier, 18 route de Narbonne, 31062 Toulouse, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Rajaraman (ed.), Computational Modelling of Molecular Nanomagnets, Challenges and Advances in Computational Chemistry and Physics 34, https://doi.org/10.1007/978-3-031-31038-6_2

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2.1 Introduction Our understanding of magnetic anisotropy or more specifically zero-field splitting (ZFS) is intimately related to the development of the electron paramagnetic resonance (EPR) spectroscopy. The premises of this technique may date back to the key experience of Stern and Gerlach in 1922 [1]. It was shown that a beam of silver atoms (with no orbital momentum) subjected to the inhomogeneous magnetic field prevailing in the air gap of an electromagnet was deflected into two observable spots. This beam splitting was attributed to the spin of the electron and more specifically to the lift of degeneracy of its magnetic components (characterized by the quantum numbers ms = ± ½) induced by the magnetic field (Zeeman effect). Two decades later, the first EPR experiments were carried out by Bleaney in the UK and Zavoisky in the Soviet Union (independent works), paving the way for the observation of real transitions between these two levels. Conceptually, EPR and nuclear magnetic resonance (NMR) spectroscopies are quite similar and it is not surprising to see that the beginnings of both techniques occurred at the same period and also that famous scientists such as Abragam have worked on both techniques [2, 3]. In both cases, a model Hamiltonian is invoked to interpret the data, typically acting on spin degrees of freedom. In fact, the terminology “spin Hamiltonian,” first appeared in a paper by Abragam and Pryce [4], in the context of “hyperfine” coupling (the effective coupling between the nucleus and electron spins). Many other experimental techniques were developed and used to characterize magnetic anisotropy, such as magnetometry, inelastic neutron scattering, frequency domain magnetic resonance, and magnetic circular dichroism [5–7]. Let us also mention AC magnetic measurements that make it possible to quantify the magnetic relaxation time and to evaluate the effective spin-reversal barrier which is a crucial ingredient for nanomagnets. As examples, some recent studies revealed slow relaxation times in transition metal complexes and hybrid magnetic systems, particularly promising for technological applications [8–11]. Between 1922 and 1944 (i.e., between the Stern and Gerlach experiment and the first EPR experiments), two important fields emerged, (i) quantum mechanics and relativistic theory (the Dirac equation was derived in 1928), closely followed by (ii) crystal field theory (CFT), attributed to Bethe [12] and van Vleck [13]. In practice, performing relativistic quantum mechanical calculations to understand the properties of transition metal, rare earth or actinide complexes was yet to come, and naturally the CFT was extensively used in the following decades, leading to pen-and-paper derived analytical expressions. CFT was later improved with molecular orbital theory to include subtler effect, leading to ligand field theory (LFT) which is still widely used. These theories allowed to somehow justify the shape of the spin Hamiltonian (in fact, deriving equations compatible with it without systematically questioning it), and also to rationalize the parameter values (in analyzing the data, those parameters are free and thus adjusted), in particular the so-called axial D and rhombic E ZFS parameters, generally used to describe the ZFS of orbitally non-degenerate ions (for S = 1 ground states and above).

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Currently, the design of new anisotropic molecules aims at obtaining complexes with a very large negative D value since the relaxation barrier (of the Orbach process only) is proportional to this parameter. While for a long time D did not exceed a few cm−1 , we now find systems where it reaches values above 100 cm−1 [14]. Let us mention that in case of degeneracy, i.e., when a first-order SOC occurs (the D and E parameters no longer apply), quite long relaxation times have been measured. Note that alternative situations have been reported, for which the observed slow relaxation of the magnetization is for instance associated with a positive D and an odd number of electron [15] or even with quenching/unquenching of the orbital momentum and not with a spin transition [16]. Nowadays, many suites of programs (see for instance [17–20]) are capable of performing accurate quantum mechanical calculations on real complexes. Several quantum chemists have contributed to the birth of the study of anisotropic magnetic systems, some of them by (i) the conception of methods to accurately reproduce the magnetic anisotropy parameters either based on Density Functional Theory (DFT) (see for instance [21–28]) or Wave Function Theory (WFT) (see for instance [29–33]) while other used these methods to calculate and also rationalize observed properties (see for instance [34–42]). It is then the perfect time to revisit our classics and make explicit connections between the spin Hamiltonians, which are usually phenomenological, and an accurate quantum mechanical model on the one side, and also on the other side question or validate simple CFT formulae based on such calculations. We have the philosophy of performing pen-and-paper derivations in the same vein as what was done in the early days of the EPR spectroscopy, essentially because we believe that a quantitative and understandable description cannot be fully replaced by technique and numbers (also for fun). However, our strong added value is that we are now capable of validating CFT models and spin Hamiltonians based on accurate quantum mechanical calculations, notably by making explicit bridges between those pictures. This has allowed us on the one side to fully appreciate the beauty of the work that was done at that time (that was of exceptional relevance) and also to extend these works in specific cases (read improve the CFT models and improve the spin Hamiltonians). We hope that we will transmit our enthusiasm and also guide step-by-step the readers to master the several key points that should be understood to revisit past and present literature in the field. In fact, as soon as the size of a system exceeds two or three magnetic centers, the use of the exact electronic Hamiltonian is impossible and experimentalists and theoretical physicists must resort to model Hamiltonians, be them spin Hamiltonian or more sophisticated. The latter must in principle contain the key microscopic information and somehow open the way to the exploration of collective properties. Indeed, they can be used to treat large (even infinite with some approximations) systems. Nevertheless, their interactions are complex effective interactions that often (if not always) result from complicated mechanisms. Among the tools needed to extract the effective magnetic interactions, correlated and relativistic ab initio calculations combined with the effective Hamiltonian theory [43, 44] are particularly useful [45]. In some cases, the anticipated model is not appropriate and a suitable alternative

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model has to be determined [46–48]. In such cases, analytical derivations are mandatory. They usually start from a more sophisticated (eventually the exact) Hamiltonian and use the quasi-degenerate perturbation theory [49] up to the second or fourth order of perturbations. Analytical derivations of model Hamiltonians also allow to determine the mechanisms at work in the effective interactions and thus to provide rationalizations of their nature and magnitude. Combined with ligand field theory [50], this reasoning allows theorists to guide synthetic chemists in the elaboration of systems with improved and controlled properties. This work is in line with the work of Abragam and Bleaney who, in their famous book [51], proposed qualitative rationalizations for all d n configurations in distorted octahedral geometries and of Racah [52]. Both approaches are based on the crystal field theory and both have their advantages and drawbacks. In the approach of Abragam and Bleaney, the crystal field is treated as a perturbation of the free ion Hamiltonian. Only the effects of the field at the first order of perturbation are considered, i.e., the splitting of the free ion ground multiplet due to the crystal field. The SOC is then treated, but the contribution to the ZFS of the free ion excited multiplets is generally neglected, which is not always a valid approximation [53]. Racah’s approach allows to include such contributions, but is often too complicated for a handmade treatment of the problem because it may involve the diagonalization of a too large Hamiltonian. The approach used here takes into account the best of both worlds, by including only the first-order effects of the crystal field (like Abragam and Bleaney), but by allowing to introduce states coming from any multiplet of the free ion (like Racah). For this purpose, the spin–orbit-free states are expressed in the real orbital basis and coupled by a spin–orbit coupling single-electron operator. For the sake of pedagogy, we will first show in the simplest case that both our approach and the one of Abragam and Bleaney may lead the exact same results provided that the same states are accounted for and if their expressions match. For systems containing more than one magnetic centers, not only can each of these centers be anisotropic, but so are their interactions. In such cases, additional terms accounting for the anisotropy of exchange must be introduced in the model multi-spin Hamiltonian. They are usually called symmetric and antisymmetric as they respectively involve a symmetric and an antisymmetric tensor of rank two. The antisymmetric exchange interaction, also called Dzyaloshinskii–Moriya interaction (DMI), was introduced phenomenologically in 1958 by Dzyaloshinskii [54, 55] in order to describe the magnetic properties of R-Fe2 O3 . Two years later this interaction was theorized by Moriya [56, 57]. Prior to focusing on the derivation of the giant-spin and multi-spin Hamiltonians and their various interactions [58], we will work on mononuclear systems. We will see how the choice of appropriate ligands allows to control the nature of the anisotropy and how the first-order spin–orbit coupling allows to obtain very large anisotropies of the magnetic centers. After this, we will deal with binuclear systems of spin S = 1/2 and spin S = 1. We will show that the multi-spin model is not appropriate as soon as the spin of the magnetic centers exceeds 1/2, and we will show again that the first-order spin–orbit coupling is likely to generate particularly large exchange anisotropies.

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2.2 Rationalization of the ZFS Hamiltonian: Application to Mononuclear Complexes 2.2.1 Pedagogical Presentation for an S = 1 Complex Mononuclear complexes are the systems of choice for illustrating how to construct and how to rationalize the use of simple spin Hamiltonians to describe lowtemperature magnetism, in particular in the case of non-degenerate ground states (in terms of orbital degeneracy). In the case of an S = 1/2 ground state, for instance a mononuclear copper(II) complex, Kramers’ degeneracy prevents the occurrence of zero-field splitting (ZFS): The two spin components of the ground state, MS = 1/2 and MS = –1/2, remain degenerate in the absence of an external magnetic field. Thus, our first case of interest concern S = 1 states, e.g., (high-spin) nickel (II) complexes.

2.2.1.1

The Phenomenological Model Hamiltonian and its Application to the “Coupled” Basis

S = 1 states result from the occurrence of (at least) two unpaired electrons. With two electrons in two atomic orbitals (AOs), namely a1 and a2 , and two spin functions per electron, one may form four distinct Hartree products. If a bar denotes a “spin down” function, or β, and no bar denotes a “spin up” function, or α, one may form three “coupled” |S, MS ⟩ spin state components for the S = 1 state: |1, 1⟩ = T1 = |a1 a2 ⟩ 1 |1, 0⟩ = T0 = √ |a1 a2 − a2 a1 ⟩ 2 |1, −1⟩ = T−1 = |a1 a2 ⟩

(2.1)

where the T notation refers to “triplet” and each index denotes the corresponding M S value (in descending order, i.e., from S to −S). These are the three functions that form the basis for the phenomenological model Hamiltonian: Hˆ ZFS

⎛ ⎞⎛ ⎞ Sˆ x ) Dx x Dx y Dx z ˆ ⎝ ⎠ ⎝ = S. D. S = Sˆ x Sˆ y Sˆ z D x y D y y D yz Sˆ y ⎠ D x z D yz D zz Sˆ z Δ

(

(2.2)

where the ZFS tensor D, symmetric, only displays at most six independent components. We recall here that the Sˆ x and Sˆ y operators are not readily applicable to |S, MS ⟩ spin state components, one may thus introduce the spin raising Sˆ+ and lowering Sˆ− operators, defined as follows [59]: Sˆ+ = Sˆ x + i Sˆ y

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Sˆ− = Sˆ x −i Sˆ y

(2.3)

The action of the Sˆi operators (where i = z, + or –) on the |S, MS ⟩ functions in atomic units (è = 1) is: Sˆ z |S, MS ⟩ = MS |S, MS ⟩ √ Sˆ+ |S, MS ⟩ = S(S + 1) − MS (MS + 1)|S, MS + 1⟩ √ Sˆ− |S, MS ⟩ = S(S + 1) − MS (MS − 1)|S, MS − 1⟩

(2.4)

Naturally, the action of Sˆ z on |1, 0⟩ is null, as is that of Sˆ+ on |1, 1⟩ and that of ˆS− on |1, −1⟩. The matrix representative of Hˆ ZFS in the |1, MS ⟩ basis (where the functions are ranked in descending order of MS value) is: (

)

⎛

⎜ Hˆ ZFS = ⎝ 1 2

1 (D x + D y y ) + Dzz 2 √ x( ) 2 Dx z + iD yz ( 2 )

D x x − D y y + iDx y D x z

⎞ ) ( ) Dx z − iD yz 21 D√ x x − D y y − iD x y ( ) ⎟ D + Dyy − 22 Dx z − iD yz ⎠ √ (x x ) − 22 Dx z + iD yz 21 (D x x + D y y ) + Dzz (2.5) √

2 2

(

Several simplifications may then be done. First, one may consider a traceless ZFS tensor (D x x + D y y + Dzz = 0). In practice, one may thus subtract any scalar to each of the diagonal elements of Hˆ ZFS . Second, one may define the “magnetic anisotropy axes,” which are nothing but the principal axes of the ZFS tensor. Conventions are used to define (i) a main anisotropy axis and (ii) distinguish the other two. If Z is the main anisotropy axis, the axial parameter ZFS D is defined as follows: 1 3 D = D Z Z − (D X X + DY Y )traceless D D Z Z − − − − − − − → 2 2

(2.6)

The rhombic ZFS parameter E is then defined as: E=

1 (D X X − DY Y ) 2

(2.7)

The main anisotropy axis is chosen so that the DZZ value differs most from the DXX and DYY ones. This is easily translated into: D > 3|E|

(2.8)

Obviously, if D = 3|E|, the anisotropy tensor component along one of the anisotropy axes is the perfect intermediate between the two other ones (i.e., along the two other orthogonal orientations). We may call this the “rhombic limit,” which is the extreme case for which the D sign is irrelevant. Otherwise, this sign is relevant, positive D is associated with Z being a hard axis of magnetization (D Z Z > D X X

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and D Z Z > DY Y ), while negative D indicates an Z easy axis of magnetization (D Z Z < D X X and D Z Z < DY Y ). Two conventions prevail concerning the sign of E in the literature [19]: either E is defined positive, or the sign of E is fixed to that of D. In any case, Hˆ ZFS can be expressed in the magnetic anisotropy axis frame as: ⎛1

⎞ D 0 E ( Hˆ ZFS ) = ⎝ 0 − 23 D 0 ⎠ E 0 13 D 3

(2.9)

This Hamiltonian has three analytical solutions: 2 HZFS |1, 0⟩ = − D|1, 0⟩ 3 ) ) ) ) | s⟩ | ⟩ 1 1 D + E |1, 1s Hˆ ZFS |1, 1 = Hˆ ZFS √ (|1, 1⟩ + |1, −1⟩) = 3 2 ) ) ) ) | a⟩ | ⟩ 1 1 D − E |1, 1a Hˆ ZFS |1, 1 = Hˆ ZFS √ (|1, 1⟩ − |1, −1⟩) = 3 2

(2.10)

If E is taken positive, two situations may occur, depending on the D sign (see Fig. 2.1). Finally, it is worth mentioning that the ZFS parameters can be directly extracted from computed ab initio energies in the case of S = 1 [37], assuming that the lowest three energy levels (or, spin–orbit states) have a large projection onto the ground spin–orbit-free spin-triplet state.

Fig. 2.1 Analytical solutions of Hˆ ZFS in the magnetic anisotropy axis frame. E is defined positive (convention)

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Analytical Ground for this Model Hamiltonian: Crystal Field Derivations

So far, we have only worked within the framework of the model Hamiltonian. Of course, if this model Hamiltonian is inappropriate, its application is vain. It is therefore important to derive this model or at least derive equations that are compatible with it based on a more general framework, i.e., based on a more sophisticated Hamiltonian. This chapter being especially focused on analytical derivations, we will start our analysis by such derivations, prior to introducing additional numerical arguments (vide infra). The occurrence of ZFS is intimately related to the SOC. Depending on the SOC Hamiltonian used, the spin–orbit-free wave functions may have to be expressed in different bases. In this chapter, we always assume that we proceed in two steps, (i) the determination of spin–orbit-free states and (ii) the coupling between their M S components via a SOC operator. We will consider a spherical approximation for defining the SOC operator, meaning that we have to choose essentially between two SOC operators: ”poly” ˆ Sˆ Hˆ SOC = λ L. Σ ”mono” =ζ lˆi .ˆsi Hˆ SOC

(2.11)

i

where λ and ζ are the “polyelectronic” and “monoelectronic” SOC constant, respectively, and where summation runs on all electrons i of the d shell in the second expression. Note that a relationship exists between the two SOC constants in an atom when one only considers the first-order effect [51]: λ=±

ζ 2S

(2.12)

λ is positive for less than half-filled shells (e.g., less than 5 d electrons for a d n ion) and negative for more than half-filled ones. In the case of an exactly half-filled shell with a high-spin non-degenerate ground state, the SOC constant cannot be derived from the first-order splitting of the ground free ion term. This is why in some old tables no SOC constant was defined for such a situation. At this point, it should be recalled that lowercase quantum numbers and operators refer to single-electron quantities (closely related to the concept of orbitals), while uppercase quantum numbers and operators correspond to multi-particle states. For example, the operator (2.2) acts on the total spin S, not the individual ½ spin of each electron. This distinction is important for CFT in which the operators can be applied to the ground multiplet of an atom (low-field CFT, operators acting on the total angular momentum L) or directly on the 3d orbitals (high-field CFT, operators acting on the angular momentum of the individual electron l i ). For lanthanides and actinides, CFT is often used to describe the splitting of the basic spin–orbit state of the ion (operators effectively acting on the total angular momentum J = L + S).

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As a first example, we will show how to derive solutions of the crystal field Hamiltonian using a “polyelectronic” operator. The crystal field effective interaction is meant to split 2S+1 L terms of the corresponding reference free ion. For instance, a high-spin Ni2+ free ion has a 3 F ground term (in the absence of the spin–orbit coupling operator, which further splits it into 3 F J terms, with J ranging from 2 to 4, i.e., from |L − S| to L + S). The “spin–orbit-free” basis consists in the |L , M L ⟩ q functions. By using the so-called Oˆ k Stevens operators [60], one may quantify the 3 splitting of the F term that arise from an octahedral field. For an octahedral crystal field, the Hamiltonian writes [51]: ( ) ”poly” Hˆ CF = B4 Oˆ 40 + 5 Oˆ 44

(2.13)

where the operators (acting on L) are defined as follows: Oˆ 40 = 35 Lˆ 4z − 30L(L + 1) Lˆ 2z + 25 Lˆ 2z − 6L(L + 1) + 3L 2 (L + 1)2 ) 1( ˆ4 L + + Lˆ 4− (2.14) Oˆ 44 = 2 and B4 is the cubic crystal field parameter. Note that these operators are general and follow the same algebra, whether they act on L, S, or J. Thus, Lˆ z , Lˆ + , and Lˆ − follow ”poly” Eq. (2.1.4) (replacing S by L and M S by M L ). Applying Hˆ CF to the |L , ML ⟩ basis one gets the crystal field matrix (written in descending value of M L ): √ ⎞ 180B4 0 0 0 60 15B4 0 0 ⎜ ⎟ 0 0 0 300B4 0 −420B4 ⎜ ⎟ √0 ⎜ 0 0 0 60 15B4 ⎟ 0 0 60B4 ⎟ ( ) ⎜ ⎜ ⎟ ”poly” Hˆ CF = ⎜ √0 0 0 360B4 0 0 0 ⎟ ⎜ ⎟ ⎜ 60 15B4 ⎟ 0 0 0 60B4 0 0 ⎜ ⎟ ⎝ ⎠ 0 0 −420B4 0 0 300B4 √0 0 0 180B4 0 0 60 15B4 0 (2.15) ⎛

As before, the diagonalization of this matrix provides its eigenvalues and eigenvectors. Three solutions are actually trivial: | ⟩ ⟩ ”poly” ”poly” | Hˆ CF ψ1 = Hˆ CF |3, 2a = −720B4 |3, 2a | ⟩ ⟩ ”poly” ”poly” || 3, 2s = −120B4 |3, 2s Hˆ ψ2 = Hˆ CF

CF

”poly” ”poly” Hˆ CF ψ5 = Hˆ CF |3, 0⟩ = 360B4 |3, 0⟩

(2.16)

Analytical expressions for the four remaining roots can be further obtained (a formal calculation program may be useful; in addition, the resulting wave functions may need to be normalized). Two of these roots have the same energy as the |3, 2s ⟩ one, i.e., −120B4 :

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) √ 1 (√ ψ3 = √ 3|3, 3⟩ − 5|3, −1⟩ 2 2 ) √ 1 (√ 5|3, 1⟩ − 3|3, −3⟩ ψ4 = √ 2 2

(2.17)

Finally, the two remaining roots have the same energy as |3, 0⟩., i.e., 360B4 : ) √ 1 (√ ψ6 = √ 5|3, 3⟩ + 3|3, −1⟩ 2 2 ) √ 1 (√ 3|3, 1⟩ + 5|3, −3⟩ ψ7 = √ 2 2

(2.18)

These “spin–orbit-free” states, expressed in the |L , ML ⟩ basis, form the basis for the subsequent treatment of the SOC. The |S, MS ⟩ components of each of these states have to be considered, leading to a basis expanded in terms of the |L , ML , S, MS ⟩ functions, with possibility for labeling to factorize the orbital part of the wave functions (for instance for expressing them in terms of linear combinations of the solutions ”poly” of Hˆ CF , i.e., the previous ψi states). Again, one may build an interaction matrix ”poly” between the |ψi , M S ⟩. in the same way as before. Hˆ SOC has now to be expressed in terms of the Lˆ z , Lˆ + , Lˆ − , Sˆ z , Sˆ+ , and Sˆ− operators: [ )] 1( ˆ ˆ ”poly” ˆ ˆ ˆ ˆ ˆ L + S− + L − S+ HSOC = λ L z Sz + 2

(2.19)

It is somehow easy to check that none of the MS components of ψ1 is coupled to any of the MS components of ψ5 , ψ6 , and ψ7 . One can also reach the same conclusion by symmetry arguments since the product A2g ⊗ T1g ⊗ T1g (for ground state ⊗ excited state ⊗ real space rotation symmetries) does not contain the totally symmetric irreducible representation (irrep), i.e., A1g [61]. Thus, in the context of the ZFS (here, actually of no ZFS because of the octahedral symmetry) one can restrict the space to the MS components of ψ1 , ψ2 , ψ3 , and ψ4 , meaning that one needs to build a 12 × 12 matrix. If the SOC is introduced at second order of perturbations, computing only a 3 × 12 part of it is sufficient to reach the effective SOC-induced stabilizations of the |ψ1 , MS ⟩ components (see Table 2.1). Some matrix elements will actually arise from the Lˆ z Sˆ z term: | | | | ”poly” |⟨ψ1 , ±1| Hˆ SOC |ψ2 , ±1⟩| = 2λ

(2.20)

Note that for the sake of simplicity, only the absolute values are displayed (this is enough to|apply second-order perturbation theory). The |ψ1 , ±1⟩ functions also ⟩ couple with |ψ3,4 , 0 components via the Lˆ + Sˆ− or Lˆ − Sˆ+ terms, respectively: | | | | | | | | ”poly” ”poly” |⟨ψ1 , 1| Hˆ SOC |ψ3 , 0⟩| = |⟨ψ1 , −1| Hˆ SOC |ψ4 , 0⟩| = 2λ

(2.21)

|ψ1 , 1⟩

0

0

0

”poly” Hˆ SOC

⟨ψ1 , 1|

⟨ψ1 , 0|

⟨ψ1 , −1|

0

0

0

|ψ1 , 0⟩

0

0

0

|ψ1 , −1⟩

0

0

2λ

|ψ2 , 1⟩

0

0

0

|ψ2 , 0⟩

2λ

0

0

|ψ2 , −1⟩

0

0

0

|ψ3 , 1⟩

0

0

2λ

|ψ3 , 0⟩

0

2λ

0

|ψ3 , −1⟩

0

2λ

0

|ψ4 , 1⟩

2λ

0

0

|ψ4 , 0⟩

0

0

0

|ψ4 , −1⟩

Table 2.1 3 × 12 part of the SOCI matrix between the lowest-four solutions of the crystal field Hamiltonian within the “polyelectronic” framework (see text)

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| ⟩ Finally, the |ψ1 , 0⟩ function couples with the |ψ3,4 , ±1 ones: | | | | | | | | ”poly” ”poly” |⟨ψ1 , 0| Hˆ SOC |ψ3 , −1⟩| = |⟨ψ1 , 0| Hˆ SOC |ψ4 , 1⟩| = 2λ

(2.22)

All the other terms in the 3 × 12 part of the matrix vanish because of the inapplica”poly” bility of Hˆ SOC to couple such components by all the possible channels ( Lˆ z Sˆ z , Lˆ + Sˆ− , and Lˆ − Sˆ+ ). In practice, one thus has to deal with 6 nonzero couplings. Each |ψ1 , M S ⟩ component couples with two components of the first excited orbital triplet, and all the coupling are equal in absolute value. As a consequence, in the octahedron, the |ψ1 , M S ⟩ components remain degenerate, and their energies at second order of perturbations are all equal to: 8λ2 (2λ)2 ] = −720B4 − E[ψ1 , M S ] = E[ψ1 ] − 2 [ 600B4 E ψ2,3,4 − E[ψ1 ]

(2.23)

Of course, a symmetry lowering is needed to lift the degeneracy of the |ψ1 , MS ⟩ components, as for instance an axial distortion. One may approximate the solutions of the crystal field Hamiltonian by the solutions of the cubic field [51], which is correct up to the neglect of specific couplings between some of them that arise from the added terms in the crystal field Hamiltonian (see Fig. 2.2): ( ) ”poly” Hˆ CF = B4 Oˆ 40 + 5 Oˆ 44 + B20 Oˆ 20 + B40 Oˆ 40

(2.24)

If one considers the previous solutions of the | ⟩ cubic field, again the |ψ1 , MS ⟩ components are only coupled to the |ψ2,3,4 , MS ones, with the same SOC matrix elements as before. Also, the ψ2 and ψ3,4 roots are now split into an orbital[ singlet ] and an orbital doublet. If one introduces Δ0 = E[ψ2 ] − E[ψ1 ] and Δ1 = E ψ3,4 − E[ψ1 ], one may then express again the energies of the |ψ1 , MS ⟩ components: 4λ2 4λ2 − Δ0 Δ1 4λ2 8λ2 E[ψ1 , 0] = E[ψ1 ] − 2 = E[ψ1 ] − Δ1 Δ1 E[ψ1 , ±1] = E[ψ1 ] −

(2.25)

Finally, D may be expressed according to rearranged Eq. (2.10) [51]: D = E[ψ1 , ±1] − E[ψ1 , 0] = −

4λ2 4λ2 + Δ0 Δ1

(2.26)

Thus, it is clear that the use of a crystal field Hamiltonian finally gives rise to the definition of D, i.e., in fact supports the model Hamiltonian expressed in Eq. (2.2). Indeed, one can also introduce the rhombic E term by a further distortion (hence,

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Fig. 2.2 Splitting of the ground state, 3 F, of the free Ni2+ ion in cubic and tetragonal fields of the second and fourth degree. The displayed wave functions on the right are the solutions of the cubic field Hamiltonian

by introducing new terms in Hˆ CF ), and finally, computations in an arbitrary frame may also be done by using “extended” Stevens operators [60]. For the sake of pedagogy, it is worth mentioning that one can actually perform the same reasoning and also extend it naturally to the coupling of S to S ± 1 states by working in the framework of the “monoelectronic” SOC Hamiltonian (see Eq. 2.11) that will be called Hˆ S O in the following. For this purpose, one needs to express the wave functions of the states of interest in terms of the five d atomic orbitals (AOs). If one restricts the space to the previous one, four triplets must be introduced. By omitting the doubly occupied d AOs, the wave function of the M S = 1 component of the ground triplet of a high-spin octahedral nickel(II) complex (3 A2g ) reads: ”poly”

| ⟩ T10 = |dx 2 −y 2 dz 2

(2.27)

Wave functions for the three components of 3 T2g may then be expressed as [62]: | ⟩ T11 = |dz 2 dx y T12

√ ⟩ ⟩ 3 || 1 || dx 2 −y 2 d yz = − dz 2 d yz + 2 2

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T13

√ ⟩ 3 || 1 dx 2 −y 2 dx z = |dz 2 dx z ⟩ + 2 2

(2.28)

Actually, T11 results from the (single) excitation of an electron from the d xy orbital into the dx 2 −y 2 one, T12 is an excitation from d yz to d y 2 −z 2 (then transformed in terms of the dz 2 and dx 2 −y 2 AOs because one needs a unique basis to express all the states) and T31 is an excitation from d xz to dx 2 −z 2 . In Eqs. (2.27) and (2.28), only the maximum M S components of these states are expressed. Naturally, the other ones can be generated via Eq. (2.1). A prerequisite for applying Hˆ SO is the generation of a coupling table (Table 2.2) between the d spin AOs (again bars mean “down”; only half of it is written as the matrix is Hermitian) [63]: Out of the to be built 12 × 12 SOC matrix (one only needs to display a 3 × 12 one for the sake of applying second-order perturbation theory, as before), ten nonzero matrix elements are at play (Table 2.3). The indices represent the M S values. If T 1 , T 2 , and T 3 are degenerate (octahedral geometry), all the M S components of T 0 are effectively stabilized by a same quantity: 2ζ 2 [ ] ΔSOC = − [ 1,2,3 ] − E T0 E T

(2.29)

By taking the square of rearranged Eq. (2.12) (ζ 2 = 4S 2 λ2 with S = 1), it is easy to see that Eqs. (2.23) and (2.29) are compatible. Additionally, if one maintains

Table 2.2 5 × 10 part of the SOC coupling matrix between the various d spin-orbitals | ⟩ | | ⟩ | ⟩ | ⟩ | ⟩ ⟩ | ⟩ | ⟩ |dz 2 |dx 2 −y 2 |d yz |dx z ⟩ |dx y |d z 2 |d x 2 −y 2 |d yz |d x z Hˆ SO √ √ ⟨ | dz 2 | 0 0 0 0 0 0 0 i 23 ζ − 23 ζ | ⟨ i 1 dx 2 −y 2 | 0 0 0 0 −i ζ 0 0 2ζ 2ζ √ ⟨ | i d yz | 0 0 0 0 −i 23 ζ − 2i ζ 0 0 2ζ ⟨dx z | ⟨ | dx y |

√

0

0

− 2i ζ

0

0

0

iζ

0

0

0

3 2 ζ

0

| ⟩ |d x y 0 0 − 21 ζ

− 21 ζ

0

0

i 2ζ

0

1 2ζ

− 2i ζ

0

Table 2.3 3 × 12 part of the SOCI matrix between the spin–orbit-free states that are used within the “monoelectronic” framework (see text) Hˆ ”mono” T 0 T 0 T 0 T1 T1 T1 T2 T2 T2 T3 T3 T3 SOC

T10

1

0

T00

0

0 T−1

0

−1

0

0 0 0

0 0 0

1

iζ 0 0

−1

0

0

0

1

0

i √

0 0

0 −i ζ

i 0

2 2 ζ

−1

0 √

2 2 ζ

0

0

√

i

2 2 ζ

0 √

i 0

1

2 2 ζ

√

2 2 ζ

0

0

−

2 2 ζ

0 √

−1

√

2 2 ζ

0 − 0

√

2 2 ζ

2 Analytical Derivations for the Description of Magnetic Anisotropy …

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the wave functions while considering a symmetry lowering to an axially distorted system, only may derive the following equation: [ ] [ 0 ] ζ2 ζ2 − E T00 = − D = E T±1 + Δ0 Δ1

(2.30)

where Δ0 and Δ1 relate to the T 0 → T 1 and T 0 → T 2,3 excitations, respectively. Again, Eq. (2.30) is compatible with Eq. (2.26), meaning that working in both frameworks may lead to the same results if the nature of the states is the same (and of course, if the SOC operator is defined in such a way that Eq. (2.12) is valid, i.e., if a unique “monoelectronic” SOC constant is defined and if spherical approximations are used in both expressions of the SOC operator). Therefore, one may choose to work in the basis of the d AOs for the sake of simplicity and easier connection with experimentalists and also with actual computations.

2.2.1.3

Ab Initio Basis for This Model Hamiltonian: Computations and Application of the Effective Hamiltonian Theory

We have seen that the application of a crystal field theory allows us to derive the ZFS Hamiltonian. To go further in the demonstration of the validity of this Hamiltonian model, one may confront it with the results of ab initio calculations. Indeed, calculations allow one to check the tensorial character of the ZFS through the extraction of the ZFS tensor in various coordinate frames, of course provided that an extraction scheme has been defined. Such a scheme was proposed in our seminal work in 2009 [39]. It had two main advantages compared to previous second-order perturbation theory implementations, (i) we can check the quality of the model space by looking at the norms of the projections of the ab initio vectors onto the model space and (ii) we do not need to stick with a supposed model Hamiltonian; if necessary, we can refine the model Hamiltonian (which is constituted of model parameters and of ad hoc operators). The spin–orbit configuration interaction (SOCI) wave functions are complex. In our extraction scheme, we first start by defining a model space, for instance the three M S components of an S = 1 spin state. As mentioned earlier, this is meant to be correct for an orbitally non-degenerate ground state, e.g., for a distorted octahedral high-spin nickel(II) complex. The first step thus consists in projecting the ab initio wave function onto the model space, which in practice is done by simply truncating them to the M S components of the ground spin–orbit-free state (by construction, the three model vectors thus have the same orbital wave functions, which we actually omit within the spin Hamiltonian approach). In typical situations, the projection norms are within the 95–100% range (NB: computationally speaking, it depends on the size of the SOCI space).

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After this, we then build an effective Hamiltonian matrix [43] defined in such a way that its diagonalization leads to the “exact” ab initio energies and to the orthonormalized projected wave functions [44] (up to phase factors within the complex plane): des Cloizeaux = Hˆ effective

n | ⟩ ⟨ | Σ | ˜L | |ψk E k ψ˜ kL |

(2.31)

k=1

where n is the size of the model space (3 for the M S components of an S = 1 state), the E k ’s are the ab initio energies (and also the effective ones, by construction) and the “L” label stands for “Löwdin orthonormalized” [64] (performed by applying 1 S − 2 to the projected ψ˜ k vectors; note that this process requires two change-of-basis transformations, one to transform the projected vectors in the orthogonal basis, and the other to transform back the outcome of the actual orthonormalization operation to the initial–non-orthogonal–basis [65]). It is now time to present an example to illustrate the power of this approach. We will focus on a model of the [Ni(HIM2 -Py)2 NO3 ]+ (HIM2-py = 2-(2-pyridyl)-4,4,5,5tetramethyl-4,5-dihydro-1H-imidazolyl-1-hydroxy) complex [66] (see Fig. 2.3), as in previous presentations [39, 67]. After performing a reference SOCI calculation [39], the following projected vectors were obtained: | ⟩ | 0 ⟩ | ⟩ |˜ − (0.668 − i0.724)|T00 |ψ1 = (0.045 + i0.092)|T−1 | 0 ⟩ + (0.096 + i0.037)|T+1 Fig. 2.3 Ball-and-stick representation of a model of the [Ni(HIM2 -Py)2 NO3 ]+ complex and its main magnetic axes. Ni in light blue, C in gray, O in red, and N in dark blue. The “external” methyl groups have been modeled by hydrogen atoms; all hydrogen atoms are omitted for clarity. Reprinted with permission from [39]; copyright 2009 American Chemical Society

2 Analytical Derivations for the Description of Magnetic Anisotropy …

| ⟩ | ⟩ | 0 ⟩ |˜ + (0.062 + i0.088)|T00 |ψ2 = −(0.395 − i0.578)|T−1 | 0 ⟩ − (0.678 − i0.173)|T+1 | ⟩ | ⟩ | 0 ⟩ |˜ − (0.090 + i0.037)|T00 |ψ3 = (0.701 + i0.026)|T−1 | 0 ⟩ − (0.519 + i0.472)|T+1

79

(2.32)

The projection norms are 99.1, 99.1, and 99.4%, respectively, which already validates the choice of the model space. These projected vectors are not perfectly orthogonal, as expected. After performing the Löwdin orthonormalization, the following vectors are generated: | ⟩ | ⟩ | 0 ⟩ |˜L − (0.671 − i0.727)|T00 |ψ1 = (0.045 + i0.093)|T−1 | 0 ⟩ + (0.096 + i0.037)|T+1 | ⟩ | ⟩ | 0 ⟩ |˜L + (0.062 + i0.089)|T00 |ψ2 = −(0.396 − i0.581)|T−1 | 0 ⟩ − (0.681 − i0.173)|T+1 | ⟩ | ⟩ | 0 ⟩ |˜L − (0.090 + i0.037)|T00 |ψ3 = (0.703 + i0.026)|T−1 | 0 ⟩ − (0.520 + i0.474)|T+1

(2.33)

The ab initio energies are 0.000, 1.529, and 11.396 cm–1 , respectively. We now have all the ingredients to build the effective Hamiltonian matrix: (

des Cloizeaux Hˆ effective

)

⎛

⎞ 6.386 −0.690 + i0.376 −3.734 + i3.134 = ⎝ −0.690 − i0.376 0.125 0.690 − i.0.376 ⎠ −3.734 − i3.134 0.690 + i.0376 6.386 (2.34)

This matrix has to be compared with Eq. (2.5). Element-to-element comparisons show that the ZFS Hamiltonian captures all the features of the effective Hamiltonian (all the terms that are meant to be equal are equal, the ones that are meant to be opposite are opposite, the ones that are meant to be complex conjugates are complex conjugates, etc.). It is interesting to note that this example is quite general in the sense that the axis frame is arbitrary and that no symmetry element constraint some of the ZFS tensor elements to display special relationships (such as equality). One can then extract the ZFS tensor components by solving the systems of equation that arise by imposing Eqs. (2.5) and (2.34) to be equal. The resulting tensor is expressed in the arbitrary axis frame that was used for the ab initio calculation: ⎛

⎞ −3.671 3.134 0.976 (D) = ⎝ 3.134 3.797 −0.532 ⎠ 0.976 −0.532 6.323

(2.35)

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Diagonalization of this tensor leads to the magnetic anisotropy axes (the ones of the ZFS tensor, which may not coincide in the general case with the ones of g) and to its expression in this frame (conventions may apply, see Eq. (2.8) and related explanations): (

D

) diag

⎞ 6.448 0 0 = ⎝ 0 4.920 0 ⎠ 0 0 −4.919 ⎛

(2.36)

Finally, by applying Eqs. (2.6) and (2.7), one can get the usual D and E parameter values (here, D = –10.604 cm–1 and |E| = 0.764 cm–1 ). Also, the tensorial character of (D) may be verified by repeating the ab initio calculations and the entire extraction procedure in this new coordinate frame. In this case, (D) will almost perfectly correspond to the previous (D diag ), meaning that the extracted tensor indeed transforms as a tensor with respect to transformations of the coordinate frame. Also, this guarantees that the outcomes of our calculations are not affected by the initial choice of the coordinate frame (as expected!). In this example, the model Hamiltonian perfectly reproduced all the features of the effective Hamiltonian, which is in fact the ideal situation. If not, two possibilities arise depending on the magnitudes of the observed deviations and of the accuracy that is targeted (and even reachable depending on the quality of the reference computations). If the observed deviations are thought negligible, one can choose to neglect them and maybe the error done by the model can be assessed. If the deviations cannot be neglected, one may refine the model Hamiltonian up to capturing most if not all the features of the effective Hamiltonian.

2.2.1.4

Generalization to Larger Spins

In the previous example, we were considering an S = 1 ground state. Thinking of mononuclear d complexes, only three other possibilities arise: S = 23 , 2 or 25 . If S = 23 , the same Hamiltonian as before applies (see Eq. 2.2). Naturally, Kramers’ doublets are observed, and this model Hamiltonian fully respects it. In practice, this means that one cannot shortcut the determination of the D and E parameters by simply looking at the ab initio energies [37]. The energy difference between the two lowest Kramers’ doublets effectively involves both parameters [19]: √ Δ K 1 −K 2 = 2 D 2 + 3E 2

(2.37)

As a consequence, the effective Hamiltonian theory is even more powerful in this case, because it notably leads to the univocal determination of the D sign, if applicable (i.e., if we are not in the pathological “rhombic limit” for which |D| = 3|E|). For S ≥ 2, effective terms of higher-rank may pop up in the Hamiltonian, as for instance higher-rank Stevens operators. These terms are in practice larger in the case

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81

of near-orbital degeneracy. For instance, we have studied in the past nearly octahedral high-spin (S = 2) Mn(III) model complexes and derived analytical equations for the higher-rank B40 and B44 terms involving the energy difference between the nearly degenerate quintet states (in the denominator) [53]. However, it is clear that the effective Hamiltonian theory can be used to extract these terms, directly in the correct axis frame (it is easy for symmetric systems and in particular the studied model complexes) or in a less direct one in an arbitrary frame (one first needs to determine the extended Stevens parameters in this frame and then transform the parameters in the correct frame). Finally, it should be mentioned that the S = 5/2 case conceptually mixes the difficulties of the two previous cases, (i) it leads to Kramers’ doublets as in the S = 3/2 situation and (ii) it may require the introduction of fourth-rank Stevens terms, as in the S = 2 case. However, if we treat a pure mononuclear d complex, a typical S = 5/2 state is not expected to exhibit near-orbital degeneracy and one can then simply apply (at least in the first instance) Eq. (2.2).

2.2.2 Example of Rationalization of the ZFS As already mentioned, if the energy gap between the ground and excited states coupled by SOC is large, a perturbative evaluation of the stabilization of the M S components of the ground state may be accurate. At the second order of perturbation, these stabilizations are proportional to: | ⟩2 ⟨S, Ms | Hˆ SO | S , , MS, ΔE

(2.38)

| ⟩ where ⟨S, Ms | Hˆ SO | S , , MS, is the SOC coupling between the M S component of the electronic ground state S and the MS, component of an excited state S , and ΔE the energy difference between these electronic states. Consequently, tuning the anisotropy parameters | of the ⟩ giant-spin model Hamiltonian is possible by modulating either ⟨S, Ms | Hˆ SO | S , , MS, , or ΔE or both terms simultaneously. We first present a series of Fe(II) complexes where D can be significantly modified by changes of the magnetic center coordination sphere that affect ΔE. Next, we show how changing the symmetry point | group⟩ (SPG) of Co(II) and Ni(II) complexes can significantly affect ⟨S, Ms | Hˆ SO | S , , MS, and even change the sign of D. 2.2.2.1

Tuning D by Changing ΔE

A study of a series of three Fe(II) complexes [68] illustrates how D may be tuned by changing the energy gap between the ground and excited states. These complexes are formed from N–N, -chelating amino-pyridine ligands. The differences between

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the ligands involve groups that are far from the magnetic center. At first glance, one may therefore think that the coordination sphere is identical in the three complexes. However, as the steric hindrance is different, so are the inter-molecular interactions, which leads to a different crystal structure that reflects on the structure of the Fe coordination sphere (see Figs. 2.4 and 2.5) and changes the value of the D parameter (see Table 2.4).

Fig. 2.4 Representation of the three complexes (1 on the left, 2 in the center, and 3 on the right) aligned along the magnetic axes, X in red and Y in green. Fe is in brown, N in red, C in black, H is not represented

Fig. 2.5 Angles and distances of the Fe coordination sphere in complexes 1 (green), 2 (red), and 3 (blue)

2 Analytical Derivations for the Description of Magnetic Anisotropy … Table 2.4 Experimental and theoretical D values for the three Fe(II) complexes. CAS(6/10)PT2 uses the CASPT2 method implemented in MOLCAS while CAS(6/10)NEVPT2 uses the NEVPT2 method implemented in ORCA

cm−1

83

Complex 1

Complex 2

Complex 3

D

D

D

CAS(6/10)PT2

−10.7

−16.6

15.8

CAS(6/10)NEVPT2

−13.7

−19.3

18.2

Experiment

−12.3

−16.9

16.5

The energy of the five excited quintuplet states and a perturbative estimate of their contribution to D are given Fig. 2.6 for the three complexes. These results clearly show that most of the magnetic anisotropy comes from the coupling with the first excited state and, to a lesser extent, with the second excited state. The wave functions of the ground state and of the first and second excited states are almost mono-configurational. In the ground state, the dx 2 −y 2 orbital is doubly occupied while the dx y , dx z , d yz , and dz 2 are singly occupied. In the first excited state, the doubly occupied orbital is dx y while in the second excited state it is dx z . Figure 2.7 schematizes the interactions between the different M S components of the ground state and the first two excited states as well as the resulting stabilization of the M S components of the ground state. It clearly shows that the interactions with the first excited state that comes from the lˆz sˆz part of the SOC operator contribute negatively to D and lift the degeneracy between the ground state components much more than the interaction with the second excited state. Concerning the latter, it has a positive contribution to D that comes from the lˆ+ sˆ− + lˆ− sˆ+ part of the SOC operator. Indeed, the interaction with the first excited state stabilizes the M S = ± 2

Fig. 2.6 Energy spectrum and contributions to D of the excited S = 2 states in Fe(II) complexes

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Fig. 2.7 Scheme of the interactions between the ground state M S components and the M S components of the first (left) and the second (right) excited states. In black energies before SOC (thousands of cm−1 ), in red, SO couplings (dashed arrows) and in blue energy stabilization after SOC (few cm−1 or tens of cm−1 ) 1 ξ2 , respectively, while the interaction with the 4 ΔE 1 1 ξ2 5 ξ2 3 ξ2 and 16 the M S = ± 2, ± 1 and state stabilizes by 16 ΔE2 , 32 ΔE 2 ΔE 2 respectively (ΔE 1 and ΔE 2 are the energy differences between the

and ± 1 components by

ξ2 ΔE 1

and

second excited 0 components, ground state and the first and second excited states). Note that the interaction with the first excited state, via the lˆz sˆz part of the SOC operator, does not generate any contribution to the rhombicity E; i.e., they do not couple the different M S components of the ground state contrarily to the interactions with the second excited state. From the second-order perturbative derivation, the contribution to D decreases when ΔE 1 increases. The ground-to-excited state excitation mainly corresponds to an excitation from the dx 2 −y 2 to the dx y orbital (see Fig. 2.8). It is possible to connect ΔE 1 to the angle θ between the pyridine planes (see Fig. 2.4). When θ is small, dx y experiences a greater destabilization from the ligand field than dx 2 −y 2 . Consequently, when θ increases (from 0° to 16° and 23° for complexes 1, 3, and 2, respectively) the energy gap between these orbitals and thus ΔE 1 decreases (1840, 1450, and 1340 cm−1 ) reinforcing the negative contribution to D (−12.3 to −16.5 and −16.9 cm−1 ) respectively for complexes 1, 3, and 2. In conclusion, the origin of the D change in this series of complexes is an angular changes θ which affects the crystal field and therefore the orbital energy. Few recent examples of the application of this type of approaches on coordination complexes with or without transition metals, lanthanide complexes and organic molecules can be found in the literature [69–75].

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85

Fig. 2.8 dx 2 −y 2 (left) and dx y (right) orbitals involved in the ground-to-first-excited-state excitation. dx 2 −y 2 is doubly occupied and dx y singly occupied in the ground state, the occupations are the opposite in the first excited state. The drawing is done for complex 2

2.2.2.2

Tuning D by Changing the Symmetry Point Group

In the free Co(II) ion, selection rules impose that SOC is only possible between 4 F ground term components and between 2 F and 2 G components (i.e., 4 P components are not coupled with 4 F ones and couplings with other terms, very high in energy, are neglected). In complexes, this feature is largely maintained and states that mainly interact with the lowest spin state are those related to the atomic 4 F and 2 G states while interactions with states coming from 4 P are much weaker. Moreover, for a given SPG, states that may interact through SO are those for which the tensor product of the irrep of both states and with at least the irrep of either Lˆ x , Lˆ y , or Lˆ z contains the totally symmetric irrep. Since Lˆ x , Lˆ y , or Lˆ z pertain to the same irrep as Rx , Ry, and Rz , respectively, SPG character tables are very useful to anticipate the main SOC couplings [61]. For example, for a D3h Co(II) complex, the main contribution to the ZFS of the 4 A,2 ground state components is expected to come from the coupling with the excited 4 E ,, state (dx z → dz 2 or d yz → dz 2 excitations) and, to a lesser extend from the coupling with 2 A,1 (dx 2 −y 2 ↔ dx y excitations). In C3v, the lowering of the symmetry induces that the 4 A2 ground state may interact by SO with 4 E (dx z → dz 2 or d yz → dz 2 excitations), 2 A1 (dx 2 −y 2 ↔ dx y excitations) but also with 4 A1 (dx z , d yz ↔ dx y , dx 2 −y 2 excitations). Finally, it is also possible to show that dx 2 −y 2 ↔ dx y or dx z ↔ d yz excitations provide a negative contribution to D if they occur between states of the same spin multiplicity and a negative contribution if the spin multiplicities are different. On the contrary, dx 2 −y 2 , dx y ↔ dx z , d yz and dz 2 ↔ dx z , d yz have a positive contribution to D for couplings between states of the same multiplicity and a negative one for couplings between states of different multiplicities. In this section, we illustrate the impact of the SPG on the magnitude and sign of D by studies of real or model complexes (that are particularly interesting to emphasize

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the role of the mechanisms at work in real complexes) of D3h , C3v, or D3d SPG. In the first example presented below, starting from a D3h complex, C3v structures are built by slight changes of angles of the coordination sphere of a Co(II) magnetic center. We will then show that a similar reasoning applies to a real quasi-D3h Ni(II) complex. The first calculations are performed on the high-spin [Co(NCH)5 ]2+ model complexes [61], (see Fig. 2.9). In all structures, the NCH ligands are bound to the Co center by the N atom, the Co–N distances are fixed to 2 Å and two ligands are along the vertical Z-axis. For the D3h symmetry, the three other ligands are in the equatorial plane, the NCH ligands are aligned with the Co ion, and the N–Co–N angles are 120°. Starting from this structure, C3v structures are obtained by applying a progressive deformation angle (3°, 6°, and 9°) between the three equatorial ligands and the xOy plane. In order to “break even more” the D3h symmetry, another study is performed on the complex [Co(NCH)4 Cl]+ where one of the apical ligand is substituted by a Cl− ion and where the deformation angle is 9°. This last model complex has a coordination sphere very similar to the one of the [Co(Me6 tren)(Cl)]+ complex which D value is around −9 cm−1 , [68] similar to the calculated one. The main contributions to D of the excited states are reported in Table 2.5. The energy gaps between the ground state to the three excited states increase with the deformation. Indeed, they are around 2800 cm−1 , 5200 cm−1 , and 22,000 cm−1 at 0° (first and third excited triplet states and first excited doublet state, respectively), 3800 cm−1 , 6300 cm−1 , and 23,000 cm−1 at 9° for [Co(NCH)5 ]2+ and 3900 cm−1 , 6500 cm−1 , and 23,500 cm−1 for [Co(NCH)4 Cl]+ . As expected from the reasoning exposed above, for the D3h complex, the coupling of the ground state with 4 E ,, is positive and twice larger than with 2 A,1 while there is no coupling with 4 A,,1 . For C3v complexes, a negative contribution of the coupling with 4 A1 appears while couplings with 4 E ,, and 2 A1 provide a positive contribution. In the D3h SPG, dx z and d yz belong to a different irrep than dx 2 −y 2 and dx y while they all belong to the same irrep in C3v (see | state | Fig. 2.9).| Then,| the ground wave function is only a combination of the |dx 2 −y 2 dx y d|z 2 | and |dx z d|yz d| z 2 | config| urations (for the M S = 3/2 component) in D3h while the |dx 2 −y 2 dx z dz 2 |, |dx z dx y dz 2 |,

Fig. 2.9 Scheme of the energy and symmetry of the MO in D3h and C3v SPG

2 Analytical Derivations for the Description of Magnetic Anisotropy …

87

Table 2.5 Contribution to D (cm−1 ) of the first (4 A,,1 in D3h and 4 A1 in C3v ) and third (4 E ,, /4 E) excited quartet states and of the first excited doublet state (2 A,1 /2 A1 ) obtained at the CAS(7,5)SCF level of calculations. At 0° the SPG is D3h , it is C3v for other structures D value Contribution to D

4 A,, /4 A 1 1 4 E ,, /4 E 2 A , /2 A 1 1

0°

3°

6°

9°

[Co(NCH)4 Cl]+

+ 34.6

+ 20.7

+ 7.6

+ 0.1

− 12.8

0

− 11.4

− 18.9

− 21.9

− 28.2

+ 26.3

+ 22.4

+ 18.8

+ 16.4

+ 13.7

+ 13.0

+ 10.8

+ 8.2

+ 5.6

+ 6.9

| | | | |dx 2 −y 2 dyz dz 2 | and |d yz dx y dz 2 | configurations appear in the M S = 3/2 component of the ground state in C3v symmetry. The intensity of the SO interactions between the ground state and the excited states can be evaluated from the coefficients of these configurations. For the D3h structure, the ground state is only coupled with the 4 E ,, state via the lˆx sˆx and lˆy sˆy operators (positive contribution to D) and, to a lesser extent, with the 2 A,1 state via the lˆz sˆz operators (positive contribution). When the structure is distorted to C3v , a coupling with 4 A1 through lˆz sˆz operators appears, leading to a negative contribution to D. The more the structure of the complex is far from the D3h one, the larger the weight of the configurations coupled by lˆz sˆz in the ground state and 4 A1 and then the larger the negative contribution to D. The impact of the removal of a symmetry element can be also illustrated by the evolution of D in the hexacoordinated mononuclear complex of [Ni(tacn)2 ]2+ (tacn = triazacyclononane, see Fig. 2.10) [76]. Also in this case, a fine analysis of the wave function allows to rationalize the intensity of the interactions at work and their evolution with the change of the structure from a quasi-D3d to a quasi-D3h structure. For this purpose, starting from the crystallographic structure, several structures were obtained by rotating one of the two ligands around the pseudo-C3 axis, the angle θ changing from 60° (experimental quasi-D3d structure) to 45°, 30°, 15° (quasi-C3v structures) and 0° (quasi-D3h structure). Then, CAS(8,5)SCF + NEVPT2 calculation [77, 78] were performed to extract the D and E values. Results are summarized in Tables 2.6, 2.7 and 2.8. For the quasi-D3h structure (θ = 0°), among the ten triplets generated by all d-d excitations, two excited states (T 3 and T 4 ) provide the largest contribution, by far, to D (see Table 2.7). Indeed, since {dx z , d yz } and {dx 2 −y 2 , dx y } pertain to different irrep. (e,, and e, , resp.), the ground| state |wave function is mostly singly configurational with of 0.96 on |d yz dx z | (unpaired electrons in d yz and | coefficients | | | dx z ) and 0.24 on dx 2 −y 2 dx y (unpaired electrons in dx 2 −y 2 and dx y ), see Table 2.8. These two configurations cannot couple to any other through lˆz sˆz , all contributions ˆ ˆ to D are thus positive. Considering | | the l x|sˆx and |l y |sˆy operators, | | they can | | couple the | 0 | | | | | | | | | | 2 2 d d d d T ( d ) with d , d , d main configuration of yz x z yz x y yz x −y x y x z , dx 2 −y 2 dx z , | | |d yz dz 2 |, and |dz 2 dx z |. Since these configurations appear with large coefficients in the four lowest excited states (Table 2.8), one may expect a large positive contribution to D of all these states while the contributions of T 1 and T 2 are actually quite small.

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Fig. 2.10 [Ni(tacn)2 ]2+ complex. Real D3d (left and center) and model D3h (right) complexes. The pseudo-C3 axis passes by the center of the tacn ligands. Ni in blue, N in red, C in gray, H are not represented Table 2.6 CAS(8,5)SCF evaluation of the D parameter in the [Ni(tacn)2 ]2+ for different angles (see text) θ D

(cm−1 )

60° (D3d )

45°

30°

15°

0° (D3h )

4.3

5.8

12.7

34.4

73.6

Table 2.7 Energies (cm−1 ) and contributions to D of the four lowest excited triplet states in the real quasi-D3d [Ni(tacn)2 ]2+ complex and in the quasi-D3h model complex D3d struct D3h struct

T1

T2

T3

T4

Energy

13,376

13,523

13,735

20,581

Contrib. to D

+ 13.5

+ 7.3

− 15.9

0.0

Energy

7487

7534

8186

8223

Contrib. to D

+ 4.0

+ 0.8

+ 35.4

+ 34.7

Table 2.8 Wave functions of the M s = 1 components of the lowest triplet states for the D3d and D3h (in bold) complexes. Dashed lines are represented when coefficients are very small D3d /D3h | | |d yz dx z | | | |dx 2 −y 2 dx z | | | |d yz dx y | | | |dx y dx 2 −y 2 | | | |d yz dx 2 −y 2 | | | |d yz dz 2 | | | |dx 2 −y 2 dz 2 | | | |d x y d z 2 | | | |d z 2 d x z | | | |d x y d x z |

T0

T1

T2

T3

0.70/0.96

T4 0.65/–

0.43/–

– /−0.64

0.30/−0.19

−0.51/−0.54

−0.46/0.27

0.38/–

– /−0.60

−0.60/0.35

– /0.22

−0.50/−0.53

0.66/−0.43

– /−0.64

−0.41/−0.40

0.26/0.23

– /−0.43

– /−0.40

0.52/0.08

−0.31/−0.42

– /0.44

0.36/0.65

0.66 / 0.37

– /0.49

−0.31/−0.27 −0.55/0.19 −0.36/– 0.27/–

−0.29/–

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Indeed, these two contributions almost vanish due to the cancelation of components because of the sign of the coefficients in the wave functions. This is not the case in the third and fourth excited states for which a very large contribution of about + 35 cm−1 is observed. For the quasi-D3d structure, due to the lower symmetry, all the dx z , d yz , dx 2 −y 2 and dx y MO pertain to the same eg irrep, the ground state wave function is much more complex with four major configurations (Table 2.8). Then, SOC couplings are possible with many others configurations which appear in the first four excited states but in fact only the first three excited states have a non-negligible contribution to D (as expected from symmetry arguments since D3d is a subgroup of Oh , see Sect. 2.2.1). However, in D3d , these three states provide contributions to D that mostly compensate leading to a small D value.

2.2.3 Approaching First-Order SOC A promising approach to obtain complexes with a very strong D comes for complexes with an almost degenerate ground state, more precisely complexes with a ground state strongly coupled to an almost degenerated excited state. This is the case of the penta-coordinated [Ni-(Me6 tren)Cl]+ complex (see Fig. 2.11) described below where an extremely large D value of −180 cm−1 was measured (High-Field HighFrequency EPR HF-HFEPR experiments) and where a competition between Jahn– Teller distortions and SO interactions occurs [79]. X-ray diffraction experiments suggest that the structure of the complex presents a C3 SPG. Then, dx 2 −y 2 and dx y (Z being the Ni–Cl axis) would be degenerate and the ground state would be degenerate (either one electron in dx 2 −y 2 and dz 2 , or one electron in dx y and dz 2 ), leading to the occurrence of Jahn–Teller distortions. Fig. 2.11 [Ni-(Me6 tren)Cl]+ complex. Ni in blue, N in red, C in black, Cl in green, and H in pink

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Fig. 2.12 Low-energy spectrum of the Ni(II) complex computed at the two states SO-CAS (8,10)PT2 level of correlation. The horizontal axis represents a linear distortion between the C3 (0) and DFT C1 minimum (100). Reprinted (adapted) with permission from J. Am. Chem. Soc. 2013, 135, 8, 3017–3026. Copyright 2013 American Chemical Society

A unrestricted DFT (UDFT) geometry optimization show that Jahn–Teller effects split the C3 structure into three equivalent C1 equilibrium structures connected by low-energy transition states. The energy of these transition states is around 75 cm−1 above the equilibrium structure after considering the zero-point energy (ZPE). This very weak energy barrier explains why X-ray experiments, determined at 100 K, i.e., above the energy barrier, are modeled by a C3 average structure. SOC plays a role antagonist to the Jahn–Teller distortions since their stabilization effect is maximum for a degenerate ground state and then decreases along the C3 –C1 deformation. This competition was evaluated by CAS(8,10)PT2 + SO calculations for structures linearly interpolated between the experimental C3 and equilibrium C1 UDFT structures (see Fig. 2.12). The minimum energy is observed for a structure almost halfway between the C3 and C1 ones for SO-CAS(8,10)SCF calculations (involving interactions only between the two almost degenerate states) and 20–80% of C3 –C1 structures for SO-CAS(8,10)PT2, i.e., closer to the DFT minimum. In any case, two Ni–N bonds are larger and the third one is almost the same as in the C3 structure (the X-axis is along this Ni–N axis). A model Hamiltonian, more sophisticated than the model Spin Hamiltonian used so far in this text and adapted to the description of the behavior of the complex when its ground state is (almost) degenerate, is given in Table 2.9. It is based on configurations differing by their space part (unpaired electrons in dx 2 −y 2 and dz 2 , or in dx y and dz 2 ) and their M S value (+1, 0 and −1), leading to a 6 × 6 Hamiltonian. Note that it is expressed in terms of configurations (the electronic Hamiltonian is not diagonal in this basis). It considers the evolution with the structure of the following mechanisms: (i) the ligand field Δ parameter, identical for the two configurations for a given structure; (ii) the Jahn–Teller stabilizing/destabilizing | | | effects (δ| 1 parameter, in fact also originating from the ligand field) on the |dx y dz 2 |/|dx 2 −y 2 dz 2 | configurations (and on the corresponding M S = 0 and −1 configurations); (iii) the electronic + SO coupling (δ2 ± i ζ ) between configurations of the same Ms value (ζ is the spin–orbit constant of Ni2+ reduced by the ligand field).

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Table 2.9 Representative matrix of the model Hamiltonian suited for a degenerate ground state of the [Ni-(Me6 tren)Cl]+ complex. d orbitals that do not appear in the determinants are doubly occupied |dx y dz 2 |dx 2 −y 2 dz 2 δ2 + i ζ

|dx y dz 2 −dz 2 dx y √ 2

|dx 2 −y 2 dz 2 −dz 2 dx 2 −y 2 √ 2

|dx y dz 2 |dx 2 −y 2 dz 2

0

0

0

0

d x y dz 2 |

−Δ + δ1

dx 2 −y 2 dz 2 |

δ2 − iζ −Δ − δ1

0

0

0

0

dx y dz 2 −dz 2 dx y | √ 2

0

0

−Δ + δ1

δ2

0

0

dx 2 −y 2 dz 2 −dz 2 dx 2 −y 2 | √ 2

0

0

δ2

−Δ − δ1

0

0

d x y dz 2 |

0

0

0

0

−Δ + δ1

δ2 − i ζ

dx 2 −y 2 dz 2 |

0

0

0

0

δ2 + iζ −Δ − δ1

δ1 and δ2 are zero for the C3 structure and increase with the distortion toward the UDFT structure. The Δ value is in fact not very relevant here since it is extracted as the mean energy of the six energy levels (it thus only depends of the arbitrary zero of energy). SO-CAS (8,5)PT2 evaluation gives δ1 = 299 and δ2 = 645 cm−1 for the halfway structure and δ1 = 186 cm−1 and δ2 = 485 cm−1 for the 20–80% structure. For both, ξ is similar (around 630 cm−1 ). Finally, evaluations of D were performed. Since contributions to D of other d-d excited states may be important, SO-CAS (8,10)PT2 calculations were performed for all the 10 triplet and 14 singlet d-d states. D values ranging between −100 cm−1 and −200 cm−1 are obtained, in agreement with the very large HF-HFEPR value (−180 cm−1 ). The best-computed magnetization versus magnetic field and susceptibility versus temperature curves, compared to experiment, are obtained for the halfway structure where D = −152 cm−1 . Therefore, the first-order Jahn–Teller effect is here a key point since it leads to a sufficiently strong removal of the orbital degeneracy in such a way that the spin Hamiltonian approach becomes relevant while not too strong, keeping the interest of the near-orbital degeneracy to maintain an exceptionally large D value.

2.3 Binuclear Complexes 2.3.1 Multi-spin and Giant-Spin Model Hamiltonians Two model Hamiltonians can be used to describe polynuclear systems: multi-spin and giant-spin Hamiltonians. In the former model, all states resulting from the interactions between the local spins are generated while in the latter only the ZFS of the ground

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state is treated. The basis on which the multi-spin Hamiltonian is spanned consists of the product states |S A , Ms A ⟩ |S B , Ms B ⟩ . . . where SI is the pseudospin of center I (called spin in the following for simplicity) and all magnetic pseudospin quantum numbers MsI are considered. In the case of a binuclear system constituted of two spins S A and S B , the multi-spin Hamiltonian is written: Hˆ Multi-Spin = J AB Sˆ A . Sˆ B + Sˆ A . D A A . Sˆ A + Sˆ B . D B B . Sˆ B + Sˆ A . D AB . Sˆ B + d AB . Sˆ A × Sˆ B

(2.39)

where J AB is the isotropic exchange between the two magnetic centers, i.e., the spin coupling of the Heisenberg Dirac van Vleck Hamiltonian [80–83], D I I is the local anisotropy tensor of center I (A or B), D AB is the symmetric tensor of anisotropic exchange, and d AB the Dzyaloshinskii–Moriya (DM) vector [55, 56]. The last term of Eq. (2.39) can be written as Sˆ A .T AB . Sˆ B , where T AB is an antisymmetric tensor and is usually referred as the antisymmetric exchange anisotropy. If one does not consider N-body (N > 2) operators, this model is usually generalized for polynuclear systems having more than two centers by simply summing local anisotropies of each center and intersite interactions on interacting pairs of spins. It must be noted that the spatial part of the wave functions described using spin Hamiltonians is assumed to be common to all states and is therefore factorized so that only the spin degrees of freedom are considered. Of course, as it will be shown in the section dedicated to the DM interaction (DMI), the spatial part is never identical in the various spin states and it is sometimes necessary to derive the models using different sets of orbitals for different states. The alternative giant-spin model is identical to the Hamiltonian used for mononuclear systems: ˆ D. Sˆ Hˆ Giant-Spin = S.

(2.40)

It is expressed in the |S, Ms ⟩ coupled basis. As in the case of mononuclear species, it can be extended with the Stevens operators for spins S greater than or equal to 2. It can be safely used if we restrict the study to the ZFS of the ground state and if we neglect its mixing with excited states, i.e., in the strong exchange limit and disregarding the DMI. One may however note that a block-spin Hamiltonian (see Sect. 2.3.3) that sums this operator on all spin states can be derived to treat the ZFS of several states.

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2.3.2 Anisotropy of Exchange for Binuclear Centers of Spin S = 1/2 2.3.2.1

Model Hamiltonian Matrices in the Case of S = 1/2 Centers

In the case of a binuclear system of spin S = 1/2, there is no local anisotropy and the Hamiltonian writes: Hˆ Multi-Spin = J AB Sˆ A . Sˆ B + Sˆ A . D AB . Sˆ B + d AB . Sˆ A × Sˆ B

(2.41)

In an arbitrary axis frame, the symmetric tensor matrix is non-diagonal: ⎞ AB AB D D DxAB x xy xz ⎜ AB AB ⎟ ( D AB ) = ⎝ DxAB y D yy D yz ⎠ AB AB DxAB z D yz Dzz ⎛

(2.42)

and the DM vector has potentially three nonzero components: d x , d y, and d z along the x-, y-, and z-axes. When we derive the DM vector components from the spin–orbit couplings, it appears that this interaction is strictly zero for a common set of magnetic orbitals for the singlet and triplet states [84]. We will call a and b the orthogonalized local magnetic orbitals, i.e., bearing the unpaired electrons, of the triplet state and a’ and b’ those of the singlet. The model Hamiltonian matrix expressed in the coupled basis, for which S0 , T0 , and T{±1 stand| for the ⟩singlet and triplet M| S = 0⟩and M S = ± }1 | ⟩ | , , , a, | ab−ba |ab √ |ab⟩; T ; T = T = = spin components, i.e., in the S0 = | a b√+b | 1 0 −1 2 2 basis, writes:

(

ˆ Multi-Spin

H

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

)

⟨ ⟨ab| | ¯ a¯ | a b−b √ | = ⟨ 2| | | ⟨ , ab , +b, a , | a b√ | 2

T1 = |ab⟩

| ⟩ | ¯√ a¯ T0 = | a b−b 2

D AB J AB + 4zz 4 AB DxAB J AB z +i D yz √ 4 2 2 AB AB AB ( Dx x −D yy +2iDx y ) 4 d y −i d x √ 2 2

AB DxAB z −i D yz √ 2 2 D AB ( D AB +D AB ) − 4zz + x x 4 yy D AB +i D AB − x z 2√2 yz d y −i d x √ 2 2

| , ⟩ | ⟩ | , , a, T−1 = |ab S0 = | a b√+b 2 ( DxABx −D yyAB −2iDxABy ) d y +i d x √ 4 AB DxAB z −i D yz √ 2 2 D AB J AB + 4zz 4 D AB − 3J4AB − 4zz

−

⎞

2 2

− id2z dy −idx √ 2 2 AB D ( +D AB ) − x x 4 yy

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(2.43)

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One may note that the DMI only couples the singlet with the three M S components of the triplet. The magnetic exchange J AB may be affected by the spin–orbit effects; i.e., the D AB tensor matrix is not traceless, and the trace is a contribution to the isotropic exchange. Diagonalizing the tensor matrix (i.e., working in the proper axis frame of the tensor) and subtracting the trace, one has access to the axial DAB and rhombic E AB parameters. Both the Spin–Spin Coupling (SSC) and the SOC can contribute to the anisotropy. The SSC is particularly weak but it must be accounted for in cases where the SOC is also very weak [42, 85]. In the following derivation, we will only consider the contributions of the SOC.

2.3.2.2

Symmetric Tensor of Anisotropy (Illustrated on Copper Acetate S = 1/2 Centers)

The appearance of DMI is subject to symmetry conditions that will be discussed in the next subsection. Let us now focus on systems for which it is strictly zero. In such a case, the only source of anisotropy comes from the symmetric tensor symmetric tensor of exchange. The multi-spin Hamiltonian is therefore limited to the two first terms of Eq. (2.42). The matrix representative of this model is identical to that given in Eq. (2.43) except that it has zero elements between the singlet and the components of the triplets. Before deriving analytically the interaction of the symmetric exchange tensor, let us note that the multi-spin and giant-spin Hamiltonian are intrinsically related. Indeed, comparing the Hamiltonian matrix expressed in the uncoupled basis (see matrix 2.43) and that in the coupled basis (see matrix 2.5), which writes: T1 = |ab⟩ (

⟨ab|

⎛ Giant-Spin | ˆ ⟨ H = ab−ba | ⎜ √ | ⎜ ⎝ 2 )

⟨ | ab|

Dzz 2 Dx z +i D yz √ 2 ( Dx x −D yy +2i Dx y ) 2

| ⟩ | ⟩ | √ T−1 = |ab T0 = | ab−ba 2 Dx z −i D yz √ 2 D +D Dzz − 2 + ( x x 2 yy ) D +i D − x z√2 yz

( Dx x −D yy −2i Dx y ) ⎞ 2 ⎟ D −i D − x z√2 yz ⎟ ⎠ Dzz 2

(2.44) it appears that D = 21 D AB . To derive analytically the D tensor components from the spin–orbit operator, we will take the example of the copper acetate molecule [Cu(CH3 COO)2 ]2 (H2 O)2 . This molecule pertains to the D2h SPG. The unpaired electrons are located in the a and b magnetic orbitals which are essentially developed on the dx 2 −y 2 orbitals of the Cu(II) ions A and B. As the isotropic exchange is antiferromagnetic, the anisotropic effects appear on the first excited triplet state, the M S = 1 component of which is: |ϕxT21−y 2 ⟩ = |dxA2 −y 2 dxB2 −y 2 ⟩

(2.45)

2 Analytical Derivations for the Description of Magnetic Anisotropy …

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where T stands for triplet and where we only specify the main nature of the a and b orbitals to simplify the presentation. The eight excited singlets and triplets which are contributing to the tensor components are the locally singly excited states: | ⟩ | T⟩ |ϕ 1 = √1 ||d A d B2 2 + d A2 2 d B n n x −y x −y n 2 ⟩ | T ⟩ 1 || A B |ϕ 0 = |d d 2 2 − d B2 2 d A − d A2 2 d B + d B d A2 2 n n x −y x −y n x −y n 2 n x −y ⟩ | S ⟩ 1 || A B |ϕ = |d d 2 2 + d B2 2 d A − d A2 2 d B − d B d A2 2 n n x −y n x −y x −y n x −y n 2

(2.46)

where d n are the 4 other d orbitals, n = xy, xz, yz, z2 . As in the mononuclear systems, the elements of the SOC matrix between the singlet and triplet components can ) ( SO ˆ ˆ ˆ be analytically calculated using the spin–orbit operator H = ζ l 1 .ˆs1 + l 2 .ˆs1 , where lˆi and sˆ i are the angular and spin momenta respectively of electron i and ζ is the spin–orbit constant (we consider a spherical approximation of the SOC and identical magnitudes on both sites). Using both the spin–orbit operator Hˆ SO and the quasi-degenerate perturbation theory at the second order, it is possible to extract the giant-spin analytical matrix dressed by the effects of the SOC: (

) Giant-Spin Hˆ analytical =

2ζ 2

− Δϕ T − xy

ζ2

4ΔϕxTz

| ⟩ | T1 |ϕx 2 −y 2 −

ζ2

4ΔϕxSz

−

ζ2

T 4Δϕ yz

−

−

ζ2 4ΔϕxSz

−

ζ2 T 4Δϕ yz

+

ζ2

0

S 4Δϕ yz

2ζ 2

− Δϕ S −

0 ζ2 4ΔϕxTz

| ⟩ | T0 |ϕx 2 −y 2

ζ2

ζ2 S 4Δϕ yz

xy

ζ2

2ΔϕxTz

0

−

ζ2

T 2Δϕ yz

4ΔϕxTz

| ⟩ | T−1 |ϕx 2 −y 2

−

ζ2

4ΔϕxSz

−

ζ2 T 4Δϕ yz

+

ζ2 S 4Δϕ yz

0 2

2ζ − Δϕ T − xy

ζ2 4ΔϕxTz

−

ζ2 4ΔϕxSz

−

ζ2 T 4Δϕ yz

−

ζ2 S 4Δϕ yz

(2.47) where ) ( ) ( ΔϕnT,S = E ϕnT,S − E ϕxT2 −y 2

(2.48)

One should note that as the dx 2 −y 2 orbital does not interact with the dz 2 through SOC, the energy of the ϕzT,S spin–orbit-free states does not appear in the derivation. 2 All components of the tensors (both D and D AB ) can be extracted analytically from this matrix. One may also note that as this matrix could be calculated ab initio, it is of course possible to assess numerical values to these components using the effective Hamiltonian theory. Introducing the geometric means ΔE x 2 −y 2 ,n between the ΔϕnT and ΔϕnS and the magnetic couplings Jx 2 −y 2 ,n between excited states: ΔE x 2 −y 2 ,n = Jx 2 −y 2 ,n

/

ΔϕnT .ΔϕnS ( ) ( T) = E ϕn − E ϕnS ,

(2.49)

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one gets the D and E parameters: | | | | ⟩ ⟨ ⟩ ⟨ | Giant-Spin | T | Giant-Spin | T D = ϕx 2±1−y 2 | Hˆ analytical |ϕx 2±1−y 2 − ϕxT20−y 2 | Hˆ analytical |ϕxT20−y 2 2ζ 2 Jx 2 −y 2 ,x y ζ 2 Jx 2 −y 2 ,x z ζ 2 Jx 2 −y 2 ,yz − − 2 2 ΔE x 2 −y 2 ,x y 4ΔE x 2 −y 2 ,x z 4ΔE x22 −y 2 ,yz | | ⟩ ⟨ ζ 2 Jx 2 −y 2 ,yz ζ 2 Jx 2 −y 2 ,x z | Giant-Spin | T E = ϕxT21−y 2 | Hˆ analytical |ϕx 2−1−y 2 = − 2 4ΔE x 2 −y 2 ,yz 4ΔE x22 −y 2 ,x z =

(2.50)

These expressions exhibit the main (2nd order) contributions of the locally excited states to the symmetric anisotropic exchange. For the theoretical chemists, they also show that these interactions calculated from ab initio methods are very sensitive to electron correlation since they require an accurate determination of the isotropic magnetic exchanges. In reference [85], the dependence of the ZFS axial and rhombic parameters has been studied. Contrarily to the local anisotropy tensor components that can be determined using perturbative methods (either CASPT2 [86] or NEVPT2[77]), an accurate calculations of the symmetric anisotropic exchange requires the use of the difference-dedicated configuration interaction sophisticated method [87]. Furthermore, as was already the case for the components of the local anisotropy tensor, these expressions allow the values of the anisotropic interactions to be correlated with the ligand field and thus give synthetic chemists the ability to choose ligands to control the magnitude of the anisotropy. For example, it can be predicted that a quasi-degeneracy between the local dx 2 −y 2 and dx y orbitals could lead to very large values of the anisotropic exchange.

2.3.2.3

Antisymmetric Tensor of Anisotropy (Illustrated on Model Binuclear Systems of Spin S = 1/2)

While well-known symmetry conditions enable one to strictly eliminate the Dzyaloshinskii–Moriya interactions (DMI), the symmetric exchange tensor a priori exists for any binuclear system. As a consequence, for a binuclear compound constituted of two spins S = 1/2, the multi-spin Hamiltonian is the one of Eq. (2.41) and the Hamiltonian matrix is identical to that given in Eq. (2.43). The off-diagonal elements of the SOC matrix between the singlet and triplet components can here again be analytically calculated using the spin–orbit operator: /| | | || |a , b, | + |b, a , | || √ | Hˆ SO ||ab|⟩ = | 2 | /| | | | |a , b, | + |b, a , | || | ⟩ √ | Hˆ SO ||ab| = | 2

⟨ | ⟨ | ⟨ ⟩) 1 (⟨ | √ b, |ζ lˆ+ |b⟩ a , ||a⟩ − a , |ζ lˆ+ |a⟩ b, |b 2 2 ⟨ | ⟨ | ⟨ | ) 1 (⟨ | √ a , |ζ lˆ− |a⟩ b, | b⟩ − b, |ζ lˆ− |b⟩ a , | a⟩ 2 2

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|| | /| \ | | || | |ab| − |ba| |a , b, | + |b, a , | || ⟨ | ⟨ | ) 1 (⟨ , || ˆz ⟨ , || | SO a ζ l |a⟩ b b⟩ − b, |ζ lˆz |b⟩ a , | a⟩ = √ √ | Hˆ | | | 2 2 2 (2.51) The symmetry conditions that cancel the DMI can be deduced from these equations. Indeed, if for symmetry reasons a = b (thus a, = b, ) the DMI is strictly zero. To create different magnetic orbitals in a system with two symmetrically equivalent magnetic centers, it is compulsory to locally mix differently the atomic orbitals on each center A or B. For instance, a mixing between two d orbitals could be: a = αd1A + βd2A b = −αd1B + βd2B a , = α , d1A + β , d2A b, = −α , d1B + β , d2B

(2.52)

where d1A and d2A on center A and d1B and d2B on center B are two different d orbitals. Local distortions along the X, Y, or Z orbitals can generate such mixings. As shown in reference [84], the SOC of Eq. (2.51) is always proportional to: [ ( ) ( )] Δ = ζ αβ α ,2 − β ,2 − α , β , α 2 − β 2

(2.53)

This equation shows that the DMI would be strictly zero for a single set of orbitals, i.e., α = α , and β = β , (i.e., same set of MO for the triplet and singlet states). The various SOC that can be obtained by mixing 2 orbitals are reported in Table 2.10. Adding the various contributions, these results can be generalized to describe mixings of three or more local orbitals. All components of the DM vector can be deduced from these SOC. As expected, the largest DM components are obtained from the application of the lˆz sˆz operator (which generates the dz components) and for the largest (in absolute value) angular momentum L Z component quantum numbers M l = ± 2. Reference [84] explores several distortions (in model complexes represented in Fig. 2.13) in order to identify the microscopic ingredients that govern the orientation and magnitude of the DM vector. The most important conclusions are: (i) Here again the derivation provides formulas that can be used in addition to the ligand field theory to guide the synthesis of complexes (or materials) with controlled anisotropic properties. (ii) An efficient way to produce a large DMI is to obtain magnetic orbitals that result from mixing between the dx 2 −y 2 and dx y orbitals. Such mixing generates the d z component of the DM vector which can otherwise be obtained by mixing the dx z and d yz orbitals. In the latter case, note that the amplitude of the DMI is less important due to the lower values of M l = ± 1.

98 Table 2.10 Nonzero elements of the SOC matrix resulting in contributions to the DM vector dx , d y , and dz components. An interchange between d 1 and d 2 leads to a change between α(α , ) and β(β , ) in Δ

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d1

d2

Nonzero matrix elements of the SOC

dx 2 −y 2

dx y

⟨S0 | Hˆ SO |T ⟩0 =

dx 2 −y 2

dx z

dx 2 −y 2

d yz

dx y

dx z

dx y

d yz

dx z

d yz

dx z

dz 2

idz 2 = −2iΔ d ⟨S0 | Hˆ S O |T±1 ⟩ = √y = − √Δ 2 2 2 iΔ ⟨S0 | Hˆ SO |T±1 ⟩ = ∓ id√x = ∓ √ 2 2 2 iΔ ⟨S0 | Hˆ SO |T±1 ⟩ = ∓ id√x = ± √ 2 2 2 dy Δ SO ˆ ⟨S0 | H |T±1 ⟩ = √ = − √ 2 2 2 ⟨S0 | Hˆ SO |T0 ⟩ = id2z = −iΔ √ d ⟨S0 | Hˆ SO |T±1 ⟩ = √y = − Δ√ 3 2 2 2

d yz

dz 2

⟨S0 | Hˆ S O |T±1 ⟩ = ∓

id √x 2 2

√ 3 2

= ± iΔ√

Fig. 2.13 Distortions of the [Cu2 Cl6 ]2− model complex that generates DMI along the Y-axis (top) or Z (middle and bottom)-axis and the corresponding magnetic orbitals. The angle φ (bottom) allows to tune the local degeneracy, i.e., the energy gap between the ground state and the first excited state (φ), for the [Cu2 Cl5 ]− model complex. Reprinted from [84], with the permission of AIP Publishing

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(iii) By increasing the covalence, i.e., for larger values of the isotropic magnetic coupling, we can further increase the DMI, since the differences between the sets of singlet and triplet orbitals play a role in the amplitude of the DMI. (iv) Finally, the recipe to reach giant DMI is to generate first-order SOC, in particular in nearly degenerate orbitals situations (ideally between the dx 2 −y 2 and dx y orbitals). This is the subject of the next subsection. 2.3.2.4

Approaching Local First-Order SOC in a Model Complex of Spin S = 1/2: Derivation of the SOC Matrix

In this subsection, we will derive the first-order SOC matrix that couples the states responsible for the DMI interactions in the ideal case where a giant d z component of the DMI is generated, i.e., in a geometry that mixes the dx 2 −y 2 and dx y orbitals. This presentation follows the derivation proposed in the article [88]. Let us consider a molecule like the one presented in Fig. 2.13 (down). We define the uncoupled states as those obtained for an angle θz = 180°, i.e., in absence of mixing between the d orbitals. Let us call a1 (a2 ) and b1 (b2 ) the dx 2 −y 2 (dx y ) orthogonal orbitals located on centers A and B, respectively. Of course, these orbitals have tails on the ligand and their mixing is thus determined by the angle θz . The four lowest triplet and singlet uncoupled states can be written as linear combinations of determinants expressed as functions of the a1 , a2 , b1, and b2 orbitals: (

T11 3 B2

)

=

(

|a1 b1 |; T01 3 B2

)

| | | | |a1 b1 | − |b1 a 1 | ( ) |a1 b1 | + |b1 a 1 | 1 1 = ; S A1 = √ √ 2 2

( ) |a1 b2 | + |b1 a2 | T12 3 B2 = ; √ 2 | | | | ( ) |a1 b2 | + |b1 a 2 | − |a2 b1 | − |b2 a 1 | 2 3 T0 B2 = ; 2| | | | ( ) |a1 b2 | + |b1 a 2 | + |a2 b1 | + |b2 a 1 | S 2 1 B2 = 2 ( ) | |b | b a −|a + 1 2 1 2 ; T13 3 A1 = √ 2 | | | | ( ) −|a1 b2 | + |b1 a 2 | − |a2 b1 | + |b2 a 1 | 3 3 T1 A1 = ; 2 | | | | ( ) −|a1 b2 | + |b1 a 2 | + |a2 b1 | − |b2 a 1 | S 3 1 A1 = 2| | | | |a2 b2 | − |b2 a 2 | ( ) ( ) ( ) |a2 b2 | + |b2 a 2 | 4 3 4 3 4 1 ; S A1 = T1 B2 = |a2 b2 |; T0 B2 = √ √ 2 2 (2.54)

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where only the neutral valence bond components (Cu 2+ –Cu2+ ) have been considered. When the geometrical distortion induces a mixing of the local dx y and dx 2 −y 2 (i.e., a1 (b1 ) with a2 (b2 )) orbitals, these states are coupled through the electronic Hamiltonian Hˆ El. . For symmetry reasons the T 3 state does not interact with the three other triplets just as S 2 does not interact with the three other singlets. For the triplet states, these interactions are: | ⟩ ⟨a1 b1 | Hˆ El. |a1 b2 ⟩ − ⟨a1 b1 | Hˆ El. |a2 b1 ⟩ ⟨ | h 1 = T01 | Hˆ El |T02 = √ 2 ⟨ 4 | El | 2 ⟩ ⟨a2 b2 | Hˆ El. |a1 b2 ⟩ − ⟨a2 b2 | Hˆ El. |a2 b1 ⟩ h 2 = T0 | Hˆ |T0 = √ 2

(2.55)

Their values differ in the singlet and triplet states due to the(exchange integrals | ⟩ ⟨ | and we will call h, the interactions between the singlet states h ,1 = S 1 | Hˆ El. | S 3 | ⟩) ⟨ | and h ,2 = S 3 | Hˆ El. | S 4 . The bielectronic integrals, expected to be small, between T01 and T04 and between S 1 and S 4 will not be reported. Let E 1 , E 2 ≈ E 3, and E 4 be the energies of the singlet functions and 2K 1 , 2K 2 ≈ 2 K 3, and 2K 4 the energy difference (exchange integrals) between the uncoupled singlet and triplet states of the same electronic configuration. The off-diagonal elements of the SOC matrix can be calculated analytically using the spin–orbit operator Hˆ SO . In the M S = 0 subspace, the Hamiltonian matrix involving both electronic and spin–orbit interactions writes: ⟨ 1| | ⟨S3| | ( ) ⟨S | SO+el Hˆ Analyt. = S4| ⟨ 1| | ⟨T02 | | ⟨T04 | T | 0

| 1⟩ | 2⟩ | 1⟩ | 3⟩ | 4⟩ |S |S |T |T | 0 0 ⎛S , E1 h1 0 0√ ⎜ h, , E ∼ E h i ξ 2 3 2 ⎜ 1 2 ⎜ , E 0 0 h ⎜ 4 2√ ⎜ ⎜ 0 2 0 −i ξ √ √ E 1 + 2K 1 ⎜ ⎝ −i ξ 2 0√ −i ξ 2 h1 0 −i ξ 2 0 0

| 4⟩ |T

0 √ ⎞ iξ 2 0√ 0√ iξ 2 ⎟ ⎟ ⎟ iξ 2 0 ⎟ ⎟ ⎟ h1 0 ⎟ ⎠ E 2 + 2K 2 h2 h2 E 4 + 2K 4 (2.56)

This matrix shows that the SOC between S 1 and T01 is strictly zero. Hence, the d z component of the DM vector is not a direct interaction but arises from complex mechanisms that go through both a SOC between S 1 and T02 (T01 and S 3 ) and an electronic coupling between T02 and T01 (S 3 and S 1 ).When these states are far in energy from the lowest singlet and triplet states, a second-order estimation of the matrix , element between the coupled singlet S 1 and triplet T0,1 states (i.e., after diagonalizing the electronic Hamiltonian) would be given by: ⟨

S

|

,1 |

⟨ 1 | SO | 2 ⟩⟨ 2 | El. | 1 ⟩ ⟨ 1 | El. | 3 ⟩⟨ 3 | SO | 1 ⟩ | ⟩ S | Hˆ |T0 T0 | Hˆ |T0 S | Hˆ | S S | Hˆ |T0 SO | ,1 (2) ˆ H T0 = + E 2 + 2K 2 − E 1 E3 − E1 (2.57)

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In Ref. [88], we show that using Eq. (2.57) and taking the energy difference between T02 (S 3 ) and the mean energy of the S 1 and T01 states provides good estimate of the DMI. As the electronic interactions involved in this expression are proportional to the overlap between the magnetic orbitals and the ligand ones, an appropriate choice of both the structure and the ligand should allow us to control the DMI value. Let us finally recall that this expression is only valid far from the first-order SOC regime. In conclusion, it should be kept in mind that the multi-spin Hamiltonian of Eq. (2.41) is only valid in the absence of first-order SOC. Indeed, in such a case the spectrum would be quasi-degenerate and the second family of states would not be reproduced by the Hamiltonian. Fortunately, the Jahn–Teller effect would induce a lift of degeneracy of the local d orbitals. We can therefore hope to approach the firstorder SOC without fear of reaching it in the design of new real compounds. Another important conclusion of this work is that here again one may expect a strong dependence of the DMI to the level of ab initio calculations as both electronic interactions and relative energies are sensitive to electron correlation.

2.3.3 Key Lessons from the Previous Section and Extension of Our Approach to Higher Spins and Higher Nuclearities In this section, we abruptly rise up the difficulty of the exposed concepts starting from the outcomes of the previous section. For the sake of simplicity and owing to the already reached length and difficulty of the present chapter, we have chosen to present the main outcomes without entering much into equations. The interested reader may find more details and key equations in a previous chapter [67] and also of course in references therein. Actually, the previous situation corresponds to the easiest (interesting) one in terms of binuclear complexes since it already allows to introduce the isotropic, symmetric, and antisymmetric exchange couplings within the multi-spin picture, and check the influence of these terms in the coupled basis: • One can easily determine the size of the “Heisenberg” space: two M S components times two M S components leads to 22 = 4. • The isotropic coupling, J, effectively splits the singlet and the triplet components (and maintain the degeneracy between the M S components of the triplet in the coupled basis). • The symmetric exchange only acts in the triplet block. • The antisymmetric exchange couples the singlet with the triplet components. Consequently, two interesting situations may occur, (i) the DMI is null due to the SPG of the system or (ii) the DMI can be neglected (either because we are in the “strong-exchange limit,” meaning that J largely rules the spectrum, or because it is

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much smaller than both J and DAB ). In this case, and in the magnetic axis frame, the multi-spin model matrix largely simplifies in the coupled basis (also note that we consider a traceless tensor on top of Eq. 2.43). For the sake of pedagogy, we will consider the E AB = 0 case: | | ⟩ ⟩ | ⟩ , +b, a , | √ |ab S0 = || a , b√ T = T1 = |ab⟩ T0 = | ab−ba −1 2 2 (

Hˆ Multi-Spin

)

⟨ ⟨ab| | | √ | = ab−ba 2 ⟨ | | | ⟨ , ab , +b, a , | a b√ | 2

J AB 4

+

AB Dzz 4

0 J AB 4

0 0

−

0 AB Dzz 2

0

0 J AB 4

0

0

0

+

0 AB Dzz 4

0 − 3J4AB

0

(2.58) Since DzzAB = 23 D AB (similarly as in Eq. 2.6) and D AB = 2D (see Eqs. 2.43 and 2.44), DzzAB = 43 D and one may thus rewrite the matrix as follows and call it a “block-spin” matrix since this matrix is block-diagonal (here diagonal since E AB = 0) and deals with all the spin states that are generated by the Heisenberg Hamiltonian: | | , ⟩ ⟩ | ⟩ , +b, a , | √ | a b√ | T T1 = |ab⟩ T0 = | ab−ba = ab S = | −1 0 2 2 (

Hˆ Multi-Spin

)

⟨ ⟨ab| | | √ | = ab−ba ⟨ 2| | | ⟨ , ab , a b√ +b, a , | | 2

J AB 4

+ 0 0 0

D 3

0 J AB 4

− 0 0

0 2 D 3

0

0 J AB 4

+ 0

0 D 3

0 − 3J4AB (2.59)

It is then easy to define a block-spin Hamiltonian expression that will generate this matrix (recall, S A = S B ): Hˆ block-spin = {

1 S Σ Σ S=0 MS =−S

]} [ J S(S + 1) 2 ˆ [S(S + 1) − S A (S A + 1) − S B (S B + 1)] + D Sz − 2 3

(2.60)

When one introduces higher local spins, local anisotropies may also be at play [89]. If and only if one local anisotropy is at play (let us take for instance S A = 1 and S B = ½), things should remain more or less tractable (not shown). To illustrate the true complexity that may arise with higher spins, it is necessary to introduce two local anisotropies. The simplest situation consists in principle in coupling two S A = S B = 1 sites (in a centrosymmetric fashion). In this case, the Heisenberg space consists in three M S components times three M S components, i.e., 32 = 9. In the coupled basis [19], it concerns the spin components of a quintet, a triplet and a singlet spin state. In

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the strong-exchange limit, the corresponding spin blocks remain well separated (and thus do not mix much). However, in the weak-exchange limit, the spin components of distinct spin states may mix, a situation usually called “spin-mixing” [90]. The definition of model Hamiltonians in such situations is not trivial. By applying our effective Hamiltonian theory approach, we had first notice that the standard multi-spin Hamiltonian was not enough to capture all the features of the effective Hamiltonian [48]. We recall here that this standard model Hamiltonian is a simple extension of Eq. 2.41 for which local anisotropy terms have been added. Actually, we studied in deep details a real centrosymmetric dinickel(II) complex [91], showing that the introduction of a fourth-rank symmetric exchange tensor was necessary. However, dealing with such a beast was not easy at all, and we somehow later managed to extract parameters in specific situations based on a combination of cluster and full system calculations [92]. Therefore, it is clear that it rapidly becomes intractable and that easy transformations from the multi-spin to either the block-spin or the giant-spin pictures are quickly practically impossible. Equations reported for such purposes [93] are based on the neglect of the fourth-rank exchange tensor and thus should in our opinion not be taken for granted. In the strong-exchange limit, it is readily possible to define block-spin and giantspin models in the same vein as in Eq. 2.21. Naturally, in this case, one needs to distinguish the D parameters for the S = 1 and for the S = 2 states, which are not equal at all, and which in the general case do not follow simple relationships already proposed [93]. However, if J is not so large, spin-mixings are allowed even in the centrosymmetric situation. In the magnetic axis frame, three coupled spin state components are effectively coupled [48]: |2, 2s ⟩, |2, 0⟩ and |0, 0⟩. The spinmixings thus perturb the simplest block-spin picture, in particular the ZFS of the ˆ D. Sˆ term (see Fig. 2.14a). Starting quintet block cannot be described by a sole S. from the “intractable” multi-spin Hamiltonian, we derived a perturbed giant-spin Hamiltonian for the S = 2 block that effectively accounts for the effect of the spinmixings on the energies [94] (see Fig. 2.14b). Indeed, the |2, 2s ⟩ and the |2, 0⟩ states are effectively stabilized (in fact |2, 0⟩ couples with |0, 0⟩, and the |2, 2s ⟩ and the |2, 0⟩ states are already coupled by the rhombic E term in model a). Perhaps the most striking result is that the |2, 0⟩ state now lies in between the |2, 1a ⟩ and |2, 1s ⟩ ones, ˆ D. Sˆ giant-spin model is meaning that the pattern of states that arises from the S. not correct anymore. By symmetry, the triplet spin components are not coupled with the quintet and singlet ones. However, if we consider a block-spin model, of course these generated coupled states have to be modeled (the green levels in Fig. 2.14c, d). Naturally, in both models (c includes effective spin-mixings, unlike d) those green levels have exactly the same energy. As already mentioned, only the |2, 2s ⟩, |2, 0⟩, and |0, 0⟩ states are affected by this effect. Of course, the energies of the |2, 2s ⟩, |2, 0⟩ states are the same as for the giant-spin models, but now it becomes possible to see that the |0, 0⟩ state is effectively destabilized by its couplings with |2, 2s ⟩ and |2, 0⟩. Because of the spin-mixings that arise in the weak-exchange limit, it is not possible to extract correct J values after introducing the spin–orbit coupling by taking the mean values of the energies of each spin-block components (the singlet is pushed up by spin-mixing and two components of the quintet are pushed down).

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Fig. 2.14 Spin-mixing and the spectrum. Left: giant-spin description of the ZFS of the S = 2 state (a: standard Hamiltonian with no spin-mixing vs. b: model with effective spin-mixings. Right: block-spin description (c: model with effective spin-mixings vs. d: standard Hamiltonian with no spin-mixing). Reprinted figure with permission from [94] Copyright (2010) by the American Physical Society (the bottom legend has been adapted for consistency)

Thus, extracting well-grounded parameters in the weak-exchange limit requires a careful analysis (e.g., effective Hamiltonians). In dicopper(II) complexes, we have seen that spin-mixing between S and S + 1 blocks can only arise from the DMI. In centrosymmetric dinickel(II) complexes, we have just seen that spin-mixings between S and S + 2 blocks are possible. What happens if we do lose the centrosymmetry in such complexes? Naturally, the DMI will couple S and S + 1 blocks [92]. However, things may actually be even more subtle. In the absence of any symmetry element (or at least of specific ones), the main local anisotropy axes may neither be collinear nor parallel. Naturally, the construction of a giant-spin or block-spin model (from the multi-spin one) requires expressing the local tensors in a common frame. Without dealing with a full multi-spin model (which is, we recall, not very practical…), one can build a model with only J couplings and local DA and DB terms [95]. Assuming a specific symmetry (C2v molecular symmetry point group with coplanar local Z-axes, see Fig. 2.15), it is possible to derive such a model and simplify equations. It is interesting to note that in the end only two types of terms appear in the coupled basis, both simply related to the mismatch between the local anisotropy axes (derivation not shown): • Couplings between the singlet and the triplet blocks are proportional to DA sin 2α (see Fig. 2.15). Note that in centrosymmetric complexes, 2α = 180°; i.e., those terms vanish (as for the DMI). • Couplings between the singlet and quintet blocks are created or perturbed (with respect to the centrosymmetric situations) by terms that are proportional to DA

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Fig. 2.15 Definition of the 2α angle between the local anisotropies of an ideal C 2v complex. The local anisotropy parameters, DA and DB, are symmetry equal. Reprinted from reference [95]. Copyright (2002), with permission from Elsevier (with minimal alteration concerning the atom labels)

sin2 α. Again, in a centrosymmetric complex, those perturbations vanish, and we recover the previous situation (for which couplings are still possible between the singlet and quintet blocks). To conclude, it is important to note that spin-mixings between S and S + 1 blocks can arise from both the DMI and the mismatch between local anisotropies. Also, these terms should not be neglected in the weak-exchange limit and added to the mechanisms taken into account in the inspiring work of Ostrowski et al. [95]. Finally, it is important to discuss extension to complexes of higher nuclearities. Recently, a complex with four S = 1 nickel(II) sites that was well-characterized experimentally [96] has been studied theoretically [97] (see Fig. 2.16). Naturally, a multi-spin description has not even been targeted. However, the giant-spin description of the ground S = 4 state has been done based on SOCI wave functions and energies. It is very interesting to see that with just a single parameter (in fact, D), one may describe the full features of the effective Hamiltonian, which is especially true when the calculation only accounts for S = 4 states. When some spin-mixing is a priori allowed (calculation with both S = 4 and S = 3 states), small deviations between the model and ab initio spectra start to appear, which is an indirect sign for spinmixing. Without entering more into details, it is our intuition that the giant-spin Hamiltonians somehow drastically simplify in complexes of high nuclearities that ultimately display close to cubic symmetries, unless strong spin-mixing is at play. How to introduce the spin-mixing in the models is still a matter of debate, in particular, we have shown in a dinickel(II) complex that the Stevens operators should not be used for this purpose. Therefore, a natural perspective of this first work on complexes with higher nuclearities would be to carefully define a giant-spin model in the weakexchange limit (the previous example was not a net example for this due to the smallness of the spin-mixing effects).

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Fig. 2.16 Ball-and-stick representations of the molecular structure of the studied cubane-like Ni4 complex, derived from the crystallographic one. Color code: gray stands for Ni, red for O, blue for N, and pink for H. Reprinted (adapted) with permission from [97]. Copyright 2021American Chemical Society

2.4 Conclusions and Perspectives We have here focused on the derivation of models to describe transition metal complexes and therefore we have only considered d magnetic orbitals. Generalization to f orbitals (lanthanides and actinides) is of course possible, keeping the same reasoning. Nevertheless, as the orbital contribution is usually very important, pseudospins, which can be very different from the actual spins of the complexes, may be used and in fact both the model and its effective interactions may have to be adapted. Derivations of model Hamiltonians, as presented here, were very early a preoccupation of physicists and theoretical chemists. They allow to establish the physical basis of the models but also to understand the mechanisms at work that determine the values of their effective interactions. This is particularly valuable to guide synthetic chemists in the design of new compounds with improved and controlled properties. We have also shown here and in other works that combined with the theory of effective Hamiltonians and ab initio calculations, they allow to question the reliability of usual models and especially to propose new models when the latter are not valid.

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Unfortunately, it is feared that in the field of magnetic anisotropy, where the interactions are of very small amplitude, reliable models imply an inordinate number of interactions, perhaps impossible to extract experimentally, as is currently the case for the 4-rank tensor for interacting S = 1 spins. The alternative of preferring the block giant-spin model to the multi-spin one is relevant in the context of a binuclear system but deprives us of a model transferable to systems with more than two centers. With the steady increase in computational power and the recent implementation of specific algorithms based on configuration state functions, [98, 99] we believe that the brute-force computation of giant- or block-spin parameters of experimental relevance (with for instance four magnetic sites or even more) is now possible and that this opens the way for direct comparisons with experiments, even if the multi-spin picture may be lost in the process. Therefore, further conceptual and computational work in this area is needed, and we hope that this chapter will help to bring new life to this field. Acknowledgements The authors deeply thank Benjamin Cahier and Talal Mallah for their attentive reading of the chapter and for useful suggestions.

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18. Aquilante F, Autschbach J, Carlson RK, Chibotaru LF, Delcey MG, De Vico L, Fdez, Galván I, Ferré N, Frutos LM, Gagliardi L, Garavelli M, Giussani A, Hoyer CE, Li Manni G, Lischka H, Ma D, Malmqvist PÅ, Müller T, Nenov A, Olivucci M, Pedersen TB, Peng D, Plasser F, Pritchard B, Reiher M, Rivalta I, Schapiro I, Segarra-Martí J, Stenrup M, Truhlar DG, Ungur L, Valentini A, Vancoillie S, Veryazov V, Vysotskiy VP, Weingart O, Zapata F, Lindh R (2016) J Comput Chem 37:506 19. (n.d.) 20. Aa HJ. Jensen, Bast R, Gomes ASP, Saue T, Visscher L, Aucar IA, Bakken V, Chibueze C, Creutzberg J, Dyall KG, Dubillard S, Ekström U, Eliav E, Enevoldsen T, Faßhauer E, Fleig T, Fossgaard O, Halbert L, Hedegård ED, Helgaker T, Helmich-Paris B, Henriksson J, van Horn M, Iliaš M, Jacob ChR, Knecht S, Komorovský S, Kullie O, Lærdahl JK, Larsen CV, Lee YS, List NH, Nataraj HS, Nayak MK, Norman P, Olejniczak G, Olsen J, Olsen JMH, Papadopoulos A, Park YC, Pedersen JK, Pernpointner M, Pototschnig JV, Di Remigio R, Repiský M, Ruud K, Sałek P, Schimmelpfennig B, Senjean B, Shee A, Sikkema J, SunagaA, Thorvaldsen AJ, Thyssen J, van Stralen J, Vidal ML, Villaume S, Visser O, Winther T, Yamamoto S, Yuan X (2022) DIRAC22. Zenodo 21. Reviakine R, Arbuznikov AV, Tremblay J-C, Remenyi C, Malkina OL, Malkin VG, Kaupp M (2006) J Chem Phys 125:054110 22. Pederson MR, Khanna SN (1999) Phys Rev B 59:R693 23. Pederson MR, Khanna SN (1999) Phys Rev B 60:9566 24. Arbuznikov AV, Vaara J, Kaupp M (2004) J Chem Phys 120:2127 25. de Souza, Farias G, Neese F, Izsák R (2019) J Chem Theory Comput 15:1896 26. Ganyushin, Neese F (2006) J Chem Phys 125:024103 27. Neese F (2007) J Chem Phys 127:164112 28. Sørensen LK, Olsen J, Fleig T (2011) J Chem Phys 134:214102 29. Michl J (1996) J Am Chem Soc 118:3568 30. Havlas Z, Michl J (1999) J Chem Soc Perkin Trans 2:2299 31. Knecht S, Aa HJ, Jensen, Fleig T (2008) J Chem Phys 128:014108 32. Neese F (2005) J Chem Phys 122:034107 33. Ganyushin, Neese F (2013) J Chem Phys 138:104113 34. Ruiz, Cirera J, Cano J, Alvarez S, Loose C, Kortus J (2007) Chem Commun 52 35. Loose, Ruiz E, Kersting B, Kortus J (2008) Chem Phys Lett 452:38 36. Chibotaru LF, Hendrickx MFA, Clima S, Larionova J, Ceulemans A (2005). ACS Publications 37. de Graaf and C. Sousa, International Journal of Quantum Chemistry 106, 2470 (2006). 38. Bolvin H (2006) Chem Phys Chem 7:1575 39. Maurice R, Bastardis R, de Graaf C, Suaud N, Mallah T, Guihéry N (2009) J Chem Theory Comput 5:2977 40. Singh SK, Rajaraman G (2014) Chem A Euro J 20:5214 41. Atanasov M, Comba P, Helmle S, Müller D, Neese F (2012) Inorg Chem 51:12324 42. Neese (2006) J Am Chem Soc 128:10213 43. Bloch (1958) Nucl Phys 6:329 44. des Cloizeaux J (1960) Nucl Phys 20:321 45. Malrieu JP, Caballol R, Calzado CJ, de Graaf C, Guihéry N (2014) Chem Rev 114:429 46. Bastardis R, de Graaf C, Guihéry N (2008) Phys Rev B 77:054426 47. Bastardis R, Guihéry N, de Graaf C (2007) Phys Rev B 76:132412 48. Maurice R, Guihéry N, Bastardis R, de Graaf C (2010) J Chem Theory Comput 6:55 49. Primas R (1963) Mod Phys 35:710 50. Telser J (2006) J Braz Chem Soc 17:1501 51. Abragam, Bleaney B (1986) Electron paramagnetic resonance of transition ions. Dover Publications, Dover 52. Griffith J (2009) The theory of transition-metal ions. Reissue édn. Cambridge University Press, Cambridge 53. Maurice R, de Graaf C, Guihéry N (2010) J Chem Phys 133:109901 54. Dzyaloshinskii IE (1964) JETP 19:960

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Chapter 3

Calculations of Magnetic Exchange in Multinuclear Compounds Giang Truong Nguyen and Liviu Ungur

Abstract The magnetic exchange interaction between metal centers in multinuclear compounds plays a defining role in their low-temperature magnetic behavior. Predicting magnetic exchange by computational methods is of high importance for the development of novel magnetic, spintronic materials, and devices that drives further the development of advanced computational methods. In this chapter, we give a brief overview of the origin of magnetic exchange and describe various microscopic theories accounting for it. Several computational approaches which allow the estimation of magnetic exchange in multinuclear compounds are reviewed. Computational challenges of the existing methods for accounting for the magnetic exchange will be discussed in detail in subsequent sections. The combination between accurate on-site eigenstates obtained in ab initio calculations (CASSCF + SOC + ANISO) and theoretical modeling of the magnetic dipole–dipole and exchange interactions within the Lines model, and the recently developed kinetic exchange model is described alongside several recent examples. The last section offers a review of several results obtained recently and gives our perspective on how to achieve an efficient interplay between strong magnetic exchange and a highly axial crystal field on metal sites for designing better exchange-coupled multinuclear single-molecule magnets. Keywords Magnetic exchange · Broken-symmetry DFT · Ab initio calculations · Magnetic anisotropy · Mechanisms of magnetic interaction · Lines model · Multireference methods

3.1 Origin of Magnetic Exchange In the language of quantum chemistry, the wave function of a quantum system contains all the required information for the description of its properties. For a manyelectron system, the wave function is obtained by solving the electronic Schrodinger equation: G. T. Nguyen · L. Ungur (B) Department of Chemistry, National University of Singapore, Singapore 117543, Singapore e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Rajaraman (ed.), Computational Modelling of Molecular Nanomagnets, Challenges and Advances in Computational Chemistry and Physics 34, https://doi.org/10.1007/978-3-031-31038-6_3

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Helec Φelec = E elec Φelec , where Helec =

N ∑

ˆ )+ h(i

i=1

and

N ∑

(3.1)

rˆi−1 j ,

(3.2)

i=1

( ) ∑ ZA ˆh(i ) = − 1 ∇i2 − , 2 ri A A rˆi−1 j

(3.3)

N ∑ 1 = . r j>i i j

ˆ ) describes the kinetic and nuclear N is the number of electrons in the system, h(i −1 potential energy of the electrons, and r12 represents the Coulomb repulsion. The antisymmetric principle states that “a many-electron wave function must be antisymmetric with respect to the interchange of the coordinate (both space and spin) of any two electrons”, i.e., Φ(x1 , . . . , xi , . . . , xj , . . . xN ) = −Φ(x1 , . . . , xj , . . . , xi , . . . xN ),

(3.4)

where | |χ1 (x1 ) χ2 (x1 ) | |χ1 (x2 ) χ2 (x2 ) 1 || . .. Φ(x1 , x2 , . . . , xN ) = √ | .. . N ! || (x ) χ (x χ 1 n 2 n) | |

| . . . χn (x1 )|| . . . χn (x2 )|| .. | .. . . || . . . χn (xn )|| |

(3.5)

is a Slater determinant. χi (xj ) denotes the molecular orbital i spanned by the electron j. The index xj stands for four variables: three spatial r = (x, y, z) and one spin variable ω of the electron j. Each molecular orbital χ (x) may be written as a product of a spatial ψ(r ) and a spin σ (ω) functions. The spatial (orbital) part describes the position of the electron such that ( P = ⟨ψr |ψr ⟩ =

|ψ(r )|2 dr,

(3.6)

is the probability of finding the electron in an infinitesimal volume around the position r . The spin of the electron represents an intrinsic angular momentum of the electron and only has two projections along a certain quantization axis (usually z), i.e. [1],

3 Calculations of Magnetic Exchange in Multinuclear Compounds

{ σ (ω) =

α(ω) β(ω),

and α(ω) and β(ω) are orthonormal: ( ( ⟨α|α⟩ = ⟨β|β⟩ = α ∗ (ω)α(ω) = β ∗ (ω)β(ω) = 1, ( ( ∗ ⟨α|β⟩ = ⟨β|α⟩ = α (ω)β(ω) = β ∗ (ω)α(ω) = 0,

113

(3.7)

(3.8)

where α(ω) is often used to denote spin up (↑), while β(ω) denotes spin down (↓). A molecular orbital is represented as a product between spatial and spin functions: { χ (x) =

ψ(r )α(ω) ψ(r )β(ω).

(3.9)

According to the Pauli exclusion principle, each spatial orbital ψr may be populated with a maximum of two electrons with opposite spins. This restriction is nicely embedded into the Slater Determinant representation of the multi-electronic wave function (Eq. 3.5) such that the determinant vanishes if two rows/columns are identical. Furthermore, the Slater determinant also ensures the indistinguishability of electrons. Let us consider a simple system, in which there exist two electrons, whose one-electron wave functions are χi (r1 , ω1 ) and χ j (r2 , ω2 ). The antisymmetric twoelectron wave functions for the whole system are ) 1 ( Φ1 (x1 , x2 ) = √ ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ) (α(ω1 )α(ω2 )), 2 ) 1 ( Φ2 (x1 , x2 ) = √ ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ) (β(ω1 )β(ω2 )), 2 ) 1( ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ) (α(ω1 )β(ω2 ) + α(ω2 )β(ω1 )), Φ3 (x1 , x2 ) = 2 ) 1( ψi (r1 )ψ j (r2 ) + ψ j (r1 )ψi (r2 ) (α(ω1 )β(ω2 ) − α(ω2 )β(ω1 )). Φ4 (x1 , x2 ) = 2 (3.10) In Φ1 , Φ2 , and Φ3 , the spatial components are antisymmetric with respect to interchange of orbital variables of electrons (r1 and r2 ), while their spin part is symmetric, i.e., the wave functions remain the same, if spins ω1 and ω2 are exchanged. In reverse, the spatial component of Φ4 is symmetric, whereas its spin is antisymmetric. In other words, in the former three wave functions, the spins of the two electrons are parallel to each other (↑↑), while in the latter, the electron spins are antiparallel (↑↓).

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The electronic Hamiltonian for the system is −1 ˆ ˆ Helec = h(1) + h(2) + rˆ12 .

(3.11)

The energy of the system, when its wave function is Φ1 , can be calculated as the expectation value of the Hamiltonian: E 1 = ⟨Φ1 |Helec |Φ1 ⟩ | / | 1 −1 ˆ ˆ = √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 ))|| h(1) | + h(2) + rˆ12 2 | \ | 1 | √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 )) | 2 | / | 1 ˆ = √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 ))|| h(1)| 2 | \ | 1 | √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 )) | 2 | / | 1 ˆ + √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 ))|| h(2)| 2 | \ | 1 | √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 )) | 2 | / | −1 1 + √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 ))|| rˆ12 | 2 | \ | 1 | √ (ψi (r1 )ψ j (r2 ) − ψ j (r1 )ψi (r2 ))(α(ω1 )α(ω2 )) | 2 \ 1/ \ 1/ ˆ ˆ (r ) + (r )| h(1)|ψ (r ) = ψi (r1 )|h(1)|ψ ψ i 1 j 1 j 1 2 2 \ \ / / 1 1 ˆ ˆ + ψi (r2 )|h(2)|ψ ψ j (r2 )|h(2)|ψ i (r2 ) + j (r2 ) 2 2 ⟩ ⟨ ⟩ 1⟨ −1 −1 + ψi (r1 )ψ j (r2 )|ˆr12 |ψi (r1 )|ψ j (r2 ) − ψi (r1 )ψ j (r2 )|ˆr12 |ψ j (r1 )ψi (r2 ) 2 ⟩ ⟨ ⟩ 1⟨ −1 −1 |ψ j (r1 )|ψi (r2 ) − ψ j (r1 )ψi (r2 )|ˆr12 |ψi (r1 )ψ j (r2 ) + ψ j (r1 )ψi (r2 )|ˆr12 2 (3.12) −1 ˆ ) and rˆ12 are independent of the spin components of It is noteworthy that h(i −1 −1 = rˆ21 . Therefore, all the spin terms in Eq. 3.12 can be integrated electrons and rˆ12 out as ⟨α(ω1 )α(ω2 )|α(ω1 )α(ω2 )⟩ = ⟨α(ω1 )|α(ω1 )⟩ ⟨α(ω1 )|α(ω1 )⟩ = 1.

(3.13)

3 Calculations of Magnetic Exchange in Multinuclear Compounds

115

Using the following notations: / \ / \ ( ˆ j = χi |h|χ ˆ j = dx1 χi∗ (x1 )hχ j (x1 ), i|h| ( ⟨ ⟩ −1 ⟨i j|kl⟩ = χi χ j |χk χl = dx1 dx2 χi∗ (x1 )χ ∗j (x2 )r12 χk (x1 )χl (x2 ),

(3.14)

one can simplify Eq. 3.12 as / \ / \ ˆ + 2|h|2 ˆ + ⟨12|12⟩ − ⟨12|21⟩ E 1 = 1|h|1

(3.15)

Similarly, the energies corresponding to the other wave functions are / \ / \ ˆ + 2|h|2 ˆ + ⟨12|12⟩ − ⟨12|21⟩ , E 2 = ⟨Φ2 |Helec |Φ2 ⟩ = 1|h|1 / \ / \ ˆ + 2|h|2 ˆ + ⟨12|12⟩ − ⟨12|21⟩ , E 3 = ⟨Φ3 |Helec |Φ3 ⟩ = 1|h|1 / \ / \ ˆ + 2|h|2 ˆ + ⟨12|12⟩ + ⟨12|21⟩ . E 4 = ⟨Φ4 |Helec |Φ4 ⟩ = 1|h|1

(3.16)

Based on the calculated results, it is concluded that Φ1 , Φ2 , and Φ3 are degenerate, i.e., E 1 = E 2 = E 3 and represent the three components of a spin triplet state (S = 1), with projections of the total spin on the z axis as + 1, − 1 and 0, respectively. The state Φ4 (x1 , x2 ) with energy E 4 corresponds to the total spin singlet (S = 0). Because ⟨12|21⟩ is positive, E 4 is larger than E 1,2,3 . Thus, the system’s total energy depends on the wave function’s total spin. While ⟨12|12⟩ is called Coulomb integral as it corresponds to the Coulomb repulsion between spatial distributions of electrons 1 and 2, the integral ⟨12|21⟩ does not connect to any classical interpretation and is called exchange integral. This exchange integral arises due to a combination of the Coulomb inter-electron repulsion and the antisymmetric nature of the wave functions. Intuitively, one can see the exchange interaction as follows: If two electrons possess different spins, they can share the same spatial component to construct a molecular orbital; on other hand, due to the Pauli exclusion principle, two electrons with the same spin cannot have the same spatial component; thus, there exists a “quantum potential” pushing the same-spin electrons away from each other, which is exchange interaction. By this simple example, one can gain some insights into the origin of exchange interaction. Details about its mechanisms will be discussed in the next section. In the case of multicenter metal compounds, the frontier (magnetic) electrons residing in 3d or 4f orbitals localized on metal sites weakly interact with each other. The on-site electron repulsion, crystal field, and spin–orbit coupling effects set up the localized wave functions and energy states. The subsequent weak inter-site magnetic exchange establishes the final magnetic configuration of the entire compound. The inter-site exchange integrals ⟨12|21⟩ involving (non-orthogonal) magnetic orbitals (i.e., the orbitals containing paired and unpaired electrons near the HOMO-LUMO

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domain) are of major interest since they define the ground state total spin of the investigated system. These integrals are very numerous, even if only ground spin eigenstates of individual sites are used. Non-orthogonality between the local wave functions corresponding to different sites and the spin–orbit effects induce additional exchange mechanisms and, as result, the effective inter-site magnetic exchange interactions may acquire any sign: ferro, antiferro, anisotropic character, and even higher-order interaction terms. Thus, the full inter-site magnetic exchange interaction defines the system’s ground state. Ferromagnetic interaction leads to the stabilization of the high-spin state, which interacts with the external magnetic field via the Zeeman interaction, whereas antiferromagnetic interaction stabilizes the low-spin ground configuration and thus shows weaker or none Zeeman splitting in an external magnetic field. Given the direct link between exchange interaction and the response to a magnetic field, the name “exchange interaction” is often equivalent to “magnetic interaction”.

3.2 Exchange Mechanisms The theory of exchange interaction was firstly considered by Kramers, developed by Anderson [2–4], and formulated by Goodenough [5–7] and Kanamori [8]. For the last decades, many other authors have elaborated and extended the theory to consider various exchange types in different compounds [9–12]. In this section, we discuss some common exchange mechanisms responsible for promoting magnetic interaction. One of them, which has been observed in many transition-metal compounds, is the direct exchange [13]. It is the direct interaction between magnetic orbitals of interacting magnetic centers that occurs through the space between magnetic sites and becomes important when magnetic centers are relatively close to each other. When open-shell metal sites are relatively distant, the magnetic exchange between them can involve a doubly occupied orbital of a bridging ligand atom. Such mechanism is called superexchange and was proposed by Anderson [3, 4, 14–17]. Another mechanism called double exchange becomes operative frequently in mixed-valence compounds [18, 19], where the highest occupied molecular orbital (HOMO) is delocalized over several metal sites. Finally, we consider magnetic dipole–dipole interaction [20–23] that is of significance in Ising-like lanthanide-based molecular magnets possessing large uniaxial magnetic moments on individual metal sites.

3.2.1 Direct Exchange To describe the direct exchange, we examine herein a system containing two magnetic centers, each of them possessing an unpaired electron in a molecular orbital ψi, j . The direct exchange Hamiltonian for the system can be described by the generalized Hubbard model [12, 24, 25]:

3 Calculations of Magnetic Exchange in Multinuclear Compounds direct HHubbard = Ui nˆ iα nˆ iβ + U j nˆ jα nˆ jβ + Ui j

+ Ki j

∑ σ,σ ,

∑ σ,σ ,

aˆ i†σ aˆ †j σ , aˆ iσ , aˆ jσ + ti j

117

nˆ iσ nˆ jσ ,

) ∑( † aˆ iσ aˆ j σ + aˆ †jσ aˆ i σ ,

(3.17)

σ

/ \ ˆ j . where Ui = ⟨ii|ii⟩, U j = ⟨ j j| j j⟩, Ui j = ⟨i j|i j⟩, K i j = ⟨i j| ji⟩, and ti j = ψi |h|ψ aˆ † and aˆ are creation and annihilation operators, respectively; nˆ = aˆ † aˆ is the occupation number operator; σ and σ , indicate the spin components of the electrons; Ui and U j are called the on-site Coulomb repulsion when the two electrons occupy the same orbital; Ui j is the inter-site Coulomb repulsion when the electrons are in different orbitals; K i j is the exchange integral for the electrons with parallel spins; and ti j is the transfer integral. The energy difference between the singlet E(↑↓) and triplet E(↑↑) states is as follows: ( ) 2 2 2 + (3.18) . ΔST = E(↑↓) − E(↑↑) = 2K i j − ti j U j − Ui j Ui − Ui j If ΔST is positive, the energy of the singlet state is higher than that of the triplet, and thus the direct exchange stabilizes the parallel spin alignment. If ΔST is negative, the direct exchange stabilizes the antiparallel alignment. On the right-hand side of Eq. 3.18, the first term originated from the Coulomb interaction and the Pauli exclusion principle is termed potential exchange [4] that favors the ferromagnetic interaction. The second term arising from electron delocalization is kinetic exchange [4]. Since the on-site Coulomb interaction (Ui and U j ) is always larger than the intersite Coulomb interaction (Ui j ), the second term favors antiferromagnetism. Hence, the competition between these two terms determines whether the direct exchange is ferromagnetic or antiferromagnetic. It is noteworthy that the inter-site transfer integral ti j is strongly dependent on the direct mixing (overlap) between the orbitals, mainly by the kinetic energy operator. The direct exchange has been employed in the theoretical description of transition-metal and radical metal complexes when the magnetic orbitals are relatively diffuse and offer large direct overlap [26–30].

3.2.2 Superexchange To investigate the superexchange mechanism, let us consider a system, where two magnetic centers, each of them containing an electron in a molecular orbital, interact with each other through a doubly occupied (diamagnetic) molecular orbital. This system may be simple, but in fact, the resultant exchange is a combination of multiple exchange mechanisms. Nevertheless, if only superexchange is considered, the Hamiltonian for the system can be represented by the following Hubbard model [31]:

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( ) super HHubbard = Um nˆ iα nˆ iβ + nˆ jα nˆ jβ + Δmk (nˆ kα + nˆ kβ ) ) ∑( † † † aˆ i σ + aˆ †j σ aˆ kσ + aˆ kσ aˆ jσ , aˆ iσ aˆ kσ + aˆ kσ + tmk

(3.19)

σ

where m ∈ {i, j} denotes the magnetic centers, Δmk and tmk are the energy gap and the transfer integral between the paramagnetic and diamagnetic orbitals, respectively. The energy gap between the singlet and triplet states is given as ΔST = −

4 4tmk (Um + Δmk )2

(

1 1 + Um + Δmk Um

) .

(3.20)

Since all the terms on the right-hand side of Eq. 3.20 are positive, ΔST is negative, leading to the antiferromagnetic exchange. If more exchange mechanisms are included in the Hamiltonian (Eq. 3.19), the ferromagnetic contribution can be of significance [5, 6, 32, 33]. Superexchange mechanism has been employed for the description of many lanthanide and transition-metal complexes where the magnetic centers indirectly interact with each other through diamagnetic ligand atoms [34–45].

3.2.3 Double Exchange The double exchange mechanism becomes important in systems where some magnetic electrons can be easily transferred from one metal site to another, a process associated with a change of the metals’ oxidation states upon the metal-to-metal charge transfer process. The transferred electron typically does not reside on the bridging ligand atoms [18, 19, 46–49]. Let us consider a two-center system, where each center contains two orbitals. ψi and ψ j are singly occupied while an electron resonates between ψi, and ψ ,j . The Hubbard model for this system is [18]: double = (Ui + Ui j )(nˆ i , α + nˆ i , β ) + (U j + Ui j )(nˆ j , α + nˆ j , β ) HHubbard ∑ † † ∑ † † aˆ iσ aˆ i , σ , aˆ iσ , aˆ i , σ + K j j , aˆ jσ aˆ j , σ , aˆ jσ , aˆ j , σ + K ii , σ,σ ,

+ ti , j ,

∑ σ

σ,σ ,

(3.21)

(aˆ i†, σ aˆ j , σ + aˆ †j , σ aˆ i , σ ),

where the direct exchange between ψi and ψ j is neglected. The pair of electrons in ψi and ψ j are coupled with each other indirectly by the electron resonance between ψi , and ψ j , (Fig. 3.1). This exchange mechanism is called double exchange and was introduced by Zener [19, 50]. K ii , and K j j , are potential exchange arising from the parallel spins within the same site according to the Hund’s rules, and ti , j , is the transfer integral. By assuming the two centers are equivalent, the energy gap between the lowest quartet and doublet states is

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Fig. 3.1 Scheme of various exchange mechanisms in multinuclear compounds

ΔQD = E(↑↑↑) − E(↑↑↓) =

|ti , j , | . 2

(3.22)

As ΔQD is always positive, the double exchange favors the ferromagnetic coupling between ψi and ψ j . This result is in contrast to the superexchange mechanism, where ferromagnetic or antiferromagnetic interaction occurs between two atoms with a stable valence (the number of electrons on metal sites does not change).

3.2.4 Magnetic Dipole–Dipole Interaction Magnetic dipole–dipole interaction refers to the direct interaction between magnetic dipoles localized on different metal sites. Magnetic dipole moment on metal sites is the sum of the spin and orbital momenta as a result of on-site spin–orbit interaction:

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Fig. 3.2 Scheme of various dipole–dipole interactions in multinuclear compounds

→ → + ge S), → = −μB ( L µ

(3.23)

→ is the orbital moment admixed via spin–orbit coupling, ge ≈ 2.0023193 where L → is the spin moment, μB ≈ 0.46686448 [cm−1 /T] is the is the electronic g-factor, S Bohr magneton. → 2 are two magnetic dipole moments, localized on two differ→ 1 and µ Suppose µ ent metal sites. The magnetic dipole–dipole interaction is given by the following Hamiltonian operator: Hdip =

→ 1µ → 1n → 2n → 2 ) − 3(µ → 12 )(µ → 12 )] μ2B [(µ , 3 r12

(3.24)

→ 12 is the directional vector connecting the two magnetic dipoles and r12 is where n the distance between them. The dipole–dipole interaction is particularly noticeable in the case when the ground state displays a large orbital contribution to the magnetic moment. As an example, the non-magnetic ground states of Dy3 triangles and Dy6 compounds are already set by the dipole–dipole interaction alone [22, 51, 52]. Figure 3.2 shows three cases of the effect of dipole–dipole interaction in uniaxial (Ising) anisotropic metal sites. Besides, there are other exchange effects arising from spin polarization. These effects correspond to electron excitations within a magnetic center which has been observed occasionally in magnetic resonance experiments [53–57].

3.3 Phenomenological Exchange Hamiltonians Spin Hamiltonian approach was first introduced by Heisenberg [58] and later discussed by Dirac [59] and Van Vleck [60, 61]. The prerequisite for the derivation and application of this model is that magnetic orbitals are non-degenerate and the spin quantum number S is a good quantum number.

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Let us consider a two-center isotropic system in which both of them are characterized by their spins Si, j . The Hamiltonian describing the interaction for this system is written as iso = −Ji j Sˆi · Sˆ j , (3.25) Hex where Ji j is an isotropic exchange coupling constant, Sˆi and Sˆ j are the spin operators. Since the spin operator for the coupled spin system is Sˆ = Sˆi + Sˆ j , the Hamiltonian in Eq. 3.25 can be rewritten as: iso =− Hex

Ji j ( ˆ 2 ˆ 2 ˆ 2 ) S − Si − S j , 2

(3.26)

where 2 Sˆi Sˆ j = Sˆ 2 − Sˆi2 − Sˆ 2j . When this operator is expanded in the basis of eigenstates of Sˆ 2 , Sˆi2 , Sˆ 2j , it is obtained diagonal with the eigenvalues corresponding to the energy levels of the coupled system [62]: E(S) = −

Ji j S(S + 1), 2

(3.27)

where S = |Si − S j |, |Si − S j | + 1, . . . , Si + S j . For systems containing more than two magnetic centers, the spin Hamiltonian can be generalized: ∑ iso =− Ji j Sˆi Sˆ j , (3.28) Hex i< j

where the sum runs over all the possible pairwise exchange interaction. If Ji j is positive, the exchange between Si and S j is ferromagnetic. If Ji j is negative, the exchange is antiferromagnetic. It is noteworthy that in some contexts, the spin Hamiltonian can be represented in J Sˆi Sˆ j or − 2J Sˆi Sˆ j formalism. A more general way of expressing magnetic exchange between two interacting spins is given by Eq. 3.29 [54]: Hex = Si · J · Sj = −Ji j Sˆi · Sˆ j + Sˆi · Di j · Sˆ j + di j · Sˆi × Sˆ j ,

(3.29)

where the fully anisotropic interaction matrix J (3 × 3 tensor) is decomposed in three terms: First term is the isotropic contribution (Ji j , scalar), the second term is the symmetric anisotropic contribution (Di j , 3 × 3 symmetric matrix), and the last term is the antisymmetric contribution (di j , 3 × 3 antisymmetric matrix) [9, 63]. The last two terms represent the anisotropic exchange interaction. In this Hamiltonian, the isotropic term favors either the parallel or antiparallel spin alignment; the symmetric term tends to align the spins along a particular axis; while the antisymmetric term tends to align them perpendicular to each other. For Sˆi, j > 1/2, higher-order terms such as biquadratic symmetric and antisymmetric terms can be of importance, as well as the zero-field splitting effect on individual metal ions [64].

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For strongly anisotropic systems, the spin of the metal sites is not a good quantum number and therefore cannot provide a good description of the ground state. Such situations arise when the spin–orbit coupling on metal sites introduces strong mixing between several low-lying (near-degenerate) spin states. In such cases, the orbital moment is unquenched ( Lˆ /= 0). If this interaction is larger than crystal field, the ground energy multiplet is well characterized by the total angular moment quanˆ This situation usually occurs in lanthanide and actinide tum number Jˆ = Lˆ + S. complexes in which the ground multiplet splits into 2J + 1 energy levels under the influence of ligand field. Owing to the anisotropy within the magnetic ions, the exchange coupling between them is also anisotropic. The anisotropic exchange was first indicated by Stevens [65] and has been developed by Moriya [66] and many others [10, 67–69]. As a result, the spin Hamiltonian is usually incapable to describe the anisotropic exchange interaction. The following approaches are applicable in such circumstances: • to employ the spin Hamiltonian as in Eq. 3.29 by using the pseudospins on metal sites. For example, the pseudospin of Dy3+ ion is typically s˜ = 1/2 for a Kramers doublet [70], arising from the crystal–field splitting of the ground J = 15/2. • to employ the full total moment J of the ground state of the interacting ions. However, in this case, the number of parameters is very large, and consequently, such J -Hamiltonians are not easily applicable in phenomenological description of experimental data. The anisotropic exchange between two J -multiplets is far more complex than the isotropic exchange between two S-multiplets. The reason for complexity is the large number of parameters required to describe the on-site crystal field, alongside with a large set of parameters required to describe the full multi-polar exchange interaction. Recently, the J –J exchange was derived analytically by Iwahara [14] which is represented in the following Hamiltonian: aniso Hex

=

∑ kqk , q ,

,

q q O ( Jˆi )Ok , ( Jˆj ) , Jkqk , q , k0 Ok (Ji )Ok0 ( J j )

(3.30) ,

q q where Jkqk , q , is the exchange coupling constant, Ok ( Jˆi ) and Ok , ( Jˆj ) are the extended , Stevens operators [71] with rank k and k and component q and q , , respectively. Since aniso is invariant with respect to the time inversion operator, the sum of k and k , is Hex even [72]. As this model corresponds to the direct exchange between the magnetic ions, the exchange constant is the sum of the potential and kinetic contributions: PE KE Jkqk , q , = Jkqk , q , + Jkqk , q , .

(3.31)

The anisotropic exchange between these ions can be reduced to the Ising exchange of two interacting doublets [11, 73]:

3 Calculations of Magnetic Exchange in Multinuclear Compounds Ising Hex = −Ji j s˜zi · s˜z j ,

123

(3.32)

where s˜zi and s˜z j are the projection of the pseudospin operators along the magnetic anisotropic axis. The conditions to obtain the Ising exchange is discussed thoroughly in Ref. [11].

3.4 Computational Approaches for the Estimation of Magnetic Exchange The computational study of exchange interaction in multinuclear compounds is required to accurately predict low-lying electronic and magnetic properties of the compounds to develop better single-molecule magnets or molecules displaying toroidal magnetization, or estimate structure-property relations, etc. For these purposes, in this section, we will discuss two major classes of computational methods available nowadays for general users: (i) methods based on density functional theory, mainly on the broken-symmetry approach and (ii) methods based on ab initio multiconfigurational wave function theory. In addition, a practical semi-ab initio hybrid approach that is a combination of the ab initio determination of localized wave functions of each metal site and a phenomenological estimation of the magnetic exchange will also be discussed.

3.4.1 Broken-Symmetry Density Functional Theory Density Functional Theory (DFT) is a very popular computational method [74–78], mostly suitable for cases where the ground state is well characterized by a single Slater determinant, following the fundamental theorems by Hohenberg and Kohn [79]. This method has been adapted for the evaluation of exchange interaction in a broken-symmetry scheme as follows: Consider the case of a weakly exchangecoupled system of two spins s1,2 = 1/2, the energy of the high-spin configuration S = 1 (ferromagnetic coupling) can be directly evaluated since its wave function can be represented by a single determinant (Φ(ω1 , ω2 ) = |α1 α2 ⟩). On the other hand, the correct wave function of the low-spin state (antiferromagnetic coupling) has a clear √ multiconfigurational character (Φ(ω1 , ω2 ) = |α1 β2 ⟩ − |β1 α2 ⟩)/ 2) that makes the direct DFT application impractical. In order to overcome this issue, it was proposed to evaluate “a half” of the low-spin wave function, e.g., the energy of the determinant |α1 β2 ⟩, which is not a true spin eigenstate, and the spin symmetry is no longer retained. To have a practical application, this method makes full use of spin-unrestricted formalism, which means that the spin-up and spin-down molecular orbitals are optimized separately. The exchange parameter for the Heisenberg model iso = −2 J Sˆ1 Sˆ2 is evaluated from the computed results of the two configurations. Hex

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For systems containing isotropic magnetic ions, broken-symmetry density functional theory (BS-DFT) has been widely applied for the estimation of exchange interaction [80–84]. There are several formulas allowing to estimate the exchange parameter J from spin-unrestricted BS-DFT calculations [85–89]. Noodleman’s equation for J reads [85–87]: E HS − E BS , (3.33) J =− 2 Smax where E HS and E BS are the total energies for the fully relaxed high-spin and (broken2 is the square of the total spin symmetry) flipped-spin configurations, while the Smax (maximal value). Yamaguchi proposed the following modified formula [88, 89]: J = −/

E HS − E BS / \ \ − Sˆ 2 Sˆ 2 HS

/ \ where Sˆ 2

,

(3.34)

BS

/ \ and Sˆ 2

denote the expectation values of the spin-squared operator for the high-spin and flipped-spin wave functions. It is noteworthy that the expectation values might vary significantly from the ones predicted or expected, mainly due to spin contamination problem in spin-unrestricted calculations [90–92]. HS

3.4.1.1

BS

Challenges of BS-DFT Method

The most common issues regarding the application of BS-DFT method are related to the convergence of the self-consistent field optimization for the broken-symmetry configuration [93]. For instance, in dinuclear Co2 II compounds, both CoII ions have a local spin state S = 3/2. The spin-flip on one of the Co sites may lead to stabilization of the low-spin (S = 1/2) on one, or even on both CoII sites, rather than reversing the full S = 3/2 on one of them. The above issue is generally related to the general complexity and multiconfigurational character of the total wave function, mainly when the broken-symmetry configuration is considered. Similar difficulties are met in connection to the evaluation of magnetic exchange in organic bi- and poly-radicals. Another limitation of BS-DFT relates to its inadequacy in describing the intrinsic multiconfigurational nature of the ground states of systems with near-degeneracy and strong spin–orbit coupling effects on individual metal sites. As an example, consider a binuclear Co2 II compound, where both CoII sites are in quasi-Oh symmetry. The ground state of each CoII located in such a ligand environment is a (slightly split) 4 T2 state, displaying an effective orbital moment ( L˜ = 1, i.e., three states with S = 3/2, near-degenerate). A correct wave function for such a ground state requires several Slater determinants, and, therefore, cannot be correctly described by a single-determinant wave function. Consequently, BS-DFT methods may fail in these circumstances [94]. In order to offer a validity test for the BS-DFT wave functions, some software packages (e.g., ORCA [95]) evaluates the overlap

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between the orbitals of the high-spin and low-spin configurations [96] following the unique definition of the corresponding orbital transformation [97, 98]. If the BS-DFT calculation is valid, the orbital overlaps are either close to one or zero. In contrast, the calculation cannot be trusted in the case of fractional values of the orbital overlap. It is advised to tighten convergence thresholds for the self-consistent field calculations and carefully inspect the results of BS-DFT calculations, namely the calculated charges and spins on individual atoms. Nevertheless, in spite of all the above challenges, the broken-symmetry approach [85] provides a reasonable approximation for magnetic exchange in compounds containing isotropic magnetic ions [99, 100].

3.4.1.2

A Few Examples of Successful Application of BS-DFT

For systems containing more than two magnetic centers, there are two strategies for BS-DFT: (i) to substitute some of the paramagnetic centers with their diamagnetic analogs in order to simplify the total exchange to a set of pairwise interactions; and (ii) to generalize Yamaguchi’s formula by Shoji’s approach [101]. As shown in several works, BS-DFT method can accurately determine the sign of the exchange parameter [102]. However, the magnitude of the parameter may be incorrect due to the overestimation of self-interaction error [103, 104]. While the spin quantum number S has been mostly used to quantify the energy states of transition-metal ions, among the paramagnetic trivalent lanthanide ions, only the energy structure of Gd(III) can be represented by its spin. The reason is that the angular momentum of Gd(III) is zero (L = 0), resulting in the absence of spin– orbit coupling. A noticeable example is a computational investigation of a dinuclear Gd(III)–Cu(II) complex reported by Rajaraman et al. [105]. The spin Hamiltonian for this two-center system was iso = J SˆGd SˆCu , Hex

(3.35)

where SGd = 7/2 and SCu = 1/2. The calculated exchange parameter Jcal was − 5.8 cm−1 (Jexp = − 4.42 cm−1 ). Note that a negative exchange parameter corresponds to ferromagnetic coupling within this Hamiltonian (Eq. 3.35). The report indicated that the account of relativistic effects together with the hybrid B3LYP functional offered an accurate estimation of exchange parameters in this compound [105]. In combination with Molecular Orbital (MO) and Natural Bond Orbital (NBO) analysis, the exchange mechanism in the complex was elucidated in Fig. 3.3. In the Gd(III) center, not only the 4f but also the 5d orbitals play an important role in the magnetic coupling with the Cu(II) ion. The reason for this is that 5d orbitals of the Gd(III) are so diffuse that they gained spin density due to charge transfer from the Cu(II) 3dx 2 −y 2 orbital and spin polarization from the 4f orbitals. This exchange can be considered as a generalization of the Goodenough–Kanamori mechanism [6, 8, 106] giving rise to the ferromagnetic coupling between the magnetic ions. On the other hand, the direct exchange between the 3d and 4f orbitals possessed both ferroand antiferromagnetic contributions. Due to the symmetry of the complex, two of

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Fig. 3.3 Mechanism for the magnetic coupling in {Gd–Cu} [105]. Printed with permission from Ref. [105]. Copyright 2009, Royal Society of Chemistry

the 4f orbitals overlapping with the 3dx 2 −y 2 gave rise to the kinetic exchange that resulted in the antiferromagnetic coupling. The other five 4f orbitals were orthogonal with the 3dx 2 −y 2 leading to the ferromagnetic coupling via the potential exchange. Subsequently, the study was extended to a trinuclear Ni(II)–Gd(III)–Ni(II) complex {Ni–Gd–Ni} [107]. The spin Hamiltonian for the system was iso = −J1 SˆNiA SˆGd − J2 SˆNiB SˆGd − J3 SˆNiA SˆNiB , Hex

(3.36)

where SNiA = SNiB = 1 and SGd = 7/2. The spin configurations considered in this work are listed in Table 3.1. Despite of the large distance between the two Ni ions, the Ni–Ni exchange could not be neglected because (i) the {Ni–Gd} exchange was weak, and (ii) the {Ni–Ni} exchange could be induced indirectly via the diffuse empty 5d and 6s orbitals of the Gd ion [108, 109]. The computed results showed that the {Ni–Gd} exchange interaction was ferromagnetic with J1 ≈ J2 and the {Ni–Ni} exchange was antiferromagnetic. Both electron delocalization and spin polarization took place in the exchange mechanism (Fig. 3.4). Due to the near orthogonality between the 3dx 2 −y 2 /3dz 2 orbitals of the Ni ions and the 4f orbitals of the Gd ion, the antiferromagnetic kinetic exchange in {Ni–Gd} was minor leading to the dominant ferromagnetic interaction. Importantly, the magneto-structural correlation for the {Ni–Gd} exchange was derived that the Ni–O–Gd angles played an important role in the nature of exchange interaction.

3 Calculations of Magnetic Exchange in Multinuclear Compounds Table 3.1 Spin configurations of {Ni–Gd–Ni} Spin configuration NiA Gd HS BS1 BS2 BS3

↑ ↓ ↑ ↓

↑ ↑ ↑ ↑

127

NiB

Spin number

↑ ↑ ↓ ↓

11/2 7/2 7/2 3/2

Fig. 3.4 Mechanism for the magnetic coupling in {Ni–Gd–Ni} [107]. Printed with permission from Ref. [107]. Copyright 2009, Royal Society of Chemistry

3.4.1.3

Application of BS-DFT for the Estimation of Magnetic Exchange in Strongly Anisotropic Compounds

As mentioned earlier, the BS-DFT method is applicable for cases when the ground state is well described by a single Slater determinant. This is true for metals ions such as high-spin Fe3+ , Gd3+ , and Mn2+ . However, the method becomes less adequate for metal ions with near-degenerate ground states, e.g., Co2+ , Dy3+ , Er3+ , etc. The low-spin states of compounds containing the latter metal ions have a true multiconfigurational character and orbital near-degeneracy that renders nonnegligible spin–orbit coupling to first order of perturbation theory. As a result, their ground states are not characterized by the total spin quantum number, but rather by their pseudospin [110]. In order to apply BS-DFT for the estimation of the magnetic exchange in these compounds, the following procedure is proposed: 1. Substitute computationally the strongly anisotropic metal ions by their spinisotropic equivalents, while keeping the ligand frame intact. For instance, Dy3+ whose ground state originates from a slight splitting of the free ion 6 H term (S = 5/2, L = 5) may be computationally replaced by Gd3+ which has a clear non-degenerate ground state 8 S term (S = 7/2, L = 0). 2. Evaluate all magnetic exchange for the metal-substituted compound. 3. Re-scale the exchange parameter for the interacting spins of the original metal ions. For example, the rescaling factor for the magnetic exchange of Dy2 compound, while evaluated for the substituted compound Gd2 is

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JDy–Dy =

2 (7/2)2 49 SGd × JGd–Gd = × JGd–Gd = × JGd–Gd 2 25 (5/2)2 SDy

(3.37)

The above method has been successfully applied in several studies [111–113]. The metal ion substitution can also be a solution for cases where the transition from high-spin configuration to low-spin configuration occurs. There are examples of Mn2+ and Fe2+ displaying low spin on one metal upon spin-flip in the BS-DFT method. To overcome for this issue, the metal ion may be substituted by, e.g., Cu2+ , which has a well-defined S = 1/2 in the ground state.

3.4.2 Multiconfigurational Wave Function-Based Methods for Magnetic Exchange Computational prediction of magnetic exchange by multiconfigurational methods has quite some history. Among first studies refer to the estimation of magnetic exchange in dinuclear Cu2 complexes through the evaluation of singlet-triplet energy difference [114]. Standard Hartree–Fock calculations gave results that were quite far from the ones extracted from the spin Hamiltonian fitting of the measured magnetism. Configuration interaction and multiconfigurational self-consistent field calculations were applied to further improve the agreement [115–117]. It was reported that the electron correlation played a crucial role for the quantitative estimation of spin state energetics of multicenter magnetic compounds and subsequently the magnetic exchange. An accurate account of the electron correlation in exchange-coupled systems is still an open question that further drives the development of novel methods. We may mention here the difference-dedicated configuration interaction (DDCI) method [118] which is a simplified version of the full Configuration Interaction Singles and Doubles (CISD) where excitations that do not contribute directly to the singlet-triplet gaps are eliminated from the CI expansion. In addition, the roles played by various classes of electron excitations were also investigated [115, 116, 119]. While CI methods constitute a systematically improvable approach allowing the calculation of exchange coupling constants through the direct evaluation of spinstate energies [120], their practical application for medium and large molecules is precluded due to the high computational cost required in terms of memory, CPU, and disk space. To some extent, the complications introduced by the (large) basis set expansion of molecular orbitals have been gradually reduced by switching the calculations to either Cholesky representation of bielectronic integrals [121–123] or various methods related to resolution-of-identity (RI) [124] or density fitting (DF) methods alongside with “chains-of-spheres” approaches [125]. More serious difficulties lie in the dimension of the Hilbert space defined by the enlarged active space of the multiconfigurational method. In many cases, the 3d and 4f partly filled orbitals lie at the HOMO-LUMO frontier; therefore, all of them should be included in active space. When following this requirement, the active space grows signifi-

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cantly, rendering the number of Slater determinants obtained from the distribution of active electrons among active orbitals too large to handle. Various methodological developments to reduce the number of excitations within a multiconfigurational framework are (i) restricted active space self-consistent field (RASSCF) and (ii) generalized active space self-consistent field (GASSCF) methods [126]. The main idea of these methods is to divide the total active space into several groups of active orbitals while setting the minimum and the maximum number of electrons occupying them. Another alternative to explicitly building a large multiconfigurational wave function is methods based on a different representation of the wave function, for example, density matrix renormalization group or stochastic approaches such as Quantum Monte Carlo. Among the most promising novel correlated methods along this path, we can mention the density matrix renormalization group self-consistent field (DMRG-SCF) [127–131] and its link to multiconfigurational perturbation theory (either NEVPT2 [132] or CASPT2 [133] and their multi-state variants). As an alternative to DMRG-SCF, there are recent reports about spin-adapted Quantum Monte Carlo self-consistent field (FCIQMC-SCF) and promising studies regarding spin state energetics and exchange interactions [134–137]. One of the key advantages offered by these methods is the ability to include a large number of valence orbitals of the metal ions and of the bridging atoms into the variationally optimized active space of the multiconfigurational self-consistent field method. This large active space allows a balanced description of the local crystal field as well as permits a suitable adaptation of the active orbitals as a function of the total spin of the respective states, which are responsible for the inter-center magnetic interaction. The effect of the electron correlation on natural magnetic orbitals [62] is manifold: The magnetic orbitals become slightly larger, offering a slightly larger hybridization with the ligand orbitals, thus improving the accuracy of the local crystal field (and local spin state energies) and indirectly, increasing the role of the superexchange mechanism. Besides, the direct overlap between magnetic orbitals [138] (which are non-orthogonal in the valence bond representation) of neighboring metal sites is also improved by electron correlation, thus increasing the role of the kinetic exchange mechanism. It is expected that electron correlation contributes essentially to the energies of the metalto-ligand, ligand-to-metal, and metal-to-metal charge transfer configurations, which may contribute to various mechanisms responsible for promoting magnetic exchange [62]. Moreover, electron correlation also contributes to the spin–orbit coupling, as well as the local electric and magnetic dipole moments, which are responsible for the magnetic dipole–dipole interactions (Fig. 3.5). Herein we briefly review recent work by Bogdanov et al. as an example of the application of modern computational approaches for magnetic exchange in cuprates with corner-sharing CuO4 plaquettes [137]. The work reported a series of CASSCF/CASPT2 calculations on model cluster fragments [139–141]. The CASSCF was used to account for the static electron correlation, while the CASPT2 was employed to account for the remaining dynamic correlation energy. It is noteworthy that the exchange parameter Jex is strongly dependent on the accuracy of the wave function. The results clearly showed that the obtained wave function could not reproduce the experimental exchange parameters within small active spaces [142–145].

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Fig. 3.5 Orbitals in active spaces in CuO4 [137]

For instance, Jex obtained from CASSCF(2,2) and CASSCF(8,10) was only ≈ 20% and 80% of the experimental data, respectively. To obtain an accurate wave function, the active space was enlarged significantly to consist of all 3d and 4d orbitals of the copper atoms and the 2p and 3p orbitals of the bridging oxygen (CASSCF(24,26)). The wave functions of such a large active space are impossible to optimize (solve) within the CAS framework. Therefore, for this large active space, the authors adapted full configuration interaction quantum Monte Carlo (FCIQMC) [146, 147] and density matrix renormalization group (DMRG) [148, 149] as an approximation. Combined with the multiconfigurational second-order perturbation theory (CASPT2), the obtained exchange parameter Jex was in reasonable good agreement with the corresponding experimental results.

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To shed light on the exchange mechanism, the computed results were projected on the following effective Hamiltonian [137]:

Heff

⎤ ⎡ A,B ∑ 2 2 2 ∑ ( ) ∑ ∑ ∑ ⎣ = ∈i n Li σ + ti j c†Ai σ c B jσ + c†Biσ c A jσ ⎦ σ

+

L

A,B ∑

[

∑

] n L1σ n L2σ ,

(3.38)

σσ,

A,B ∑ ∑ L

i=1 j=1

U11 n L1↑ n L1↓ + U22 n L2↑ n L2↓ + U12

L

+

i=1

σ

) ( (K 1 n L1σ + K 2 n L2σ ) c†L1σ¯ c L2σ¯ + c†L2† c L1σ¯ ,

where only one 3d and one 4d orbital on each copper atom were included. i = 1, 2 referred to the 3d and 4d orbital, respectively. Diagonalizing the matrix representation of Heff within a simplified basis set illustrated that the correlated breathing-enhanced hopping mechanism plays an important role in the superexchange in this compound. This breathing mechanism could be essentially thought of as the radial expansion of the 3d orbitals of the copper atoms that was described as a mixing of the 3d and 4d orbitals [150, 151]. The breathing effect reduced the Coulomb energy U but increased the transfer integral t; thus, it strongly influenced the magnitude of the exchange parameter. This was the reason why to accurately describe exchange interaction, the active space was extended to include the 4d orbitals, resulting in CASSCF(24,26) [137]. This ab initio approach was also applied in several other works [25, 152–154].

3.4.2.1

Current Challenges of Correlated Computational Methods for Prediction of Magnetic Exchange

While the new methods are attractive in terms of the number of correlated orbitals, their benchmark for the exchange interaction is yet to be reported. One of the most important challenges of novel computational methods for exchange-coupled systems arises from the fact that the magnetic interaction is quite small in magnitude. As an example, in most lanthanide-based compounds, the magnetic exchange is usually smaller than 1.0 cm−1 (1.0 cm−1 = 4.55633 × 10−6 Ha = 1.23981 × 10−4 eV = 1.42879 K); while in most transition-metal compounds, magnetic exchange is usually slightly larger, ranging from a few cm−1 to about 30 cm−1 . Metal-radical compounds usually display larger magnetic exchange interactions. Thus, for any computational method aiming at describing the magnetic exchange correctly, it must be suitable for describing very small energy differences. This requires extremely tight convergence thresholds for optimization of the wave function. For instance, most of the current quantum chemistry programs trigger the convergence of the CI/SCF procedures when the energy difference between two subsequent iterations falls below 10−7 Ha (around 0.02 cm−1 ), which is already on the same energy scale as the

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exchange energy states we wish to describe. The situation is even more complicated by various numerical approximations aimed at speeding up the calculations, like Cholesky or RI approaches because all intrinsically possess a certain numerical error. With respect to electron correlation methods, there is another issue related to spurious splitting, which refers to the significant energy splitting between (near-) degenerate orbital states. For high-symmetry compounds, this artificial splitting is usually ignored. However, in the investigation of slight geometrical distortions which break the orbital degeneracy, this spurious splitting becomes problematic, as it is of the same order of magnitude as the true splitting arising from the geometrical distortion. This issue is not only related to the multiconfigurational second-order perturbation theory, but also to cases when the active space of the CASSCF method is increased inconsistently, breaking the balance between ligand and metal orbitals or breaking the symmetry. Another computational issue is related to the large number of spincoupled states that need to be optimized computationally to offer an accurate account of spin–orbit coupling. This issue is still tractable for transition-metal compounds but falls quickly out of hand in the case of multicenter lanthanide compounds, which are known to require a large number of spin states for spin–orbit interaction already for mononuclear compounds and fragments. Finally, the metal-to-metal, metal-toligand, and ligand-to-metal charge transfer states are also required to account for a correct magnetic exchange description. However, computational evaluation of such electronic states is not a trivial task to date.

3.4.3 Semi-ab Initio Approach for the Description of Magnetic Exchange Given the relatively small magnitude of magnetic exchange in a large class of compounds and the computational difficulties for its accurate evaluation within the multiconfigurational framework, alternative routes were developed. One of the most practical ways to estimate the magnetic exchange of multinuclear strongly anisotropic systems is a hybrid approach that combines multiconfigurational methods with phenomenologic modeling of magnetic exchange. The strategy is as follows: 1. The system is divided into mononuclear fragments. This fragmentation process does not mean removing atoms from the ligand framework, but rather replacing the neighboring metal ions containing open shells by their closest equivalents. It is expected not to alter the low-lying energy structure in each magnetic center significantly. 2. The multiconfigurational ab initio approach (CASSCF/RASSI/CASPT2) is applied to all fragments to obtain an accurate description of their energy structures. 3. The obtained eigenstates of each fragment are then mixed within a suitable exchange model (e.g., the Lines model [155, 156]). In such cases, the parameters of magnetic exchange are obtained empirically by the fitting procedure using the experimental data.

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Fig. 3.6 Fragmentation model of a multicenter compound. The scheme shows an overview of a trinuclear compound and the resulting three mononuclear fragments obtained by the diamagnetic atom substitution method. By this scheme, the neighboring magnetic centers, containing unpaired electrons are computationally replaced by their diamagnetic equivalents. As an example, divalent transition-metal sites are best replaced by either diamagnetic Zn(II) or Sc(III), in the function that one is the closest by atomic radius and charge. For lanthanides Ln(III), the same principle is applicable, and La(III) or Lu(III) is best suited to replace a given magnetic lanthanide. Individual mononuclear metal fragments are then investigated by common multiconfigurational methods (e.g., CASSCF/CASPT2/RASSI/SINGLE_ANISO). The ab initio results for each fragment are subsequently merged within a suitable model of magnetic exchange, e.g., Lines model [155–157]

4. The energy spectra and magnetic properties of the entire multicenter compound can be computed using the best-fitted parameters of magnetic exchange. 5. As an alternative to the fitting process, the parameters of magnetic exchange may be evaluated using the BS-DFT method, as described above (Fig. 3.6).

3.4.3.1

The Lines Model

The Lines model begins with an effective Heisenberg Hamiltonian when the anisotropic exchange is simplified to isotropic exchange in the absence of spin–orbit coupling [155]: ∑ Lines = −Ji j Sˆi Sˆ j , (3.39) Heff i< j

where Sˆi, j is the local spin of the ground state on the metal site i, j, respectively. For cases when the local spin–orbit coupling is strong, the effective Hamiltonian is written on the basis of the local pseudospins: Lines Heff =−

∑

J˜i j sˆ˜i sˆ˜ j ,

(3.40)

i< j

where sˆ˜i, j is the local pseudospin of the ground state on the metal site i, j, respectively. Given that the local pseudospins are rather anisotropic due to a combined effect of low symmetry and strong spin–orbit coupling effects, the Hamiltonian in

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Eq. 3.40 effectively describes the anisotropic exchange between the magnetic centers. Its advantage is that only one exchange parameter can represent a single pairwise anisotropic exchange interaction. The Lines model becomes exact in three cases: • When the two interacting metal sites are isotropic (g X = gY = g Z ). • When the two interacting metal sites are axially anisotropic (g X = gY = 0, while g Z /= 0). In this case, the exchange interaction is of the Ising type. • When one metal site is isotropic, while the other has an extreme axial anisotropy. The Lines model is regarded as a suitable approximation for all other cases when the two metal sites have an intermediate anisotropy. The Lines model has been applied successfully to a variety of compounds [158– 160]. The first example is a mixed-valence Co3 II Co4 III heptanuclear wheel [158]. Ab initio fragment calculations revealed that the ground states of all CoIII sites has S = 0, i.e., are non-magnetic. Similar calculations showed that CoII sites have a ground state characterized by unquenched orbital moment, 4 T2 , split slightly by the low-symmetry components of the crystal field, which deviates from the perfect octahedron. In this case, the spin–orbit interaction is operative in the first order of perturbation theory, leading to a strong mixing of all the three orbital spin S = 3/2 states, giving rise to six Kramers doublets. The energy separation between the ground and first-excited Kramers doublet is more than 250 cm−1 . The corresponding magnetic exchange Hamiltonian was set as follows: ( ) Lines = −J1 sˆ˜Co1 sˆ˜Co2 + sˆ˜Co1 sˆ˜Co3 − J2 sˆ˜Co2 sˆ˜Co3 , (3.41) Heff where s˜Co1 = s˜Co2 = s˜Co3 = s˜Co(II) = 1/2. The reason for the pseudospin s˜ = 1/2 in these Co fragments is the large crystal–field and spin–orbit coupling effects on individual Co sites, leading to a large energy separation between the ground and the first-excited Kramers doublets. This leads to a situation where magnetic exchange interaction couples essentially only the ground Kramers doublets of individual Co sites. In Fig. 3.7, J1 and J2 are nearest-neighbor and next-nearest-neighbor exchange parameters, respectively. Intermolecular exchange is considered in the mean-field approximation, simulated by one single parameter z J , , where z represents the number of the surrounding molecules and J , is the average intermolecular exchange parameter [62]. By the simulation of magnetic properties, the computed exchange parameters were found: J1 = 1.5 cm−1 , J2 = 5.5 cm−1 , and z J , = − 0.03 cm−1 . Interestingly, the exchange interaction between the distant Co1 and Co3 ions was found much stronger than the interaction between the neighbor Co ions (Co1–Co2 and Co1–Co3). The reason for this surprising effect could originate from the fact that this Co3 II Co4 III complex has two exchange paths with low electron-promotion energy Co(II)–Co(III)–Co(III)–Co(II) that could enhance the exchange between the distant Co(II) centers [161]. Slow magnetization of relaxation was not observed experimentally in this complex even it exhibited strong on-site magnetic anisotropy and inter-site exchange interaction. The lack of single-molecule magnet (SMM) behavior in this compound was explained by the large value of the matrix element of the transverse magnetic moments between the two components of the ground exchange

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Fig. 3.7 (Top) Exchange scheme in the Co3 II Co4 III complex, (bottom) computed (lines) and experimental (symbols) temperature-dependent magnetic susceptibility, and (inset) molar magnetization of the Co3 II Co4 III complex [158]. Reprinted with permission from Ref. [158]. Copyright 2008, American Chemical Society

states, indicating a pronounced quantum tunneling of magnetization (QTM) [162]. Consequently, it was suggested that for complexes in the weak-exchange limit, i.e., when zero-field splitting is much stronger than exchange interaction, lowering of site symmetry (i.e., enhancing the axiality) might enhance the magnetization blocking behavior. The Lines model was also applied to multinuclear lanthanide compounds [159, 163–166], for instance, a trinuclear [Dy3 (Meosalox)2 (MeosaloxH)4 (X)(Y)]·S

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Fig. 3.8 a Temperature-dependent magnetic susceptibility, (inset) molar magnetization, and b molecular structure of [Dy3 (Meosalox)2 (MeosaloxH)4 (X)(Y)]·S. In b, dot lines represent the local anisotropy axes, and arrows illustrate the magnetic moment of each Dy ion [167]. Reprinted with permission from Ref. [167]. Copyright 2011, John Wiley and Sons

(X/Y/S = OH/H2 O/MeOH·7H2 O) complex [167]. Since the energy gap between the ground and first-excited Kramers doublets of the complex was much larger than the exchange splitting, the description of the exchange interaction between the ground J -multiplets was reduced to only the ground doublet on each Dy center. Each of these doublets was represented by the pseudospin s˜ = 1/2 [70]. Since the axial component of the ground g-tensor was larger by few orders of magnitude than the transverse components, the magnetic moment of each Dy ion was directed toward its anisotropy axis as shown in Fig. 3.8b. Therefore, the exchange Hamiltonian was close to the Ising model [168, 169]: ) ( HIsing = −J s˜zDy1 s˜zDy2 + s˜zDy1 s˜zDy3 ,

(3.42)

Dyi

where s˜z is the projection of the pseudospin along the anisotropy axis of the Dyi ion. The exchange between the distant Dy ions was neglected. The parameter J was obtained as 7.5 cm−1 . By using a single parameter, the computed magnetic susceptibility and molar magnetization in Fig. 3.8a were in reasonable agreement with the experimental data. Another interesting example is an asymmetric dinuclear [Dy2 ovph2 Cl2 (MeOH)3 ]· MeCN complex [20]. Besides exchange coupling, magnetic dipole–dipole interaction also contributes to the magnetic communication between the lanthanide ions. The intramolecular magnetic Hamiltonian becomes Hˆ intra = Hˆ exch + Hˆ dip .

(3.43)

However, as opposed to magnetic exchange, magnetic dipole–dipole interaction can be computed fully ab initio, using Eq. 3.24 for which all the data are available from fragment calculations. For the case of strong axial anisotropy on metal sites, both exchange and dipolar interactions reduce to Ising magnetic interaction. The

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Fig. 3.9 Temperaturedependent magnetic susceptibility and (inset) molar magnetization of [Dy2 ovph2 Cl2 (MeOH)3 ]· MeCN [20]. Reprinted with permission from Ref. [20]. Copyright 2011, American Chemical Society

Ising parameters obtained by the fitting procedure in Fig. 3.9 were J = 5.88 and z J , = − 1.84 cm−1 ; the Ising dipolar parameter was computed as 5.36 cm−1 . In this compound, the intramolecular exchange contributed significantly less (Jex = 0.52 cm−1 ) than the dipole–dipole interaction, such that the ferromagnetic coupling between the Dy ions originated almost entirely from the latter. This phenomenon was attributed to the near parallel alignment of the local magnetic anisotropies [170]. On the other hand, even though the exchange was very weak, it created a bias field efficiently shifting the quantum tunneling steps at zero-field conditions [171, 172]. In [Cp,2 Dy(μ-X)]2 (Cp, = cyclopentadienyl-trimethylsilane anion, X = CH3 − , Cl− , − − Br , I ), the exchange interaction was modeled by the following Ising Hamiltonian [173]: Hˆ total = −(Jdip + Jex )ˆs1z sˆ2z , (3.44) where sˆz = 1/2 is the ground pseudospin of the Dy ions. The computed results indicated that both the dipolar and superexchange interactions were significant and antiferromagnetic, which was inversely proportional to the Dy–X bond length. In general, the Lines model has been a great tool in computing the magnetic exchange in Ln complexes as the exchange between them is usually small, and their magnetic anisotropy is very axial [174–176]. The Lines model was also applied for the theoretical study of Dy3 triangles in Fig. 3.10, which displayed a non-magnetic ground state, quite unexpected for an oddelectron system [177]. Fragment ab initio calculations revealed that (i) the crystal– field splitting of the ground J = 15/2 multiplet of each Dy site is quite strong, such that the ground Kramers doublet is well separated from the excited Kramers doublets (about 200 cm−1 ); (ii) the ground Kramers doublet possesses a very large magnetic anisotropy, g X ≈ gY ≈ 0, while g Z ≈ 20μB ; (iii) the main magnetic axes of each Dy sites are located almost in the plane of the Dy3 molecule, making an angle between them ≈ 60◦ . The Hamiltonian describing magnetic interaction for the system was

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Fig. 3.10 (Left) Molecular structure and (right) the two components of the ground Kramers doublet of the Dy3 molecule [177]. Reprinted with permission from Ref. [177]. Copyright 2008, John Wiley and Sons

Lines Heff =−

3 ∑

Ji Sˆi Sˆi+1 = −J

i=1

3 ∑

Sˆi Sˆi+1 ,

(3.45)

i=1

where Ji = J is an exchange parameter and Si = 5/2 is the spin of the DyIII ions in the absence of the on-site spin–orbit interaction. The addition of the spin–orbit coupling and further diagonalization of the above Hamiltonian gives the exchange spectra and exchange eigenstates for the investigated Dy3 triangle. Given that the local Kramers doublets were close to |MJ = ± 15/2⟩ kind and that excited Kramers doublets are much higher in energy with respect to the magnitude of magnetic interaction, the exchange Hamiltonian was expected to be of the Ising type between the local Kramers doublets (non-collinear): 3 ∑ HIsing = − J˜ s˜i z s˜i+1z , (3.46) i=1

where s˜i z is the z component of the ground state pseudospin. Since S = 5/2 and s˜ = 1/2, (3.47) J˜i = 25 cos φi,i+1 Ji = − 12.5Ji , where φi,i+1 ≈ 2π/3 is the angle between any pair of main axes (Fig. 3.10, cos(2π/3) = 1/2). The simulation of magnetic properties in Fig. 3.11 yielded J = − 0.6 cm−1 ( J˜ = + 7.5 cm−1 ). This ferromagnetic exchange stabilizes the local moments in a toroidal fashion, which leads to a cancelation of the total magnetic moment in the ground state. The ground exchange Kramers doublet state consisted of two components, as shown in Fig. 3.10, where all the local magnetic moments were arranged clockwise and anticlockwise, both non-magnetic. These two states ⎡+ and ⎡− in Fig. 3.10 possess a toroidal magnetic moment that is oriented perpendicularly to the plane of the Dy3 molecule, τ+ and τ− , respectively. As a consequence of the non-magnetic ground state of the molecule, the ground doublet does not inter-

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Fig. 3.11 (Left) Magnetic susceptibility and (right) molar magnetization of the Dy3 molecule [177]. Reprinted with permission from Ref. [177]. Copyright 2008, John Wiley and Sons

act with an external magnetic field applied perpendicularly to the plane of the Dy3 molecule. However, Zeeman splitting is significant between the excited (magnetic) exchange states for the in-plane magnetic fields. Due to the large magnetic moment in the excited exchange states, they quickly become the ground state already for field strengths of about 1.0 T. It is highlighted that molecules displaying stable toroidal moment in the ground states are sought for their potential in quantum computing and information storage since the toroidal state is not perturbed by the neighboring magnetic fields, and as such, may display a long lifetime. Controlling the toroidal states, write-in and read-out, of toroidal information may be undertaken by electric fields [178–180]. Besides this work, the Lines model was also applied to many other complexes to investigate their toroidal magnetic moment. Details about them can be found in Refs. [181–185]. The next class of compounds which has been studied intensively using the model is heterometallic 3d–4f molecules [157, 186–195], for example, a [Cu5 II Dy4 III ] cluster [196]. As the magnetic anisotropy axes of the Dy ions were found to be nearly perpendicular to each other, the magnetic exchange between them was neglected. These results were due to the following reasons: (i) the magnetic anisotropy on the Dy sites offered a nearly Ising exchange and (ii) the parameter of the Ising exchange was J˜Ising = 25 cos φ1,2 JLines and φ1,2 = π/2. Therefore, only three exchange interactions that were supposed to play the dominant role were included in the Lines model: Dy Dy Dy Dy Lines = −J1 ( Sˆ1 Sˆ5Cu + Sˆ1 Sˆ8Cu + Sˆ1 Sˆ9Cu + Sˆ2 Sˆ5Cu Heff Dy Dy Dy Dy + Sˆ2 Sˆ6Cu + Sˆ2 Sˆ7Cu + Sˆ3 Sˆ5Cu + Sˆ3 Sˆ6Cu Dy Dy Dy Dy + Sˆ3 Sˆ7Cu + Sˆ4 Sˆ5Cu + Sˆ4 Sˆ8Cu + Sˆ4 Sˆ9Cu ) ( ) − J2 Sˆ5Cu Sˆ6Cu + Sˆ5Cu Sˆ7Cu + Sˆ5Cu Sˆ8Cu + Sˆ5Cu Sˆ9Cu ( ) − J3 Sˆ6Cu Sˆ7Cu + Sˆ8Cu Sˆ9Cu ,

(3.48)

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Fig. 3.12 (Left) Computational model and (right) magnetic simulation of [Cu5 II Ln4 III ] [196]. Reprinted with permission from Ref. [196]. Copyright 2011, John Wiley and Sons

where the structure of magnetic ions is shown in Fig. 3.12a. Due to the axial nature of the Dy ions, the {Cu–Dy} exchange was approximately Ising type. The obtained exchange parameters were J1 = 1.0 cm−1 , J2 = 1.3 cm−1 and J3 = − 4.6 cm−1 . Therefore, the former two interactions were ferromagnetic while the latter was antiferromagnetic. The analysis of the exchange states also indicated that the main magnetic axis of the cluster was almost perpendicular to the Dy4 plane. Recently, actinide complexes have attracted some attention in the field of molecular magnetism [197–200]. While their magnetic properties are interesting, the multiconfigurational ab initio studies of them have faced significant challenges due to their intricate electronic structures already for mononuclear actinide compounds [200–206]. The main computational challenges arise due to combined effect of the strong relativistic effects, spin–orbit coupling, and metal-ligand covalency, demanding explicit account of electron correlation. The accuracy of the ab initio calculations refers here not only to the low-lying energy structure but also to the wave functions, electric and magnetic dipole moments, etc., which are challenging for actinide compounds. With respect to the application of the Lines model for the description of magnetism in exchange-coupled actinide compounds, we do not infer any major

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problem, except that the strength of the interaction may be stronger compared to exchange-coupled lanthanides. In the case, the local energy structure and magnetic properties of a single actinide ion could be accurately evaluated, then it is possible to apply the Lines model to simulate their exchange interaction. The Lines model and its applications have been discussed in the above examples. Despite its simplicity, the model has described the exchange coupling quantitatively in a diversity of compounds. Its accuracy strongly depends upon the local energy states calculated by the preceding ab initio methods. Recently, we have further improved and implemented some enhanced versions of the original Lines model in the POLY_ANISO module, which are named LIN3 and LIN9. Their matrix representation is ⎛

HLIN3

HLIN9

−J X X = (sˆ˜ X 1 , sˆ˜Y1 , sˆ˜Z 1 ) ⎝ 0 0 ⎛ −J X X = (sˆ˜ X 1 , sˆ˜Y1 , sˆ˜Z 1 ) ⎝ −JY X −J Z X

⎞ ⎛ˆ ⎞ s˜ X 0 0 ⎜ ˆ 2⎟ ⎠ −JY Y 0 ⎝ s˜Y2 ⎠ , 0 −J Z Z sˆ˜Z 2 ⎞ ⎛ˆ ⎞ s˜ X −J X Y −J X Z ⎜ 2⎟ −JY Y −JY Z ⎠ ⎝ s˜ˆY2 ⎠ , −J Z Y −J Z Z sˆ˜ Z 2

(3.49)

(3.50)

where the sˆ˜αi represents the pseudospin on-site i written in its own main axes. The LIN3 and LIN9 models are the extensions of the original Lines model that are more generalized to the full 3 × 3 exchange matrix in Eq. 3.29. These two models are for cases where the exchange cannot be simplified to the isotropic exchange in the absence of spin–orbit coupling. While these models give more freedom to describe a more complex interaction, these models may face over-parameterization issues, as there are a larger number of parameters involved.

3.4.4 Kinetic Exchange Model In 2011, a series of N2 3− -radical-bridged dilanthanide complexes [K(18-crown6]{[(Me3 Si)2 N]2 (THF)Ln}2 (μ-N·2 )] (Ln = Gd (1-Gd), Tb (1-Tb), Dy (1-Dy), Ho (1-Ho), Er (1-Er)) (Fig. 3.13) was reported [207, 208]. This series has immediately attracted huge attention due to the giant exchange interaction between the Ln3+ and N2 3− centers, which is a few orders of magnitude larger than in the conventional complexes. The exchange strength is of the same order as the crystal–field interaction; hence, instead of the ground doublets, the exchange process involves the whole ground J -multiplets [15]. Therefore, the exchange Hamiltonian for the series has to be derived from Eq. 3.30 [14]: Hex =

∑∑ i=1,2 kqq ,

Ok ( Jˆi ) Sˆq , , Ok0 (Ji )S q

Jkq1q ,

(3.51)

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Fig. 3.13 a Molecular structure and b experimental (symbols) and calculated (lines) magnetic susceptibility of 1-Ln [15]

which describes the interaction between the total angular momentum Ji of the lanthanide ions (i = 1, 2) with the spin S = 1/2 of the radical. Jkq1q , only accounts for the kinetic exchange as it is usually much stronger than the potential exchange [3, 4]. Due to the D2h symmetry of the magnetic core, the π ∗ orbital of the radical only overlaps with the 4fx yz orbital of the lanthanide ions [15]. Therefore, the maximal rank of k and |q| are 7 and 5, respectively. The set of Jkq1q , was derived by electronic parameters which were obtained from the combination of theoretical models and calculations. The transfer parameter and the energy gap of the 4f and π ∗ orbitals were extracted from orbital energies in DFT calculations via the tight-binding Hamiltonian: [ ∑ ∑ ∈f nˆ i γ˜ σ + ∈π ∗ nˆ π ∗ σ H= σ (3.52) i=1,2 ( )] † † † † + t aˆ 1γ˜ σ aˆ π ∗ σ + aˆ π ∗ σ aˆ 1γ˜ σ − aˆ 2γ˜ σ aˆ π ∗ σ − aˆ π ∗ σ aˆ 2γ˜ σ , where σ, n, ˆ a, ˆ and aˆ † are defined in Eq. 3.17; γ˜ indicated the Cartesian component of the orbital; ∈f and ∈π ∗ were the energies of the 4f and π ∗ orbitals, respectively; t and Δ = ∈π ∗ − ∈f were the transfer parameter and 4f − π ∗ energy gap, respectively. The diagonalization of the Hamiltonian in Eq. 3.52 resulted in ∈f,a = ∈f , ∈f,s = ∈f + ∈π ∗

1(

√

)

Δ − Δ2 + 8t 2 , 2 ) √ 1( = ∈f + Δ + Δ2 + 8t 2 , 2

(3.53) (3.54) (3.55)

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where a and s denoted the antisymmetric and symmetric orbitals, respectively. Once these orbital energies were obtained from DFT calculations, t and Δ were derived from the above system of equations. Similarly, the electron-promotion energy from π ∗ to 4f in 1-Gd could be derived from the Hubbard Hamiltonian: ∑ ∑∑ ∈f nˆ iγ σ + ∈π ∗ nˆ π ∗ σ H= i=1,2 γ σ

σ

) ∑ ( † + t aˆ 1γ˜ σ aˆ π ∗ σ + aˆ π† ∗ σ aˆ 1γ˜ σ − aˆ 2†γ˜ σ aˆ π ∗ σ − aˆ π† ∗ σ aˆ 2γ˜ σ σ

+

∑

∑

Uf nˆ iγ σ nˆ iγ , σ , + Uπ ∗ nˆ π ∗ α nˆ π ∗ β +

∑ ∑∑ i=1,2 γ σ

i=1,2 ⟨γ σ,γ , σ , ⟩

Uf,π ∗ nˆ γ σ nˆ π ∗ σ , ,

σ,

(3.56) where γ was the component of the 4f orbital; Uf , Uπ ∗ were on-site, and Uf,π ∗ was intersite Coulomb repulsion as defined in Eqs. 3.14 and 3.17. The energy gap between the high-spin and low-spin states was ΔE = E LS − E HS = where

/ ) ( 1 ¯ 2 U − U¯ + 8t 2 , 2

U¯ = nUf − Δ − 2nUf,π ∗ ,

(3.57)

(3.58)

in which, U¯ was the averaged promotion energy, and n was the number of 4f electrons in the Ln3+ ions. Using the high-spin and low-spin energies obtained from DFT calculations, U¯ for 1-Gd was derived. For the other Ln3+ ions, U¯ could not be derived in this way owing to their multiconfigurational nature. Nevertheless, U¯ was not suitable for the calculation of the exchange parameters since it was just an averaged value and could be inaccurate due to insufficient dynamic correlation in the functional. Therefore, the promotion energy was calculated as follows: U = U0 + ΔE α ,

(3.59)

where U0 was the minimal promotion energy and ΔE α was the excitation energies of the intermediate virtual electron transferred states. The former was obtained from the simulation of experimental magnetic susceptibility, while the latter was approximated as the excitation energies for free Ln2+ /Ln4+ ions which were computed by the multiconfigurational methods. Substituting t, Δ and U in the formula (Eq. 35 in Ref. [14]) resulted in the set of exchange parameters Jkq1q , . The calculated results show that the energy gap between the 4f and π ∗ orbitals was extremely large (Δ ≈ 104 cm−1 ), leading to small promotion energy (U0 ≈ 103 –104 cm−1 ). This was the origin of the giant exchange interaction in this series. Moreover, the (4f)n (π ∗ )1 ⇔ (4f)n−1 (π ∗ )2 promotion energy was much larger than that of (4f)n (π ∗ )1 ⇔ (4f)n+1 (π ∗ )0 ; hence, the former process was neglected. The

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Fig. 3.14 Molecular structure of (top left) 2-Ln and (top right) 3-Ln. (Bottom) (lines) calculated and experimental (symbols) temperature-dependent magnetic susceptibility (χ T ) at H = 1 T [16]. Reprinted with permission from Ref. [16]. Copyright 2021, American Chemical Society

trend of the transfer parameter was tGd > tTb > tDy > tHo > tEr because the higher the atomic number was, the smaller the ionic radius was; thus, the mixing between the orbitals by the kinetic operator decreased. In this exchange model, the only fitting parameter was U0 , whereas the other parameters were derived explicitly. The analysis of the computed exchange parameters shows that the non-negligible terms were up to rank k = 7. The first-rank terms were dominant and isotropic due to the fact that only the 4fx yz orbital participated in the exchange process. If the other 4f orbitals are involved, the anisotropy of the first-rank terms will be activated. It is conclusive that the exchange interaction intermixed the ground and excited crystal–field levels on each Ln ion. Therefore, it diminished the magnetic axiality of the low-lying exchange states and restricted the blocking barrier to the first-excited level. Following by this theoretical study, two other N2 3− -radical-bridged dilanthanide Ln(THF))2 (μ-N·2 )]− (THF = tetrahydrofuran, CpMe4H = series, which were [(CpMe4H 2 Ln)2 (μ-N·2 )]− tetramethylcyclopentadienyl, Ln = Tb (2-Tb), Dy (2-Dy)) and (CpMe4H 2 (Ln = Tb (3-Tb), Dy (3-Dy)) (Fig. 3.14) [16, 209] were investigated. The kinetic exchange model was applied to these complexes as the magnetic core remained the same. All the electronic parameters were kept fixed except the minimal promotion energy (U0 ) which was obtained by the simulation of magnetic properties.

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Table 3.2 Experimental (exp) and calculated (cal) blocking energy barriers [16] 1-Tb 1-Dy 2-Tb 2-Dy 3-Tb (exp) Ueff1 (cal) Ueff1 (exp) Ueff2 (cal) Ueff2

227 208 – –

123 121 – –

242 240 – –

110 104 – 208

276 222 564 440

3-Dy 108 113 – 210

The computed results indicated that the surrounding ligands did not significantly impact the nature of the exchange interaction provided by the radical. The reason was that the first-rank exchange terms were dominant and isotropic due to the D2h symmetry of the magnetic core. Moreover, in 2-Dy, 3-Tb, and 3-Dy, there existed two blocking energy barriers corresponding to two magnetization relaxation paths. However, the property was only observed experimentally in 3-Tb. It might be because the measured temperature in the Arrhenius plots for 2-Dy and 3-Dy was not high enough to detect the high energy path [16, 209]. As given in Table 3.2, the blocking energy barrier of the Tb congeners is always twice larger than that of the Dy analogs, which appears to be a general property of the N2 3− -radical-bridged complexes. Since the surrounding ligands and the bridging radical are identical, the property must originate from the Ln ions themselves. A thorough examination of the exchange states in 3-Tb and 3-Dy showed that the exchange splitting in the Tb complexes is approximately twofold that of the Dy complexes. It is because that (i) the π ∗ orbital of the radical only overlaps with the 4fx yz orbital(s) corresponding to |L , M L ⟩ = |3, ± 2⟩, and (ii) the kinetic exchange mainly arises from the (4f)n (π ∗ )1 ⇔(4f)n+1 (π ∗ )0 process [14, 15]. As illustrated in Fig. 3.15, for Tb3+ ion, there exist two electronic transfer paths; for Dy3+ , the orbital |3, 2⟩ is already doubly occupied that the electronic transfer only happens via the other orbital. Thus, the blocking barrier in the Tb complex is around twice larger than in the Dy analog as the magnetization in this family mostly relaxes through the first-excited level [16] (Fig. 3.16). A subsequent computational investigation of 3-Tb [17] shows that indeed the Cp ligands lowered the axiality of the complex. The reason was that besides exchange coupling, the N2 3− radical also provided a non-negligible crystal–field effect. As the radical formed the shortest bond with the Tb ions, the magnetic anisotropy axis was oriented by the radical. Thus, the Cp ligands provided the CF effect perpendicular to the anisotropy axis that diminished the axiality of the Tb ions. It is illustrated in Fig. 3.17d that if the crystal field of the surrounding ligands is collinear with the exchange interaction, the axiality will be enhanced significantly. Further analysis highlighted that the ultimate SMM performance of exchange-coupled compounds could be obtained when the crystal field was collinear with the magnetic anisotropy axis while the magnetic exchange was ferromagnetic [210, 211]. For this purpose, the energy of the magnetic orbital of the radical needed to be in resonance with the 6s/5d orbitals of the lanthanide to activate the Goodenough mechanism [212]. This

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Fig. 3.15 Kinetic exchange path(s) between the π ∗ orbital and 4fx yz orbital(s) in Tb3+ and Dy3+ ions. For clarity, the splitting between orbitals is not drawn at its true scale [16]. Reprinted with permission from Ref. [16]. Copyright 2021, American Chemical Society

Fig. 3.16 Molecular structures of a 1-Tb, b 1-Cp, c 1-N, and d 2-Tb [17]

prediction was recently confirmed by a novel (CpiPr5 )2 Dy2 I3 (CpiPr5 = pentaisopropylcyclopentadienyl) complex, which has set the record 100-s blocking temperature of 72 K [213].

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Fig. 3.17 Calculated energies of the ground J -multiplet of the Tb ions (cm−1 ) [17]. Reprinted with permission from Ref. [17]. Copyright 2022, John Wiley and Sons

3.5 Conclusions In this chapter, we have introduced the calculation of magnetic exchange in multinuclear compounds from theory to current applications. Several exchange mechanisms have been discussed along with some phenomenological exchange Hamiltonians. Three main approaches for the computational studies of magnetic exchange that have been examined thoroughly are broken-symmetry density functional theory (BS-DFT), multiconfigurational wave function-based methods, and semi-ab initio approach. Each of them has its own advantages and disadvantages that are suitable for different types of complexes. For instance, whereas BS-DFT is very robust, it is not suitable in cases where the ground state is well characterized by a single Slater determinant. Various practical approaches for the application of BS-DFT for deriving magnetic exchange in lanthanide and transition metals are discussed. Several studies where computational magnetic exchange was derived from multiple computational approaches complementing each other are discussed in detail [15, 214]. Besides reviewing various interesting results, we have also pointed out the challenges and limits of current computational methods for predicting magnetic exchange. In conclusion, we hope that this chapter will stand as a motivation for the development of novel computational methods, aiming at accurate description of electronic structure and properties of exchange-coupled systems, for the ultimate dream of computational prediction of electronic structure and properties of multicenter multifunctional materials.

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Chapter 4

Exact Diagonalization Techniques for Quantum Spin Systems Jürgen Schnack

Abstract The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groups—and their application. However, the applicability of exact diagonalization is limited by the exponential|| growth of the Hilbert space of the spin system with the number of spins: dim(H) = i (2si + 1). Nowadays, complex Hermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size si to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos method, allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011 . Hands-on experience and examples will be provided. Keywords Quantum spin systems · Exact diagonalization techniques · Finite-temperature Lanczos method

J. Schnack (B) Faculty of Physics, Bielefeld University, 33601 Bielefeld, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Rajaraman (ed.), Computational Modelling of Molecular Nanomagnets, Challenges and Advances in Computational Chemistry and Physics 34, https://doi.org/10.1007/978-3-031-31038-6_4

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4.1 Introduction Molecular magnetism is the result of interactions involving electronic spins and angular momenta. It is characterized by correlation effects encoded in the manybody quantum states of such systems [1, 2]. It turns out that the overall low-lying energy levels of such systems can be modeled by spin Hamiltonians rather accurately, in particular as long as spin–orbit interactions are small as for instance for 3d ions [3, 4]. Spin Hamiltonians contain only spin operators and model parameters that aggregate the intricate influence of the full many-electron problem. A general spin Hamiltonian up to second order in spin operators reads Hˆ =

E

/\

/\

s-i · J i j · s- j .

(4.1)

i≤ j /\

Here, operators are marked by a hat. The s-i denotes spin vector operators with respective spin quantum numbers si residing at site i. One should keep in mind that for many such operators, i denotes an abstract site and not a real position. Real positions enter the Hamiltonian only for interactions such as the dipolar interaction. J i j is a 3 by 3 matrix for each pair (ij) of spins including the case (i = j ). For (i /= j ), it provides the strength of the exchange interaction, whereas for (i = j ), it models the single-ion anisotropies that the various spins experience. The latter example demonstrates how spin–orbit interaction is taken into account in spin models—it enters the Hamiltonian via parameters and terms that could be derived from perturbation theory, see e.g., [5–7]. Higher orders in spin operators such as biquadratic terms are possible and can be derived from, e.g., multi-orbital Hubbard models or simply be postulated as formal series expansions. A special and very relevant case when considering 3d ions is the Hamiltonian of the isotropic Heisenberg model Hˆ =

E

/\

/\

Ji j s-i · s- j .

(4.2)

i< j

Here, the exchange is given by numbers Ji j which means that the interaction is isotropic. This means that there are no preferred or special directions. In a classical picture, the interaction depends only on the relative orientation of spin directions encoded in the dot product. Quantum mechanically, this leads to the conservation of total spin as discussed later. Such symmetries turn out to be very useful both for numerical calculations as well as for the discussion of physical quantities [8]. The eigenstates of spin Hamiltonians are true correlated many-body states such as spin multiplet states like singlet states. Phenomena like quantum tunneling of the magnetization require the description in terms of Hamiltonians allowing for superpositions of many-body states. In this respect, the consideration of spin Hamiltonians is complementary to density functional theory (DFT) where the correlations are embodied in energy functionals, and correlated states are not available. The goals of

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an approach via spin Hamiltonians are thus twofold: One seeks an understanding of the physical properties of the investigated material and often one wants to determine the parameters of the model from fits to experimental data; the latter is being opposite to a DFT approach. In order to achieve the desired goals, the objective is threefold: (1) evaluation of the partition function and related thermal equilibrium expectation values to understand thermodynamic properties, (2) evaluation of time evolution to understand and use, e.g., quantum manipulation, and (3) evaluation of transition rates to understand spectroscopic data such as given by electron paramagnetic resonance (EPR), nuclear magnetic resonance (NMR), or inelastic neutron scattering (INS). In the following, we focus on the partition function and equilibrium thermodynamics and provide references for the other cases. The partition function is the central object of equilibrium thermodynamics, and it is defined as ) ( ) ( ˆ Z T , B- = tr e−β H ( B ) , β =

1 , kB T

(4.3)

with T being the absolute temperature, ( ) B the magnetic induction, ( ) and kB Boltzmann’s constant. If the eigenvalues E ν B- of the Hamiltonian Hˆ B- are known, i.e., the eigenvalue equation ( ) ( ) Hˆ B- |ν> = E ν B- |ν>, ν = 1, . . . , dim(H),

(4.4)

with |ν> being the eigenvectors, could be solved, then the partition function is easily evaluated as ( ) E (4.5) Z T , B- = e−β Eν ( B ) . ν

Thermodynamic observables can then be derived via ( ) | ( )| G T , B- = −kB T ln Z T , B- , ( ) ∂ ( -) G T, B , S T , B- = − ∂T ( ) ( ) - T , B- = − ∂ G T , B- . M ∂ B-

(4.6) (4.7) (4.8)

( ) The entropy S T , B- can be related to the heat capacity via ( ) ∂ ( -) S T, B , C T , B- = T ∂T

(4.9)

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( ) - T , B- can be related to the susceptibility tensor as and the magnetization M ( ) ( ) ( ) ∂ ∂ ∂ χi j T , B- = M j T , B- = − G T , B- . ∂ Bi ∂ Bi ∂ B j

(4.10)

More general observables can be evaluated according to the following formula ( ) O T , B- =

(

1

Z T , B-

)

E −β E ν ( B- ) ˆ e .

(4.11)

ν

It is therefore immediately understandable that the determination of all energy eigenvalues and eigenvectors is of great help (if not an indispensable prerequisite) to evaluate the partition function and thus thermodynamic quantities. The same is true in a very similar way for time evolution since the time evolution operator is structurally the same as the operator under the trace in (4.3). If eigenvalues and eigenfunctions of the Hamiltonian are known, spectral functions can be evaluated as well [9].

4.2 Exact and Complete Diagonalization Techniques Before we discuss how energy eigenvalues and eigenvectors can be obtained, we need to recapitulate some basic knowledge on spin operators. The spin is a vector operator /\

s- = sˆx e-x + sˆy e-y + sˆz e-z

(4.12)

where each spatial component is again an operator; these fulfill the following commutator relation | | sˆx , sˆy = ihˆsz

(4.13)

and cyclic permutations thereof. Since the square of a spin vector operator commutes with its z-component, they have a common eigenbasis given by 2

/\

s- |sm> = h2 s(s + 1)|sm>,

(4.14)

sˆz |sm> = hm|sm>, m = −s, −s + 1, . . . , s − 1, s.

(4.15)

We will omit h in the following; it is absorbed into appropriate units. The introduction of ladder operators sˆ ± = sˆx ± i sˆy

(4.16)

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159

with sˆ ± |sm> =

/

s(s + 1) − m(m ± 1)|sm ± 1>

(4.17)

will turn out to be useful when evaluating the Hamiltonian with respect to a chosen basis. Since the Hilbert space of a system of N spins is given by the product space of the individual single-spin Hilbert spaces, a possible and easy to construct many-spin basis is the product basis |m> - = |s1 m 1 , s2 m 2 , . . . , s N m N >

(4.18)

- = m j |m> sˆ zj |m>

(4.19)

for which

applies. When spatial indices and spin number indices are used, the spatial indices are placed as superscripts. The dimension of the Hilbert space for N spins with spin quantum numbers si is dim(H) =

||

(2si + 1)

(4.20)

i

which for equal spins amounts to (2s + 1) N , i.e., the dimension of the Hilbert space grows exponentially with number of spins. This restricts the size of solvable problems considerably.

4.2.1 Complete Diagonalization Without Symmetries Very general Hamiltonians do not possess (many) symmetries. In such cases, we have to determine all eigenvalues at once, i.e., solve the eigenvalue equation for the complete Hamiltonian Hˆ |ν> = E ν |ν>.

(4.21)

To this end, we write the eigenvalue equation in matrix form using the product basis (4.18) E < | | >< | '> > < ' > - m|ν - = Eν m - . m - ' | Hˆ |m - |ν ; ∀|m m -

(4.22)

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< '| | > < ' > - as input and delivers E ν as well as m - | Hˆ |m - |ν as A computer program needs m < '| | > | | ˆ - can be easily evaluated if the Hamiltonian is - H m output. The matrix elements m expressed in terms of operators sˆiz and sˆi± only since their application onto the product < '| | > - is real symmetric for - | Hˆ |m basis is known, compare (4.15) and (4.17). The matrix m Heisenberg Hamiltonians and complex Hermitean in general. Many programming libraries such as LAPACK, MKL, or NUMPY offer routines for diagonalization. As already mentioned, linear matrix dimensions of about 100,000 are manageable. An example is given by the family of {Mn6 Cr} clusters [10] where the dimension of Hilbert space amounts to 62,500 because of s = 2 for Mn(III) and s = 3/2 for Cr(III). These molecules can be modeled with a Hamiltonian containing two exchange interactions as well as six easy-axis anisotropy terms, see Fig. 4.1a, Hˆ =

E

/\

/\

)2 E( s-i · e-i . /\

Ji j s-i · s- j + d

i< j

(4.23)

i

The anisotropy is modeled by easy-axes along non-collinear directions e-i provided by the Jahn–Teller axes of the respective manganese ions. Since all eigenvectors can |< | x | >|2 |υ can be evaluated as well. The latter be obtained, transition probabilities μ| Sˆtot characterizes the strength of the transition induced by a transversal field in this single-molecule magnet, see Lippert, Inorg. Chem. 56, 15119 (2017). If the experimental sample is a powder, then an angular (powder) average has to be performed when calculating observables. There are accurate approximations to a full integration over all angles (Lebedev–Laikov grids); however, for every combination of spherical angles ϑ and ϕ, the Hamiltonian has to be diagonalized again [11].

4.2.2 Employing Total Sˆz -Symmetry and Cyclic Point Group Symmetries The central idea of all schemes employing symmetry is to set up symmetry-related basis states that span orthogonal subspaces labeled by the group characters/quantum numbers of the symmetries used. In the case of total Sˆ z -symmetry discussed in the following, these subspaces are labeled by the total magnetic quantum number M. The z-component of the total spin is given by the sum of the z-components of all individual spins Sˆ z =

E

sˆiz ,

(4.24)

i

| | and total Sˆ z -symmetry means that Hˆ , Sˆ z = 0. The states of the product basis are already eigenstates of Sˆ z ; for later use, they are sorted according to the quantum number M [12]

Fig. 4.1 a Temperature dependence of μeff for the enantiopure chiral triplesalen-based SMM RR [MnIII 6 CrIII ](ClO4 )3 at 0.01 T and coupling scheme used. b Variable temperature—variable field magnetization measurements at 1, 4, and 7 T for RR [MnIII 6 CrIII ]3+ . Experimental data are given as circles. The solid curves correspond to simulations performed by a full-matrix diagonalization of the multi-spin Hamiltonian provided by Eq. (4.23). c Magnetic energy spectra for RR [MnIII 6 CrIII ]3+ (c, e) and for comparison for [MnIII 6 CrIII ]3+ ([{(talent−Bu2 )MnIII 3 }2 {CrIII (CN)6 }(MeOH)3 (CH3 CN)2 ](BPh4 )3 ·4CH3 CN·2Et2 O48 (d, f), obtained from the simulation of the magnetic data. Only low-lying energy levels are shown. The x-axis represents the magnetization of each eigenstate, which corresponds to the expectation value for a tiny magnetic field along the C 3 quantization axis. The blue crosses in c, d correspond to the M S states of an isolated S = 21/2 spin multiplet of the same barrier as of the respective complexes. The eigenstates are colored with respect to their symmetries (red: symmetric, black: antisymmetric). In e, f, selected values of the probabilities of coherent transitions by transversal field components (QTM) are provided. The bold green arrows summarize the “allowed” phonon-assisted thermal direct, Raman, and Orbach processes. The thin red arrows correspond to coherent QTM pathways. Reprinted with permission from K.-A. Lippert et al., Inorg. Chem. 56, 15119. Copyright 2017. American Chemical Society

4 Exact Diagonalization Techniques for Quantum Spin Systems 161

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J. Schnack

| \ | E | |m; - M> = |s1 m 1 , s2 m 2 , . . . , s N m N ; M = mi . |

(4.25)

i

The advantage using this symmetry (and symmetries in general) is that the Hilbert space decomposes into mutually orthogonal Hilbert subspaces belonging to the irreducible representations of the group, in this case into Hilbert subspaces of constant M H = ⊕ H(M).

(4.26)

M

These subspaces are typically much smaller than the complete space, which is an advantage since the eigenvalue problem can be solved for each subspace H(M) separately E< | | | ' >< > < ' > > - M m; - M|ν = E ν m - ;M . m - ' ; M | Hˆ |m; - ; M|ν ; ∀|m

(4.27)

m -

Since energy eigenvalues are degenerate in subspaces H(M) and H(−M), the diagonalization has to be performed for M ≥ 0 only. Eigenstates can be transformed by inverting all m i of the product basis. Often, it is possible to employ an additional point group symmetry on top of total Sˆ z -symmetry. The production of the new symmetry-related basis states is rather simple since one needs permutations of product basis states only. The procedure will be demonstrated for the special case of cyclic systems, i.e., systems where the spins can be translated on a loop without changing the Hamiltonian. This procedure can easily be generalized to situations where the cyclic point group is more complicated, i.e., for centered rings or sawtooth systems. We introduce a new operator who moves all spins to the next site in the loop; the operator is defined by its action on a product state, i.e., Tˆ |m 1 m 2 . . . m N > = |m N m 1 m 2 . . . m N −1 >.

(4.28)

Tˆ has got eigenvalues (characters of the cyclic point group) that are e−i

2πk N

; k = 0, 1, . . . , N − 1,

(4.29)

and the states of the product basis can be transformed into eigenstates of Tˆ via N −1 1 E ( i 2πk ˆ )ν |m; - M>. - M, k> = √ e N T |m; N ν=0

(4.30)

E - are Please note that the following rules apply: (1) M = i m i ; (2) the states |m> - that can be transformed into each grouped into equivalence classes, i.e., all states |m>

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163

other by repeated application of Tˆ form one class, and only one representative of the class is used in (4.30) to construct the eigenstates of Tˆ , since the other members of the - M, k>, compare (4.30). If a certain class has got class would yield the same state |m; D < N members only, the number of transformed states reduces as well, and only k = N /D · κ, κ = 0, 1, D − 1, yield new basis states. In this way, we obtain basis states that are already simultaneous eigenstates of Tˆ and Sˆ z . If both commute with the Hamiltonian—and among each other, which they do—we can block-structure the Hamiltonian according to the decomposition of the Hilbert space into mutually orthogonal subspaces [13] H = ⊕ H(M, k).

(4.31)

M,k

As an example, the spectrum of a Heisenberg spin ring with N = 20 spins of s = 21 and antiferromagnetic coupling between adjacent spins is presented. The data are produced with a Hamiltonian that contains an additional overall factor of “−2”—such factors being convention and a persistent problem in quantum magnetism Hˆ = −2J

N E

/\

/\

/\

/\

s-i · s-i+1 , s-N +1 = s-1 .

(4.32)

i=1

In a computer program, one would first generate the basis in H(M) and then construct H(M, k) for the various k = 0, . . . , N − 1 in that space. The Hamiltonian matrix is then only diagonalized for each combination of M and k separately. Again, only M ≥ 0 needs to be considered, and only 0 ≤ k ≤ |N /2|, where |N /2| symbolizes the greatest integer less or equal to N /2. If the Hamiltonian possesses SU(2) symmetry, i.e., total spin S is a good quantum number, then the energy levels will be energetically degenerated across multiplets and share the same k within a multiplet. This property has been used for the graphical representation of the energy eigenvalues in Fig. 4.2. Systematic studies of spin rings—not limited to s = 1/2—revealed several of their general properties. Quantum numbers of the ground state and of low-lying excited states are of particular interest [14]. Since inelastic neutron scattering often determines, e.g., the energy gaps between ground state and low-lying excited states with S = 1, the lowest levels with S = 1 are marked by x-symbols in Fig. 4.2b, compare also [15].

4.2.3 Employing SU(2) Spin Rotational Symmetry and Point Group Symmetries From a mathematical or esthetic point of view, the application of the full spin rotational symmetry is particularly pleasing. This symmetry is described by the group SU(2) and leads to large reductions in matrix sizes [16]. The popular program

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Fig. 4.2 a Energy eigenvalues for a spin ring with N = 20 spins of s = 21 and antiferromagnetic coupling between adjacent spins (nearest-neighbor coupling). The levels are grouped into multiplets of total spin S. b Low-lying energy eigenvalues as function of k. The ground state has got S = 0 and k = 0. The lowest levels with S = 1 are marked by x-symbols

MAGPACK employs this symmetry [17]. The symmetry is often used even if not fully present in a compound since it allows to obtain approximate energy eigenvalues for systems which would be too large for an exact diagonalization of the complete system. Since the details of the method are discussed extensively in the given literature, only the core idea shall be formulated here. If the Hamiltonian commutes with all components of the total spin vector operator, i.e., | | (4.33) Hˆ , S- = 0, /\

as is for instance the case for the Heisenberg model, then the eigenstates of the /\

2

Hamiltonian are also simultaneous eigenstates of S- and Sˆ z . A working basis that /\

2

is already an eigenbasis of S- and Sˆ z would thus block-structure the Hamiltonian tremendously, i.e., it would be block-structured into blocks of total spin S and each of these S-blocks would be block-structured into M-blocks. Out of these blocks, only those with M = S need to be diagonalized since the energy eigenvalues are degenerate across all M-blocks inside an S-block (multiplets), and the respective eigenstates can be generated by the application of Sˆ − . In order to obtain the Hamiltonian matrix, the method of irreducible tensor operators (ITO) is used. In this approach, the Hamiltonian is an ITO of rank zero expressed in terms of spins that are ITOs of rank one. If the ITOs making up the Hamiltonian are coupled in the same way as given by the coupling scheme of the many-spin basis, then the matrix elements can be rather easily obtained by sequential decoupling and use of the Wigner–Eckart theorem. A popular coupling scheme would be the sequential coupling (one after the other) to obtain basis states of the form |s1 s2 S12 s3 S123 s4 . . . ; S M>

(4.34)

4 Exact Diagonalization Techniques for Quantum Spin Systems

165

that would match Heisenberg Hamiltonians containing terms of the form {{ {

sˆ1(k1 )

⊗

sˆ2(k2 )

}(k12 )

⊗

sˆ3(k3 )

}(k123 )

}(k) ⊗

sˆ4(k4 )

...

.

(4.35)

All terms of a Heisenberg Hamiltonian can be written in such a form. The operators { }(k12 ) such as sˆ1(k1 ) are now single-spin ITOs, whereas objects such as sˆ1(k1 ) ⊗ sˆ2(k2 ) are compound tensor operators. The indices (k... ) are the respective ranks which can only be 0 or 1 for a Heisenberg Hamiltonian. Matrix elements with respect to the basis states (4.34) can then be decomposed into a product of reduced matrix elements of single-spin ITOs, phase factors as well as Wigner-9J symbols. For details, please consult the literature. A relatively new implementation of the ITO approach is realized in the program PHI by Chilton [18]. Although being an obvious extension, the combination of SU(2) and point group symmetries is not widespread. Invented by Gatteschi [19], it was promoted by Waldmann for simple point groups such as D2 [20]. The reason for the scarce use is given by the numerical complexity due to an incompatibility of the chosen coupling scheme and the transformations of the point group. Take as an example the cyclic permutation of the vector coupling basis (4.34) Tˆ |s1 s2 S12 s3 S123 s4 . . . ; S M> → |s N s1 S12 s2 S123 s3 . . . ; S M>.

(4.36)

Any application of a point group transformation leads to a new coupling scheme that differs from the coupling scheme according to which the Hamiltonian is expressed as an ITO. It is therefore necessary to express the generated vector coupling states as linear combinations of the working vector coupling basis. To this end, a plethora of expansion coefficients, called recoupling coefficients, has to be calculated which renders this approach impractical in many cases. For cases where it works, huge system sizes can be achieved. The full beauty of this approach and the underlying ITO method is explained in great detail in Ref. [21] including the first presentation of the full spectrum of the famous ferric wheel {Fe10 } with a dimension of Hilbert space of 60,466,176 [22]. In a recent investigation of antiferromagnetic spin rings, it was concluded that SU(2) and cyclic translations according to the group C N lead to smaller numerical effort the better the chain length can be prime factorized into small primes. The least effort is necessary for spin rings of size N = 2n [23]. Figure 4.3 presents one of the obtained results, here for a spin ring of size N = 2·2·3, which is close to optimal. All energy eigenvalues can be obtained together with their quantum numbers S, M, k, and thus, thermal observables can be evaluated numerically exactly.

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J. Schnack

(a)

(b)

(c)

(b)

Fig. 4.3 Properties of a spin ring with antiferromagnetic nearest neighbor coupling; N = 12, s = 3 2 , dim(H) = 16,777,216 [24]: a, b energy spectrum versus total spin S and momentum quantum number k, respectively. c heat capacity versus temperature for various external fields. d magnetization versus field for various temperatures

4.3 Approximate Finite-Temperature Lanczos Method Many realistic spin systems are just too large for a complete numerical diagonalization even when taking symmetries into account. For such cases, provided the Hilbert space dimension does not get too large, the finite-temperature Lanczos method (FTLM) offers very accurate approximations to thermodynamic functions [25]. It rests on the approximate evaluation of traces with the help of random vectors [26] and is intimately related to several other methods such as thermal pure quantum states (TPQ) [27] or the kernel polynomial method (KPM) [28]. These methods have a broad range of applicability, see e.g. [29], and their accuracy has been investigated in detail, see, e.g., Ref. [30]. Since this is an easy-to-implement method, it shall be discussed in detail in the following. The method rests on two approximations: (1) The trace needed for the partition function is evaluated by means of a trace estimator and (2) the exponential is approximated by a reduction to a Krylov space. This way the dimension of the problem is drastically reduced while the accuracy of thermal expectation values is, however, still very good. A trace can be approximated as an expectation value with respect to a random vector. The normalized random vector, E |r > = rν |ν>, (4.37) ν

4 Exact Diagonalization Techniques for Quantum Spin Systems

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is made of random components rν which are, e.g., distributed according to a Gaussian distribution with zero mean. A trace of an operator is then approximated as in the following example of the partition function: ) ( ) ( ˆ ˆ Z T , B- = tr e−β H ( B ) ≈ dim(H).

(4.38)

As for other Monte Carlo-like methods, this estimate can be improved by averaging over several random vectors R ( ) dim(H) E ˆ . Z T, B- ≈ R r =1

(4.39)

ˆ The matrix elements are approximately evaluated by a Lanczos procedure, where |r > serves as the seed for the generation of the related Krylov space [31]. To this end, a Krylov space and its orthonormal basis are generated together with a tridiagonal matrix representing the Hamiltonian restricted to the Krylov space. The normalized random vector |r > constitutes the first basis state, i.e.,

|ψ1 > = |r >.

(4.40)

The second vector is generated by application of the Hamiltonian to |ψ1 > and projection onto the orthogonal complement of |ψ1 >, i.e., | '> |ψ | >< |) | > | > | '> ( |ψ = 1ˆ − |ψ1 ψ1 | Hˆ |ψ1 , |ψ2 = / 2 2 < ' '>. ψ2 |ψ2

(4.41)

For the third and all further vectors, one can show that only the last two vectors need to be projected out, since they are already orthogonal to all other vectors, so that the recursion relation reads | ' > |ψ | ' > ( | | >< |) | > | | >< > k+1 | | | |ψ | | | ˆ ˆ / ψ ψ ψ − H ψ = ψ ψ , = 1 − k−1 k−1 k k k k+1 k+1 < ' >. ' ψk+1 |ψk+1 (4.42) One can prove that any new vector is orthogonal to all previous vectors. While building the basis, the matrix elements of the Hamiltonian with respect to this basis are evaluated as part of (4.42), < | | > K i j = ψi | Hˆ |ψ j .

(4.43)

This matrix is tridiagonal, i.e., K i j = 0 for |i − j| > 1. The matrix K i j can be diagonalized at any step of the procedure. This way, the convergence of, e.g.,

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the ground state energy can be monitored. After a sufficient number of steps N L , the partition function can be approximated using the eigenvectors |n(r )> and eigenvalues εn(r ) of K i j as NL R E ( ) dim(H) E (r ) Z T , B- = e−βεn ||2 . R r =1 n=1

(4.44)

Here, an average over R random vectors is already performed. The “(r )” in |n(r )> and εn(r ) shall remind us that these quantities belong to the Krylov space grown from the seed |r >. If symmetries can be used, the procedure can be executed separately for each orthogonal subspace. The thermal expectation value of an observable can then be approximated as (

) O T , B- =

( ) Rγ N L | E )> )| | >< ( (r,γ ) < ( dim Hγ E E e−βεn n r, γ | Oˆ |r r |n r, γ . R γ Z T , B- γ =1 r =1 n=1 (

1

)

(4.45) In (4.45), the version with symmetries is shown; γ = 1, . . . , | labels the irreducible representations of the employed symmetry groups. It is important to note that the partition function is evaluated with respect to the same set of random vectors as the numerator in (4.45). Figure 4.4 provides a first impression of the method and its accuracy. The lightblue curves provide FTLM estimates according to (4.45) where the Sˆ z -symmetry was used together with R = 1 in each subspace, i.e., only a single random state was used (per subspace) to evaluate the thermal equilibrium functions. The astonishing and good news is that even with a single random state, the thermodynamic functions are accurate above a certain system-specific temperature—in the presented case for kB T > 10|J |. One also sees that the fluctuations of the observable grow toward T = 0, but typically vanish at exactly T = 0, since the Lanczos method is quasiexact for the ground state. When an average over R = 100 random states is performed according to (4.45), the resulting approximations, shown by dotted curves, fall almost perfectly on top of the exact result, which is depicted by the red curve. The exact result could be calculated using SU(2) and point group symmetries, see Sect. 2.3. The number of Lanczos steps N L has only a mild influence on the accuracy as long as it is large enough, which means that it should be of the order of at least 100; a value of N L between 200 and 500 is practically always sufficient according to our experience. Then, the total numerical effort can be summarized as follows: For N = 10 and s = 5/2, the magnetic quantum number runs from 0 to 25. For each of those, an average over R = 100 random vectors is performed, where a Krylov space of dimension ∼ 200 is generated for each random vector and a matrix of ∼ 200×200 is diagonalized. In particular, prohibitive matrix dimensions do not occur, although the Hilbert space of this spin system possesses a dimension of 60,466,176. For FTLM, the limiting factor is given by the size of the Lanczos vectors, of which at least two

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169

Fig. 4.4 FTLM calculation of differential susceptibility a as well as heat capacity b for a spin ring of N = 10 spins of spin quantum number s = 25 with antiferromagnetic nearest-neighbor interaction J . Each light-blue curve depicts the result using a single random vector; Ns = 100 of these curves are shown. The dotted curves represent averages over R = 100 random vectors according to (4.45), and the red curves provide the exact result that is available using SU(2) and point group symmetries. Adapted with permission from J. Schnack et al., Phys. Rev. Research 2, 013186 (2020) under the terms of the Creative Commons Attribution 4.0 International license

need to be stored in the RAM. This restricts the size of the largest subspace to about 1011 . The matrix elements of the Hamiltonian can be calculated on-the-fly; therefore, the method can be easily parallelized on modern supercomputers, see [12]. The inaccuracies observed for the spin ring discussed in connection with Fig. 4.4 can be traced back to the rather large low-lying energy gaps characteristic for bipartite spin systems such as even-membered spin rings. For more realistic systems with competing interactions, the low-lying gaps are usually much smaller, even if the interaction strength J is of similar size. We call systems with competing interactions frustrated [32] as if this was something special, but on the contrary, this is the normal state of matter. Figure 4.5 shows the differential susceptibility as well as the heat capacity for an icosidodecahedron, an Archimedean solid with N = 30 vertices, here with spins of spin quantum number s = 1/2 and antiferromagnetic nearest-neighbor interaction J . Such beautiful structures have been realized as polyoxometalates with various ions and thus various single-spin quantum numbers, i.e., {Mo72 Fe30 } [33], {Mo72 Cr30 } [34], {Mo72 V30 } [35], and {W72 V30 } [36], altogether once termed “kagome on a sphere” [37]. This spin system taught us a lot, see [2], and in connection with FTLM, we learned that frustration works in favor of the approximation. The denser low-lying spectrum of frustrated spin systems considerably lowers the temperature down to which the approximation is accurate already using a single random state. As Fig. 4.5 demonstrates, approximation (4.45) with R = 1 is accurate for kB T > 1|J |. With R = 100, the approximation seems to have converged against the (unknown) true result as may be conjectured from the red curve in the inset of Fig. 4.5b which shows that the result for R = 500 is on top of that with R = 100. As a final example, the evaluation of the magnetic properties of {Fe10 Gd10 } shall be discussed. The dimension of the Hilbert space for 10 Fe(III) spins of s = 25 and

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Fig. 4.5 FTLM calculation of differential susceptibility a as well as heat capacity b for an icosidodecahedron, an Archimedean solid with N = 30. Vertices, here with spins of spin quantum number s = 1/2 and antiferromagnetic nearest-neighbor interaction J . Each light-blue curve depicts the result using a single random vector. The dotted curves represent averages over R = 100 random vectors according to (4.45), and the red solid curve in the inset in b shows the result for R = 500. Adapted with permission from J. Schnack et al., Phys. Rev. Research 2, 013186 (2020) under the terms of the Creative Commons Attribution 4.0 International license

10 Gd(III) spins of s = 27 is a staggering 64,925,062,108,545,024 ≈ 6.5 × 1016 rendering a straight application of many methods impossible. This compound needed the application of several methods in order to deduce and understand its singular behavior [38]. It was known that Heisenberg systems with a structure shown in Fig. 4.6d—called sawtooth or delta chain—exhibit a quantum phase transition for a certain ratio of ferromagnetic coupling J1 and the antiferromagnetic coupling J2 [39]. A quantum phase transition happens at absolute zero; here, the character of the ground state changes as function of an external parameter such as the applied field or the ratio of exchange interactions that may be influenced, e.g., by pressure. It turned out that {Fe10 Gd10 } is close to the quantum phase transition and that it shows almost critical behavior. In order to arrive at this conclusion, the exchange interactions had to be deduced from the experimental data shown in Fig. 4.6a, b. In the following, the applied methods and the rationale behind them shall be explained step by step. There is virtually no quantum method to evaluate the susceptibility of {Fe10 Gd10 } as shown in Fig. 4.6 except, maybe, for future versions of the thermal density matrix renormalization group method (thDMRG). Thus, the data were first analyzed by means of high-temperature series expansions (HTE), a method that works for temperatures higher than the characteristic scales of the problem, e.g., given by the couplings [40]. This yielded a first estimate of J1 and J2 . Quantum Monte Carlo (QMC) provides very accurate thermodynamic observables for non-frustrated systems [41]; it is freely available in the ALPS package [42] and was employed for various spin systems, e.g., [43]. For frustrated systems such as {Fe10 Gd10 }, QMC is still accurate for higher temperatures and yields the same results as HTE, see Fig. 4.6a. Classical Monte Carlo (CMC) assumes that spins are classical vectors, which might be a good approximation for large enough spin quantum numbers, as, e.g., in the present case. Taking the same exchange interactions, CMC reproduces the

4 Exact Diagonalization Techniques for Quantum Spin Systems

(a)

(b)

(c)

(d)

171

Fig. 4.6 Magnetic properties of {Fe10 Gd10 }: a experimental susceptibility (× symbols) with theoretical estimates (curves); b experimental magnetization (+ and × symbols) with theoretical estimates (curves); c theoretical isothermal entropy change calculated for various coupling scenarios for the fictitious of {Fe6 Gd6 }; d schematic structure of {Fe10 Gd10 }, Fe = blue, Gd = red. Adapted from A. Baniodeh et al., NPJ Quantum Materials 3, 10 (2018) under the terms of the Creative Commons CC BY license

full susceptibility curve, which is more than one can realistically expect. HTE does not work for the magnetization, Fig. 4.6b, taken at very low temperatures. QMC works if the applied field is large since then only a few levels contribute to the magnetic observables and the sign problem is small. The same reasoning stands behind the use of FTLM which was restricted to subspaces with |M| ≥ 45 (thin dashed red line). Summarizing, the combined use of HTE, QMC, CMC, and FTLM enabled us to deduce J1 and J2 and to understand that the system is close to the critical value of |J2 /J1 | = 0.7. FTLM was then employed to understand the caloric properties by considering a somewhat smaller fictitious {Fe6 Gd6 }, compare Fig. 4.6c. It turns out that, e.g., the isothermal entropy change (red solid curve) is almost as large as for the critical version of the spin system (red dashed curve) and for low temperatures larger than a simple nearest-neighbor ferromagnetic coupling (black solid curve). For an antiferromagnetic nearest-neighbor coupling, the entropy change would be much smaller, which demonstrates that substances close to quantum phase transitions show exceptional magnetocaloric properties [44]. Finally, the application of FTLM to anisotropic spin Hamiltonians shall be discussed. In principle, FTLM can be used as outlined in Eq. (4.45) but the lack of symmetries, in particular the lack of the Sˆ z -symmetry, leads to larger inaccuracies

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in particular of the magnetization. If Sˆ z -symmetry can be used as in the preceding examples, then the random coverage of subspaces H(M) and H(−M) is the same since the subspace H(−M) is taken as a copy of H(M). This guarantees a balance between (states )of positive and negative magnetization among all Krylov spaces, and - T , B- = 0 for B- = 0 exactly for any number of random vectors. thus, M If Sˆ z -symmetry cannot be employed, the balance between states of positive and negative magnetization is not given by construction but is achieved in the limit of a large number of random vectors R (central limit theorem). Since the convergence is possibly slow, this requires a much larger number of random vectors. Reference [45] therefore suggests to use pairs of random vectors where the first vector is truly random and the second is its time-reversed twin. This provides a reasonable and cost-effective improvement of the method; it is, however, still not fully balancing the magnetization. As outlined in the same publication, this could be achieved by using |n(r )> as well as its time-reversed twin for averaging. Since this would require to keep all basis states |ψk > and not only the final two, such an approach is too costly. Figure 4.7 demonstrates with the example of {Mn6 Cr} clusters that the achievable accuracy is sufficient for typical observables. Here, the effective magnetic moment has been calculated both numerically exactly (solid curves), compare Sect. 2.1, as well as with the modified FTLM (pairs of time-reversed random vectors, dashed curves). FTLM delivers these accurate approximations in a fraction of the computing time needed for full diagonalization. In the case shown, the exact diagonalization requires hours if not days depending on the number of CPU cores, whereas FTLM runs in minutes and can be ported to graphics cards for problems where the two Lanczos vectors fit into the RAM of the card. This is particularly helpful for fitting, i.e., searches in parameter space. FTLM also allows to deal with spin systems as large as the famous Mn12 -acetate, which can be treated quasi exactly in a few days compared to not being treatable at all by complete diagonalization.

Fig. 4.7 Effective magnetic moment of the depicted model system consisting of six spins s = 2 and a central spin s = 23 in analogy to Fig. 4.1. a Magnetic moments along z-direction for two parameterizations of Hamiltonian (4.23), b powder-averaged moment for the same parameter sets. The solid curves show the result of full matrix diagonalization, the dashed ones the result of FTLM a R = 100, N L = 100, b R = 200, N L = 150). Reprinted with permission from O. Hanebaum, J. Schnack, Eur. Phys. J. B 87, 194. Copyright 2014. Springer Nature

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4.4 Discussion and Outlook Summarizing, spin Hamiltonians are a necessary prerequisite to obtain true manybody quantum states and to understand correlation effects beyond the DFT energy functionals. However, they always constitute a reduction of the full electronic problem. One therefore has to be very careful in setting up a spin Hamiltonian since this builds the model and limits or enables what may be described. As has been demonstrated, spin Hamiltonians can be treated by various methods nowadays. Although complete numerical diagonalization using symmetries would be the method of choice, it is often not feasible. Approximate methods such as the finitetemperature Lanczos method (FTLM), quantum Monte Carlo (QMC), and density matrix renormalization group theory (DMRG), in particular its thermal extension, provide accurate thermodynamic observables. Among these approximations, FTLM constitutes an easy to set up scheme that gives access to equilibrium as well as non-equilibrium observables. Not only can standard magnetic observables be evaluated but spectral functions or cross-sections for inelastic neutron scattering can also be calculated with comparably little effort [46]. If I was asked which method will contribute the greatest progress in the foreseeable future, I tend to put my money on DMRG or tensor network methods in general and their thermal extensions. The reason is that these methods aim at a representation of the relevant quantum states in terms of special, memory-saving parametrizations, e.g., as matrix product states while simultaneously achieving a very good accuracy of lowlying energies as well as observables, see e.g. [47]. For chemically oriented readers, a recent review on “Quantum Algorithms for Quantum Chemistry and Quantum Materials Science” could be of interest [48]. The potential power of such methods might be easily imaginable when considering that molecules as large as Mn84 can be modeled [49]. Acknowledgements I thank the Deutsche Forschungsgemeinschaft DFG for funding (355031190 (FOR 2692); 355031190 (SCHN 615/25-2); 449703145 (SCHN 615/28-1)) and the Leibniz Supercomputing Center in Garching/Germany for supercomputing resources. I would also like to thank all past and present members of my group for a great collaboration and for all the results we achieved. Proofreading by Kilian Irländer and Dennis Westerbeck is gratefully acknowledged.

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Chapter 5

Modeling Magnetic Properties of Actinide Complexes Hélène Bolvin

Abstract This chapter presents different aspects of the modeling of magnetic properties in monomeric open-shell actinide complexes. Those properties are closely related to their electronic structure, which is difficult to achieve since none of crystal-field effects, electron-electron repulsion nor spin-orbit interaction is predominant. The electronic structure should be analyzed within the intermediate coupling scheme, between on one hand the Russell-Saunders coupling scheme where the interelectronic repulsion is considered before spin-orbit, and, on the other hand, the j-j coupling scheme where one-electron wave-functions including spin-orbit are used to build the many-electron wave-function. Ab initio calculations on these complexes are challenging, and SO-CAS-based methods are still the quantum chemistry tool of choice since they include a balanced description of the three effects. It is only by a close interplay between experimental data which are sparse for transuranide complexes due to radioactivity, numerical methods, and model Hamiltonians that one succeeds to unravel the electronic structure and magnetic properties of these complexes. Keywords Magnetic susceptibility · Actinide · Ab initio

5.1 Electronic Structure of Actinide Complexes Most of the actinide compounds are man-made since the twentieth century. Thorium and uranium are both long-lived and can be found in the earth in notable amounts. Actinium and protactinium exist in nature in extremely small amounts whereas transuranium elements are man-made. All the actinides are radioactive, quite strongly for some of them. This requires facilities specially equipped and approved for radioactive work. The radioactivity often plays a part in their chemistry and may H. Bolvin (B) Laboratoire de Chimie et de Physique Quantiques, CNRS, Université Toulouse 3, Toulouse, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Rajaraman (ed.), Computational Modelling of Molecular Nanomagnets, Challenges and Advances in Computational Chemistry and Physics 34, https://doi.org/10.1007/978-3-031-31038-6_5

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impact their magnetic properties. The majority of the studies concern thorium and uranium compounds, particularly the latter, due to accessibility of raw materials, ease of handling, and the long lifetimes of the relatively weakly α-emitting elements Th and U [1]. The 4f orbitals for lanthanides are core-like due to the shielding by the occupied 5s and 5p orbitals. The 5f orbitals, are less shielded by the 6s and 6p, due to a larger spatial extension, are less core-like and thus, more involved in the bonding, leading to a greater tendency toward covalent bond formation. Early in the actinide series, the near degeneracy of the 5f, 6d, and 7s orbitals means that more valence electrons can be involved in the formation of compounds and a wider range of oxidation states is observed. For the early actinides, Ac–Pu, the oxidation state ranges from +3 to +7, while the late actinides, from Bk, the +3 oxidation state is the most common one, like in the lanthanides. The electronic structures of the later actinides become more and more like those of the lanthanides, with similar chemistry. Relativistic effects are more pronounced for actinides than for lanthanides, since their atomic number is larger. Scalar relativistic effects lead to an expansion and a destabilization of d and f orbitals. 5f orbitals in actinides are more destabilized than are the 4f orbitals of the lanthanides, to be nearly degenerate with the empty 6d and 7s orbitals for early actinides [2]. As a result, those valence orbitals are more chemically active, more valence electrons can be involved in the formation of compounds, and a wider range of oxidation states is observed. Spin-orbit coupling leads to the splitting of the f orbitals; due to their small density close to the nucleus, this effect is not as important as for p orbitals, and actinide free ions are still well described within the Russell-Saunders coupling scheme, where the many-electron states are built at the spin-free level, and spin-orbit coupling calculated between many-electron states: The states of the free ion are accordingly labeled as 2S+1 L J , where L, S, and J are the orbital, spin, and total angular moments, respectively. This scheme is used when the electronic repulsion is greater than the spin–orbit splitting in contrast to j-j coupling, where the spin-orbit coupling is introduced at the one-electron level. Since spin-orbit coupling is larger in actinides than in lanthanides, one needs to go beyond the Russell-Saunders coupling scheme, by mixing states with the same value of the total angular moment J . The modeling of metal complexes is related to the relative strength of three physical interactions: (i) the crystal-field splitting which gauges the strength of the interaction between the open-shell metal orbitals with those of the ligands, (ii) the electron repulsion in the open-shell metal orbitals, and (iii) the spin-orbit coupling. In actinides, as compared to lanthanides, the former is larger as already discussed, the second smaller as a consequence of the larger expansion of the 5f orbitals, and the latter larger. All this makes that the actinide free ions are not as well modeled by the Russell-Saunders coupling as lanthanide ones are, and in general, since none of those interactions are predominant, it is more difficult to interpret the spectra and magnetic behavior of actinide compounds as in lanthanide. In this chapter, we will address the magnetic properties of molecular actinide systems, containing only one actinide center, with an open 5f shell. Magnetic properties probe the low-lying states, up to thermal energy. The various multiplets of actinide complexes can not be described within a single-configuration framework. Scalar

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relativistic DFT (Density Functional Theory) methods are generally adequate for the calculation of the ground state properties of actinide systems, including molecular geometries and vibrational frequencies [3, 4] but not suitable to describe the many excited states. Since there is not a clear predominance between the covalent effects, the electron repulsion nor the spin-orbit coupling in the 5f orbitals, all those interactions need to be adequately described, not only to achieve a quantitative but even a qualitative description. Scalar relativistic effects are easily incorporated using the non-relativistic machinery [5, 6]. SO-CAS (Spin-Orbit Complete Active Space)-based methods [7] have been successfully applied to actinide complexes since one decade. In the first step, the CASSCF (Complete Active Space Self-Consistent Field) [8] method describes the multi-configurational many-electron states, which are corrected for dynamical correlation by a perturbative method, CASPT2 (Complete Active Space Perturbation Theory 2nd order) [9] or NEVPT2 (N-Electron Valence state Pertubation Theory 2nd order) [10] in a second step. The spin-orbit coupling is calculated as state interaction between those many-electron correlated wave-functions in the last step [11]. This scheme is in line with the RussellSaunders coupling scheme. To obtain a full treatment of the spin-orbit interaction, four-component methods should be used; but four-component multi-configurational correlated methods allow only the description of molecules with few atoms [12, 13]. The first-principle calculations are still challenging for actinide complexes, because they gather many of the difficulties of quantum chemistry: They are open-shell and should be described using multi-configurational methods as soon as there is more than one unpaired electron, relativistic effects are important, both scalar and spin-orbit, and correlation effects play an important role. Magnetic properties of actinide complexes are mostly characterized by their magnetic susceptibility, either in solid state using a SQUID (Superconducting Quantum Interference Device) spectrometer which provides the average susceptibility and in a large temperature range, or in solution, using the Evans method, which probes by paramagnetic NMR (Nuclear Magnetic Resonance) a control molecule, usually around room temperature. On the other hand, paramagnetic NMR on nuclei of the ligand provides essential information about the susceptibility tensor, when the so-called dipolar contribution is dominant. We consequently devote the second section to the modeling of the magnetic susceptibility. Magnetic properties are usually described using model Hamiltonians, using phenomenological parameters. For monomeric actinide complexes, the most used model Hamiltonians are the crystal-field Hamiltonian and spin Hamiltonians in degenerate or nearly-degenerate manifolds. This will be presented in the third section of this chapter. Finally, we review shortly in the last section recent works where magnetic properties of actinide complexes where modeled either based on ab initio calculations or on crystal-field theory.

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5.2 Magnetic Susceptibility 5.2.1 Magnetic Susceptibility Tensor from Ab initio Calculations The Hamiltonian describing a molecular system in an external magnetic field B takes the form (5.1) Hˆ = Hˆ 0 + Hˆ Z where Hˆ 0 is the Hamiltonian in the absence of any external field and Hˆ Z the Zeeman term which accounts for the interaction between the magnetic field and the electronic magnetic moment ˆ ·B Hˆ Z = −m ( ) = μ B Lˆ + ge Sˆ · B

(5.2)

ˆ the magnetic where μ B is the Bohr magneton, ge the g-factor of the free electron, m ˆ results from both orbital and spin angular moments by m ˆ = dipole moment. m ˆ = Lˆ + ge S, ˆ 1 and M ˆ where M, ˆ L, ˆ and Sˆ are the total, orbital, and spin angular −μ B M moments, respectively. In the following, we will consider ge = 2. For an external magnetic field in direction u, B = Bu, the eigenfunctions of Hˆ are |Ψ I (B)< with of| the|magnetic\angular moment in corresponding energies E I (B). The component / | | direction v Mˆ v of state I is M I,uv (B) = Ψ I (Bu) | Mˆ v | Ψ I (Bu) . The magnetic susceptibility tensor χ is deduced from the thermal average at temperature T of the magnetization >m v (Bu )m v (Bu )m u (Bu )< du >m(B)m(B)m x (Bx )< < ( )> ) m y B y T + >m z (Bz )m(B)m z (Bz )m z (Bz )±1/2| Mu | ± 1/2< = ∓( 2 g + 8 g ). The cubic term is by far non-negligible and opposite in sign to the linear one. When the complex is distorted, the quartet splits into two Kramers doublets. Supplementary terms should be added to Eq. 5.27. It becomes then more intuitive to model the quartet by two interacting doublets, each with a g-tensor coupled by second-order Zeeman interaction [52].

5.4 Linear Complexes 5.4.1 Spinor Scheme An(V) and An(VI) cations form strong bonds with oxo or nitride groups, leading to linear structures. The [AnO2 ]n+ cations are named actinyl ions. In a linear point group, the 5f orbitals are f σ , f π (2) f δ (2) and f φ (2)—the degeneracy is given in parenthesis. The two former ones are engaged in σ and π bonding, respectively, with the orbitals of the O or N atoms: The bonding orbitals are mostly borne by the ligands while the corresponding anti-bonding ones are on the 5 f , which are largely destabilized (see Fig. 5.5). The two latter are non-bonding and host the unpaired electrons. With spin-orbit coupling, states are characterized by their projection ω of ˆj = ˆl + sˆ on the Z axis, leading to σ1/2 , π1/2 , π3/2 , δ3/2 , δ5/2 , φ5/2 , and φ7/2 spinors [53]. States with the same ω-values mix by spin-orbit coupling. Usually, the [AnO2 ]n+ , [AnO]n+ , [AnN2 ]n+ , or [AnN]n+ units are coordinated by equatorial ligands. And it is the interaction with the equatorial ligands which tunes the nature of the ground state and excited states and the magnetic properties. The degeneracy of the two δ and φ orbitals splits by the equatorial ligands, the fourfold symmetrical δ orbitals by ligands with quaternary symmetry, and the threefold symmetrical φ orbitals by ligands with ternary symmetry.

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Fig. 5.5 5f spinors for [AnO2 ], [AnO], [AnN2 ], or [AnN] units, linear (left) or with equatorial ligands (right). The degeneracy of the states is given in parentheses

In the free linear cations, without spin-orbit coupling, the φ are lower in energy than the δ because more distant from the oxo/nitride groups, and this reduces the electrostatic repulsion. In the presence of an equatorial ligand, the δ are often lower than the φ, because the latter are now closer from the equatorial ligands, and this is unfavorable on a electrostatic point of view and involved in σ anti-bonding with the orbitals of the ligands, while the δ have π interactions. The δ and φ orbitals have m l values of ±2 and ±3, respectively. When the degeneracy is lifted, the angular moment is quenched, but partially recovered by coupling the two orbitals: The closer the two orbitals, the larger the angular moment. As shown in Fig. 5.5, the e3/2 and e5/2 spinors are very close in energy, and one or the other one can be the ground spinor, according to the nature of the axial and equatorial ligands. The former is a pure δ3/2 while the composition of the latter is a mixing between the δ5/2 and φ5/2 . The axial ligands determine the δ-φ splitting, and the nature and the symmetry of the equatorial ligands the splitting of the δ and/or φ orbitals.

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5.4.2 g-factors Spin-orbitals are given in terms of spherical harmonics in Table A1 in Appendix and the representation matrix of the spin-orbit operator hˆ S O = ζ ˆl · sˆ in the basis of the spin-orbitals in Table A3 in Appendix. The crystal-field operator hˆ C F is diagonal, and the representation matrix of hˆ = hˆ C F + hˆ S O in the δ and φ spin-orbitals basis is | > | > |φ¯ 2 hˆ |δ1 < |δ2 < |φ¯√1 √ >δ1 |

εδ1

iζ

− 2√3ζ2 −i 2√3ζ2 √

√

>δ2 | −i ζ εδ2 i 2√3ζ2 − 2√3ζ2 √ < | √3ζ φ¯ 1 | − 2√2 −i 2√3ζ2 εφ1 −i 3ζ2 √ < | √3ζ φ¯ 2 | i 2√2 − 2√3ζ2 i 3ζ2 εφ2

(5.28)

| > |δ1 < and |δ¯ 1 denote spin-orbitals with α and β spin, respectively. By Kramers symmetry, one gets the same matrix in the set of Kramers conjugated spin-orbitals (see Tables A1 and A3 in Appendix). The ground Kramers partners issued from this matrix take the form | > | > |ψ< = a |δ1 < + b |δ2 < + c |φ¯ 1 + d |φ¯ 2 | > | > | > |ψ¯ = o ˆ |ψ< = a ∗ |δ¯ 1 + b∗ |δ¯ 2 − c∗ |φ1 < − d ∗ |φ2
g||S = 4 ψ¯ |sˆz | ψ = 2 |a|2 + |b|2 − |c|2 − |d|2 / | | \ √ | | g XL = 2Re ψ¯ |lˆx | ψ = 2 6Im (bd − ac) / | | \ √ | | gYL = 2Im ψ¯ |lˆy | ψ = −2 6Re (ad + bc)

(5.30)

( ) < | | > g⊥S = 4Re ψ¯ |sˆx | ψ = 2Re a 2 + b2 − c2 − d 2

(5.31)

g|| = g⊥ =

g||L g⊥L

+ +

g||S g⊥S

In the case of an axial symmetry, g XL = gYL = g⊥L . The axial orbital component g||L arises from δ1 /δ2 and φ1 /φ2 couplings (a ∗ b and cd ∗ ) while the transverse orbital component from δ1 /φ1 and δ2 /φ2 couplings ((bd, ad, bc and ac).

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Let us consider the case where the two φ orbitals are strongly destabilized and the model restricted to the δ orbitals. c = d = 0. The values of a and b are determined by the splitting εδ2 − εδ1 as compared to ζ , with the two limit cases • εδ2 − εδ1 is very large. |ψ< = |δ1 < is a pure spin state. g||S = g⊥S = 2, g||L = g⊥L = 0, g is isotropic, as expected. • εδ2 − εδ1 = 0. – |ψ< = √12 (|δ1 < + i |δ2 ) ( perpendicular direction are less intuitive. The eigenvectors of sˆx are |ψx < = √12 |ψ< + |ψ¯ and | >) | > ( |ψ¯ x = θˆ |ψx < = √1 − |ψ< + |ψ¯ . Written as spinors, 2

( ) | x> aδ1 + bδ2 − c∗ φ1 − d ∗ φ2 |ψ = √1 ∗ ∗ 2 a δ1 + b δ2 + cφ1 + dφ2 ( ∗ ) | x> 1 a δ1 + b∗ δ2 + cφ1 + dφ2 | sˆx ψ = √ ∗ ∗ 2 aδ1 + bδ2 − c φ1 − d φ2

(5.34)

The spin magnetization in direction x is expressed as m xS (r) = μ B ψx†∗ (r)ˆsx ψx (r) ( ) μB | ( 2) = Re a |δ1 (r)|2 + Re b2 |δ2 (r)|2 2 ( ) ( ) | −Re c2 |φ1 (r)|2 − Re d 2 |φ2 (r)|2

(5.35)

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In this case, the population of each NSO may be either positive or negative since a, b, c, and d are complex. We let as an exercise to the reader to demonstrate that the same spin magnetization is obtained in direction y: m y (r) = ψ y†∗ (r)ˆs y ψ y (r) = m x (r) with | >) | > ( |ψ y = √1 |ψ< + i |ψ¯ . 2 Let us consider again the cases mentioned above. • For the pure spin state |ψ< = |δ1 |φ¯

| | √ √ 5 b 3 ( || ¯ > || ¯ >) = a |δ2 < − √ |δ1 < + √ (i |π¯ 1 < + |π¯ 2 5 b || > 3 | ¯ ¯ = a δ2 − √ δ 1 + √ (i |π1 < − |π2 ω, τDir ∼ B kB T , τDir ∼ B 2 kB T 0 ,

(6.140) (6.141)

depending on whether the modulation of the Zeeman interaction by phonons drives relaxation or not, respectively. Next we can use Eq. (6.137) to derive an expression similar to (6.140) for the Orbach relaxation mechanism. Let us now assume that we have a system with at least one excited spin states above the ground-state pseudo-doublet, as depicted in Fig. 6.2a, b. Let us also assume that the direct relaxation is inhibited. In this circumstances, one-phonon relaxation can only occur in a two-step process with the absorption of a phonon and the subsequent emission of another one. From Eq. (6.58), we can observe that the relaxation rates of the absorption and emission transitions have the exact same mathematical structure except for the phonon population contribution, which accounts for the spontaneous emission of one phonon through the term n¯ + 1. As a consequence, the emission process will always be faster than the

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absorption one, leaving the latter to act as the rate determining step of the relaxation process. Therefore, we can focus on adapting Eq. (6.137) to the case of the transition |a⟩ → |c⟩ due to the absorption of one phonon, which leads to ( βhωca )−1 3 e −1 . Rcc,aa = V˜sph ωca

(6.142)

However, differently from the direct relaxation case, we are now in the opposite temperature versus spin splitting limit: ωca ≫ kB T , which leads to 3 −βhωca Rcc,aa = V˜sph ωca e .

(6.143)

We note that this expression has been derived under the additional assumption that the Debye model is still valid and that acoustic modes are driving relaxation, i.e. ωca < ω D . Differently from the direct case, the Orbach necessarily concerns systems with S > 1/2, where zero-field splitting interactions due to the ion’s crystal field are the dominant term of Hˆ s . Therefore, additional contributions to the field in V˜sph are not expected, and we obtain the final result τ −1 ∼ Δ3 e−Δ/kB T ,

(6.144)

where we have also assumed that Δ does not depend on the field as by construction Δ ≫ δ ∝ |B|. Two-Phonon Relaxation In Sect. 6.2, we have distinguished two different two-phonon spin transition mechanisms: one coming from second-order time-dependent perturbation theory plus second-order spin-phonon coupling and the other one coming from fourth-order time-dependent perturbation theory plus first-order spin-phonon coupling strength. Starting our analysis from the first case and using Eq. (6.85), we can apply a similar strategy as for the one-phonon case. However, this time we have a double summation over the phonon spectrum and the transition |a⟩ → |b⟩ due to the simultaneous absorption and emission of two phonons reads Rbb,aa

) ) |2 ∂ 2 Hˆ s π ∑ || | = 2 |a⟩ | nˆ αq (nˆ βq, + 1)δ(ωba − ωαq + ωβq, ). |⟨b| 4h αq/=βq, ∂ Q αq ∂ Q βq,

(6.145) By substituting the expression of Eq. (6.126) √ and accounting that each phonon derivative contributes with a term proportional to ω [see Eq. (6.133)], we obtain Rbb,aa = V˜sph

{ω D

{ω D dω

0

0

,

dω, ω3 ω,3

eβhω δ(δ − ω + ω, ). βhω (e − 1)(eβhω, − 1)

(6.146)

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A. Lunghi

The Dirac delta function in Eq. (6.146) enforces that ω = δ + ω, and, since δ ≪ ω for ordinary Zeeman splittings and phonons energies, we can assume that ω = ω, , which leads to {ω D eβhω . (6.147) Rbb,aa = V˜sph dωω6 βhω (e − 1)2 0

An important result of Eq. (6.147) is that two-phonon relaxation receives contributions from the entire vibrational spectrum without any dependence on the size of the spin splitting δ, nor involving excited spin states. In order to make the T dependence of this expression more transparent, we can make the substitution x = βω and rewrite it as {βω D ex −1 7 ˜ , (6.148) dx x 6 x τ = Vsph (kB T ) (e − 1)2 0

where the integral does not explicitly depend on temperature anymore for low T , i.e. when βω D → ∞. In the high-T limit, the ratio of exponential functions in Eq. (6.147) simplifies to (βω)−2 , thus leading to an overall T −2 . Concerning the field dependence of τ −1 , we note that it can only arise from the nature of V˜sph , which might be driven by the modulation of the Zeeman interaction, as seen for the direct relaxation. Therefore, we can conclude that Raman relaxation follows the trends for kB T < ω :

−1 −1 τRaman ∼ B 2 T 7 , τRaman ∼ B0 T 7,

(6.149)

for kB T > ω :

−1 −1 τRaman ∼ B 2 T 2 , τRaman ∼ B0 T 2,

(6.150)

or

depending on whether the modulation of the Zeeman interaction by phonons drives relaxation or not, respectively. Let us now investigate the temperature and field dependence of τ due to the use of Eq. (6.119) for a non-Kramers system, e.g. when only one excited state is available as depicted in Fig. 6.2b. By substituting the expression of Eq. (6.126) into Eq. (6.119) and √ accounting that each phonon derivative contributes with a term proportional to ω, we obtain Rbb,aa = V˜sph

{ω D 0

×

{ω D dω

| | | ⟨b|V , |c⟩ ⟨c|V |a⟩ ⟨b|V |c⟩ ⟨c|V , |a⟩ |2 | (6.151) + dω, ω3 ω,3 || | Δ−ω Δ + ω,

0 βhω,

e δ(δ − ω + ω, ). (eβhω − 1)(eβhω, − 1)

(6.152)

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249

As before, the Dirac delta function in Eq. (6.152) enforces that ω = δ + ω, and, since δ ≪ ω for ordinary Zeeman splittings and phonons energies, we can assume that ω = ω, . Additionally, we are going to assume that the only portion of the integral with Δ ≫ ω is going to contribute in virtue of the condition kB T ≪ Δ, which leads us to Rbb,aa

= V˜sph Δ−2

{ω D dωω6 0

eβhω . (eβhω − 1)2

(6.153)

Except for a different pre-factor containing the term Δ−2 , this result is identical to Eq. (6.147) and has the same T and B dependence. However, while Eq. (6.85) led to identical results for Kramers and non-Kramers systems, the use of Eq. (6.119) does not. Let us call the ground-state Kramers doublet ± p and the excited one ±q. A transition p → − p would then involve a sum over the excited states that reads ⟨− p|V , |q⟩ ⟨q|V | p⟩ ⟨− p|V , | − q⟩ ⟨−q|V | p⟩ + Δ−ω Δ−ω ⟨− p|V |q⟩ ⟨q|V , | p⟩ ⟨− p|V | − q⟩ ⟨−q|V , | p⟩ + + Δ + ω, Δ + ω, , , ∗ ⟨− p|V |q⟩ ⟨q|V | p⟩ ⟨ p|V |q⟩ ⟨q|V | − p⟩ ∗ − = Δ−ω Δ−ω ⟨− p|V |q⟩ ⟨q|V , | p⟩ ⟨ p|V |q⟩ ∗ ⟨q|V , | − p⟩ ∗ − + Δ + ω, Δ + ω, , ⟨− p|V |q⟩ ⟨q|V | p⟩ ⟨− p|V |q⟩ ⟨q|V , | p⟩ − = Δ−ω Δ−ω , ⟨− p|V |q⟩ ⟨q|V | p⟩ ⟨− p|V , |q⟩ ⟨q|V | p⟩ − , + Δ + ω, Δ + ω,

(6.154) (6.155) (6.156) (6.157) (6.158) (6.159)

where the first equality follows from the Kramers symmetry and the second one from the Hermitian property of the spin-phonon coupling Hamiltonian. Equation (6.159) can then be further simplified by enforcing the condition Δ ≫ ω = ω, (

) 1 1 − ⟨− p|V |q⟩ ⟨q|V | p⟩ Δ − ω Δ + ω, ( ) 1 1 + ⟨− p|V |q⟩ ⟨q|V , | p⟩ − (6.160) Δ + ω, Δ−ω ( ( ) ) −2ω 2ω , | p⟩ + ⟨− p|V |q⟩ ⟨q|V (6.161) = ⟨− p|V , |q⟩ ⟨q|V | p⟩ Δ2 − ω 2 Δ2 − ω2 ) 2ω ( (6.162) = 2 ⟨− p|V |q⟩ ⟨q|V , | p⟩ − ⟨− p|V , |q⟩ ⟨q|V | p⟩ . Δ ,

When the result of Eq. (6.162) is used together to Eqs. (6.119) and (6.153) turns into

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A. Lunghi

Rbb,aa

= V˜sph Δ−4

{ω D dωω8 0

eβhω . (eβhω − 1)2

(6.163)

In conclusion for Kramers systems that relax according to the fourth-order perturbation theory, τ has the following dependency with respect to field and temperature for kB T < ω :

−1 −1 τRaman ∼ B 2 T 9 , τRaman ∼ B0 T 9,

(6.164)

for kB T > ω :

−1 −1 τRaman ∼ B 2 T 2 , τRaman ∼ B0 T 2,

(6.165)

or

We have now concluded the review of the most fundamental results presented by Van Vleck [35] and Orbach [36] in their seminal works. It must be noted that these derivations were carried out assuming the very specific condition summarized in Fig. 6.2. When different conditions are probed, such as hω D < Δ or extending the Debye model to optical phonons, the phenomenology can drastically change. Indeed, it is notable that Shrivastava, in his 1987 review [41], concludes that depending on the sample’s properties and external conditions, relaxation time can exhibit a dependence on temperature as exponential or as a power law T −n with any exponent in the range 1 ≤ n ≤ 9. The impressive range of different regimes that spin relaxation can exhibit is at the heart of the challenge of interpreting experiments in a unequivocal way.

6.4 Ab Initio Spin Dynamics Simulations In the previous section, we have illustrated how the theory of open-quantum systems can be used to derive simple relations between relaxation time and external conditions such as temperature and magnetic field. However, spin relaxation can receive contributions from several different fundamental processes, as expressed by the different perturbation orders involved at both coupling strength and timescale levels. Moreover, the relation demonstrated in the previous section also depends on assumptions about the reciprocal positioning of the Debye frequency with respect to the spin energy levels and the thermal energy. As a consequence, a plethora of different experimental behaviours can be expected for different materials in different conditions, making the interpretation of experimental results hard at best. Moreover, the phenomenological models derived in the previous section depend on the validity of the Debye model itself, which makes strong assumptions on the nature of phonons and their dispersion relation. Although this model qualitatively accounts for the acoustic dispersions of simple elements, it clearly fails in accounting for the complexity of vibrations in molecular crystals. A way around all these limitations requires to exploit electronic structure theory to quantitatively predict all the terms populating Eqs. (6.58), (6.85) and (6.119) and extracts the relaxation time from their numerical solution. In this section, we will explore this strategy and how accurately

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and effectively determine phonons and spin-phonon coupling coefficients in molecular crystals. Ab Initio Simulation of Phonons The first fundamental ingredient we need for the computation of Eqs. (6.58), (6.85) and (6.119) is the notion of phonons in a molecular crystal. As anticipated in the section about solid-state vibrations, the Hessian matrix Φ is all we need to estimate harmonic frequencies and related displacement waves (i.e. the phonons) at any qpoint of the Brillouin zone. Let us recall its definition ( Φi0lj =

∂ 2 E el ∂ xli ∂ x0 j

) ,

(6.166)

where E el can be computed with standard electronic structure methods [59, 60]. It is important to note that the summation over the cells of the molecular lattice [index l in Eq. (6.166)] is in principles extended to an infinite number of them. However, the forces among two atoms decay with their distance and the elements of Φ will decay to zero for distant enough lattice cells. It is then possible to estimate all the elements of Φ by performing electronic structure simulations on a small supercell, i.e. the replication in space of the crystal unit cell [61]. The size of the supercell actually needed to converge the elements of Φ should determined case by case, depending on the size of the unit cell and the nature of atomic interactions. For instance, large unit cells with only short-range interactions will necessitate of only small supercells, such as a 2 × 2 × 2 repetition of the unit cell along the three different crystallographic directions. Polar crystals usually do not fall in this category due to the long-range nature of dipolar electrostatic interactions, but dedicated expressions to account for this effect fortunately exist [62]. Electronic structure methods often also make it possible to compute the forces acting on atoms by mean of the Hellman–Feynman theorem, → i = −⟨ψ|∇ → i E el |ψ⟩ , F

(6.167)

where ψ is the ground-state wave function for a given molecular geometry. Using atomic forces, the numerical estimation of Φ only requires a first-order numerical differentiation ( ) ∂ Fl j , (6.168) Φi0lj = − ∂ x0i 0 where the right-hand term should be interpreted as the force acting on atom j of cell l due to the displacement of atom i in the reference cell with Rl = 0. The estimation of the integral of Eq. (6.167) is only slightly more computationally expensive than the determination of the electronic ground state |ψ⟩ , making it possible to significantly speed up the numerical evaluation of Φ compared to using Eq. (6.166). It is also important to remark that the use of Eq. (6.168) only requires the displacement of the atoms in one unit cell of the entire supercell, in virtue of the translational

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symmetry of the problem. Numerical differentiation of Eq. (6.168) is often carried out with a single-step two-sided finite difference method. In a nutshell, this method involves (i) optimization of the supercell’s lattice parameters and atomic positions, (ii) displacement of every atom of the unit cell by ±δ (Å) along every Cartesian direction and calculation of Fl j (x0i0 ± δ), and (iii) estimation of Φ according to the expression ) ( Fl j (x0i0 + δ) − Fl j (x0i0 − δ) ∂ Fl j = , (6.169) 2δ ∂ x0i 0 where the size of δ should be chosen in a way that the F versus x profile is mostly linear and not significantly affected by numerical noise. Values around 0.01 Å are often a good choice for molecular crystals. Once all the elements of Φ have been computed, the definition of the dynamical matrix in Eq. (6.22) can be used to obtain phonon frequencies and atomic displacements at any q-point. The calculation of Φ by numerical differentiation invariably carries over some numerical noise. One of the main sources of numerical noise in the simulation of force constants is the partially converged nature of simulations, both in terms of self consistent cycles and geometry optimization. Unless very tight convergence criteria are implemented, forces will result affected by a small amount of error that will then results into inconsistent force constants. Due to their small size, numerical errors in the determination of forces mainly affect the low-energy part of the vibrational spectrum (from a few to tens of cm−1 ) and often manifest themselves through the appearance of imaginary frequencies (generally plotted as negative values of ωα ) and breaking down of translational symmetry. The latter in turns reflects in the breakdown of the acoustic sum rule, i.e. the condition that force constants must obey as consequence of the translational symmetry of the crystal. This condition reads ∑∑ l

Φi0lj = 0, for any i.

(6.170)

j

The relation also implies that D(q = 0) has three eigenvalues equal to zero, which correspond to the frequencies of the three acoustic modes at the ┌-point. These modes correspond to a rigid translation of the entire crystals in the three different crystallographic directions, and since they do not create any lattice strain, they do not have any energy change associated. In real calculations, Eq. (6.170), often referred to as acoustic sum rule, breaks down and the three acoustic phonons’ frequencies acquire a finite value (either positive or negative) at q = 0. The size of this deviations provides indications on the reliability of phonons and can be used to tune simulations’ convergence criteria. The maximum value of the acceptable numerical noise in the simulation of phonons’ frequencies depends on the specific application, but the energy of the first optical modes gives a reference value. More stringent requirements are in general needed if the acoustic modes at small q are of interest. Besides providing a measure of numerical noise, Eq. (6.170) also suggests a practical route towards correcting it. Generally speaking, the symmetries of the problem

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can be enforced by computing Φ as the best compromise between reproducing the values of (∂ Fli /∂ X 0 j ) and the condition expressed by Eq. (6.170) or similar expressions for other symmetries. This approach is however numerically expensive, as it requires the solution of a linear system Ax = b with A being a matrix of size (N × M), where N is the number of force constants and M is the number of DFT forces used to infer them. Even for a crystal with a small unit cell containing 60 atoms and using a 2 × 2 × 2 supercell, N ∼ 260,000 and M ∼ 500,000, requiring ∼ 1 TB of memory to only store A. Additionally, the size of A scales to the power of 4 with the number of atoms in the unit cell, rapidly leading to an intractable problem for the common sizes of molecular crystals (often in the range of a few hundred atoms). Although sparsity arguments can help reducing the complexity of this problem, this is not an efficient route for large molecular crystals. However, if the numerical noise is little, Eq. (6.170) can be used to correct the values of Φ by rescaling its value in order to enforce the acoustic sum rule. This simple method helps enforcing the correct behaviour of the acoustic phonons at the ┌-point with no additional costs. So far, we have discussed the practicality of phonons calculations assuming that electronic structure theory can provide an accurate representation of E el . This is in fact not always the case, and many different approaches to the problem of computing energy and forces for a molecular or solid-state system exist [63, 64]. When dealing with crystals of magnetic molecules, there are three main subtleties that must be carefully considered: (i) the treatment of dispersion interactions, fundamental to describe the crystal packing, (ii) the treatment of correlation of magnetic states, and (iii) temperature effects. Ideally, quantum chemistry methods such as multi-reference coupled cluster should be preferred for the task of computing atomic forces as they automatically include electronic correlation effects leading to dispersion forces as well as to correctly treating the magnetic degrees of freedom. However, their computational cost virtually limits their application to small gas-phase molecules. When it comes to periodic systems, density functional theory (DFT) is the only option. Moreover, for large systems comprising several hundreds of atoms, as often is the case for molecular crystals, DFT in its generalized gradient approximation (GGA) is often the best, if not unique, option. GGA functionals, like PBE [65], are generally recognized to provide a good estimation of bond lengths and vibrational force constants. Alternatively, hybrid functionals also provide a good choice for the simulation of vibrational and thermodynamic properties of molecular systems [66], but the computational overheads often hinder their use for solid-state systems. Relatively to point (i), common GGA or hybrid functionals fail in capturing vdW interactions, but several correction schemes are available at very little additional computational cost. Some of the most widely employed methods include a parametrization of dispersion interactions by adding a potential term V (ri j ) = −C6 /ri6j among atoms. The parametrization of the coefficients C6 is then carried out according to different schemes, such as the Tkatchenko’s or Grimme’s methods [67, 68]. These methods account for the screening effect of covalent bonds on local atomic polarizability, which reflects on a reduction of the coefficients C6 with respect to the reference gas-phase atomic values. Concerning point (ii) of correctly describing the

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strongly correlated states of d- or f -elements, DFT is known to perform poorly, and methods such as CASSCF [47] or DMRG [69] are often used to fix the problem. These methods are however not suitable for large periodic systems, and the use of GGA DFT is once again a forced choice for current algorithms and computational power. Fortunately, however, the shortcoming of GGA DFT in representing the correct magnetic states of open shell metal compounds does not seem to dramatically spread to the prediction of forces, and phonons calculations of magnetic molecules’ crystals based on DFT are routinely carried out with good success. Finally, in relation to point (iii), it should be noted that a structural optimization leads to a zero-temperature geometry. The latter does not correctly describe the compound structure in the same thermodynamics condition of the experiments. The inclusion of finite-temperature effects can be accomplished by accounting for anharmonic contributions to the crystal potential energy surface. Anharmonic effects in crystals can be modelled in different ways, including quasi-harmonic approaches, perturbative treatments, or by running molecular dynamics simulation in the presence of a thermostat [70]. The importance of including anharmonic effects into the description of phonons for spin-phonon coupling calculations has only recently be pointed out [71–73], and a systematic investigation is yet to be presented, but it represents an important area of investigation. Ab Initio Simulation of Spin Hamiltonian’s Parameters The next fundamental step in simulating spin-phonon relaxation time is the calculation spin Hamiltonian coefficients. Let us assume that we can solve the Schrödinger equation for the electronic Hamiltonian Hˆ el , inclusive of relativistic interactions such as spin-orbit and spin-spin coupling. From the knowledge of the first Nk eigenvalues, E k , and eigenvectors, |k⟩ , of Hˆ el , we can write Hˆ el =

Nk ∑

|k⟩ E k ⟨k|.

(6.171)

k=1

Very often the eigenstates |k⟩ will also be eigenstates of the operator S 2 or J 2 , for vanishing and non-vanishing orbital angular momentum, respectively. If one of these two cases applies, we can perform a mapping of Hˆ el onto an effective spin Hamiltonian by defining a projector operator Pˆ =

2S+1 ∑

|S, M S ⟩ ⟨S, M S |.

(6.172)

MS

Pˆ verifies the relation Pˆ 2 = Pˆ = Pˆ −1 in virtue of the fact that the eigenstates of the operator Sz , |S, M S ⟩ form a complete (2S + 1)-fold basis set. Similar expressions are build for the case of non-vanishing angular momentum. The operator Pˆ can now be used to project the first (2S + 1) states of the electronic Hamiltonian onto the basis set |S, M S ⟩ , thus eliminating the explicit dependence over all the electronic degrees of freedom. The resulting operator corresponds to an effective Hamiltonian

6 Spin-Phonon Relaxation in Magnetic Molecules …

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function of the sole total spin operator, which exactly corresponds to the definition of the spin Hamiltonian [49, 50, 74] Hˆ s = Pˆ Hˆ el Pˆ −1 =

2S+1 ∑ 2S+1 ∑ MS,

|S, M S ⟩ ⟨S, M S | Hˆ el |S, M S , ⟩ ⟨S, M S , |.

(6.173)

MS

Equation (6.173) can be recast in the form ⟨S M S | Hˆ s |S M S , ⟩ = ⟨S M S | Hˆ el |S M S , ⟩ =

2S+1 ∑

⟨S M S |k⟩ E k ⟨k|S M S , ⟩ ,

(6.174)

k

which leads to a linear system of equations to solve once a specific form of Hˆ s is chosen. For instance, assuming that we are dealing with a transition ∑ metal single-ion complex, the spin Hamiltonian can be chosen of the form Hˆ s = s,t Dst Sˆs Sˆt , where (s, t) = x, y, z. Under this assumption, Eq. (6.174) reads ∑ s,t

Dst ⟨S M S | Sˆs Sˆt |S M S , ⟩ =

2S+1 ∑

⟨S M S |k⟩ E k ⟨k|S M S , ⟩ .

(6.175)

k

Equation (6.175) has the from of a linear system Ax = B, where the vector x = (D11 , D12 , . . . , D33 ) spans the nine elements of D that we want to determine, and the vector B spans the (2S + 1)2 values of the right-hand side of Eq. (6.175). When the linear system is overdetermined, it can be solved with common linear algebra routines for least-square fitting, leading to a set of spin Hamiltonian coefficients that optimally map the electronic Hamiltonian’s low-lying eigenstates. This approach is valid for any spin Hamiltonian term that has its origin in the sole electronic degrees of freedom, such as the Heisenberg Hamiltonian and the Crystal Field effective Hamiltonian commonly used for Lanthanide complexes [49]. A similar approach can be used to include the effect of the electronic Zeeman interaction, which leads to the mapping of the effective Zeeman spin Hamiltonian [75, 76]. Other mapping procedures exist for interactions involving nuclear spins, such as the hyperfine one [51]. It should also be noted that the possibility to find a good solution to Eq. (6.174) lies on the assumption that we have knowledge of the function Hˆ s that fits the spectrum of Hˆ el . For common classes of compounds, this is usually known, but some care in this choice should be applied. Now that we have a procedure to map Hˆ s from the eigenstates of Hˆ el , it remains to address the question of how to determine the latter. Electronic structure methods have evolved during the years in order to address this point and have now reached a very high degree of sophistication and accuracy. Wave function-based methods, such as complete active space, are among the most popular methods to implement Eq. (6.174), as they directly provide E k and |k⟩ [47, 49]. Density functional theory can also be used to determine spin Hamiltonian coefficients but within a different

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A. Lunghi

mapping formalism from the one presented here. DFT’s solutions cannot in fact be easily interpreted as electronic excited states with a well-defined S 2 value, and a different expression from Eq. (6.174) is thus necessary. An extensive review of electronic structure and spin Hamiltonian mapping is beyond the scope of this work, and we redirect the interested reader to the rich literature on this field. [47, 49, 51, 74]. Regardless the electronic structure method used to perform the mapping, calculated spin Hamiltonian parameters are very sensitive to slight structural changes of the molecule under investigation. A common choice is to use DFT to optimize the molecular structure in gas-phase starting from the atomic positions obtained from crystallography experiments. In case of rigid coordination spheres and ligands, this approach provides a good estimation of the intra-molecular distances, thus leading to an accurate prediction of the molecular magnetic properties. This is however not always the case, and accounting for the effect of inter-molecular interactions due to the crystal packing is often of crucial importance. As detailed in the previous section, the most appropriate computational strategy for dealing with molecular structure’s optimization in condensed-phase involves the use of periodic DFT, which enables a full unit cell optimization. It is important to note that the effect of crystal packing on molecular’s magnetic properties is mostly indirect and operates through the steric effects of molecule-molecule interactions. This is a very convenient situation, as it makes it possible to assume that the electronic structure of a molecule is the same in- or outside the crystal for the same set of intra-molecular distances. It is then possible to perform the simulation of spin Hamiltonian using CASSCF methods, only available for gas-phase calculations, while steric effects of crystal packing are introduced by using the coordinates of a single molecule as obtained at the end of the periodic crystal’s optimization. Early attempts to also include long-range electrostatic effects in the simulation of magnetic properties showed that the gas-phase approximation to the simulation of spin Hamiltonian parameters might not always be appropriate [77], but more systematic investigations are needed. Linear Spin-Phonon Coupling Strength Now that we have established a procedure to predict the coefficients of Hˆ s as function of molecular geometry, and we can use it to estimate the spin-phonon coupling coefficients as numerical derivatives of the former. One possible way to approach this problem is to generate a set of distorted molecular structures according to the atomic displacements associated with each phonon Q αq by means of the inverse relation of Eq. (6.24). Once the spin Hamiltonian coefficients have been determined for each one of these configurations, a numerical first-order derivative can be estimated with finite difference methods. This method was first used by Refs. [78–80], but it can be modified to reduce its computational cost [71]. We start noting that there is an imbalance between the number of molecular degrees of freedom (3Nat ) and the number of the unit cell degrees of freedom (3N ). The two, in fact, coincide only when the unit cell contains a single molecular unit and no counter-ions or solvent molecules. Since this is rarely the case, performing the

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numerical differentiation with respect to phonons, which are proportional to the unit cell degrees of freedom, implies a redundant number of calculations. The situation becomes even more striking when a number Nq of q-points are considered to sample the Brillouin zone. In this case, the total number of phonons used to describe the crystal’s vibrational properties becomes 3N Nq ≫ 3Nat . A more convenient strategy requires (i) the calculation of the derivatives of the spin Hamiltonian parameters with respect to molecular Cartesian coordinates, (∂ Hˆ s /∂ xi ), and (ii) their transformation into the basis set of the crystal coordinates by means of the expression Ncells ∑ N ( ∂ Hˆ ) ∑ s = ∂ Q αq l i

/

( ˆ ) h q ∂ Hs eiq·Rl L αi , Nq ωαq m i ∂ xli

(6.176)

where Nq is the number of q-points used. It is important to note that the sums over l and i involve all the atoms of the crystal, but, according to our assumptions, only the derivatives with respect to the atoms of the single molecule we selected for the spin Hamiltonian calculations will be nonzero, therefore reducing the sum to only 3Nat degrees of freedom. Nonetheless, the effect of all the phonons of the Brillouin zone can be accounted for by this method, and it does not represent an approximation per se, but simply the most efficient way to perform the estimation of spin-phonon coupling coefficients for molecular crystals. To summarize, the strategy to estimate spin-phonon coupling coefficients requires distorting the molecular structure along every Cartesian degrees of freedom (one at the time) by a value ±δ or multiples. The spin Hamiltonian coefficients of each distorted molecule are computed and then fitted with a polynomial expression 2 3 f (X is ) = C0 + C1 X is + C2 X is + C3 X is + ··· ,

(6.177)

where C1 = (∂ Hˆ s /∂ X is ) and C0 = Hs evaluated at the equilibrium geometry. The order of the polynomial function should be judged based on the data and can be changed to check the reliability of the fit. Figure 6.3 shows the results presented in Ref. [71] concerning the calculation of the spin-phonon coupling of the Fe2+ single-ion magnet [(tpaPh)Fe]− [81], where the ligand H3 tpaPh is tris((5-phenyl1H-pyrrol-2-yl)methyl)amine. Finally, the excellent agreement between results obtained by differentiating with respect to Cartesian coordinates or phonon displacements [71] is depicted in Fig. 6.4. The agreement between the two sets of spin-phonon coupling predictions is excellent, and the very small differences come from the use of slightly different derivation protocol. Quadratic Spin-Phonon Coupling Strength A similar strategy can be used to estimate second-order derivatives [56]. However, in this case one needs to scan the spin Hamiltonian parameters with respect to two molecular degrees of freedom at the time and interpolate the resulting profile with a two-variable polynomial functions. The coefficients in front of the second-order

A. Lunghi 18.100 18.000 17.900 17.800 17.700 17.600 −0.003 −0.0015

1.475 D12 (cm−1 )

D11 (cm−1 )

258

0

0.0015

1.450 1.425 −0.003 −0.0015

0.003

X1,1 (˚ A) 8.350 8.325 8.300 −0.003 −0.0015

0.003

0

0.0015

0.0015

0.003

0.0015

0.003

19.000 18.900 18.800 −0.003 −0.0015

0.003

X1,1 (˚ A) D33 (cm−1 )

-7.200

-5.100 -5.150 -5.200 −0.003 −0.0015

0 X1,1 (˚ A)

-5.050 D23 (cm−1 )

0.0015

19.100 D22 (cm−1 )

D13 (cm−1 )

0 X1,1 (˚ A)

0 X1,1 (˚ A)

0.0015

0.003

-7.300 -7.400 -7.500 -7.600 −0.003 −0.0015

0 X1,1 (˚ A)

Fig. 6.3 Spin-phonon coupling coefficients. The derivatives of the anisotropy tensor for the molecule [(tpaPh)Fe]− [81] are computed by displacing the Cartesian components of the DFT optimized structure. The value of each independent component of the tensor D is plotted as function of the displacement of the Cartesian coordinate x of the central iron atom. Reproduced from Ref. [71] with permission from the Royal Society of Chemistry All Rights Reserved

Fig. 6.4 Comparison of spin-phonon coupling simulation methods. The spin-phonon coupling intensity as function of vibrational frequency for [(tpaPh)Fe]− [81] is computed starting from coefficients computed by differentiating with respect of Cartesian and phonon displacements, respectively [71]. Reproduced from Ref. [71] with permission from the Royal Society of Chemistry All Rights Reserved

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mix variables term can then used to compute the second-order spin-phonon coupling coefficients with the relation / Ncells ∑ Ncells ∑ N ,3 N ,3 ∑ ) ∑ ( ∂ 2 Hˆ h s αq eiq·Rl L is = N ∂ Q αq Q βq, ω m q αq i l is l, i ,s, / ( ∂ 2 Hˆ ) , h s iq, ·Rl , βq × e L . (6.178) ,s, , i l Nq, ωβq, m i , ∂ X is ∂ X il , s , The simulation of the second-order derivatives poses serious challenges from a computational point of view, as the number of required simulations scales quadratically with the number of atoms and the mesh used for the polynomial interpolation. In Ref. [56], a solution to this problem based on machine learning was brought forward. The idea is to first learn a general function f ML (X is , αi ) that outputs the spin Hamiltonian parameters as function of molecular coordinates, X is and then use it to predict all the data necessary to numerically estimate the second-order derivatives of Eq. (6.178). The learning of such a function is done by optimizing the built-in coefficients αi in order to reproduce the values of spin Hamiltonian of a set of reference calculations [82, 83]. It should be noted that the method provides an advantage only if the number of calculations needed to compute the training set of f ML (X is , αi ) is smaller than the number of calculations required for a brute-force finite difference calculation. There is in fact no conceptual difference from using a polynomial function to perform a regression of values of spin Hamiltonian as function of molecular distortions or the use of machine learning. The crucial difference lies in the way the two approaches account for nonlinearity and their reciprocal learning rate. In the case of polynomial functions, this is handled by increasing the polynomial order, which rapidly leads to an increase of coefficients that needs to be determined. Machine learning offers a few advantages: (i) symmetries are automatically included in the formalisms, (ii) molecular structure can be described in terms of very compact descriptors, such as bispectrum components [84], which reduces the number of independent variables that needs to be determined, and (iii) the function is represented by a (shallow) neural network able to optimally fit any function. Two flavours of machine learning methods were recently adopted to predict the spin-phonon coupling coefficients of magnetic molecules, both in the context of solid-state perturbation theory [56] and molecular dynamics of molecules in liquid solution [85]. A schematic representation of the former method, based on ridge regression and bispectrum components as molecular fingerprints is reported in Fig. 6.5. In a nutshell, a supercell approach to the calculation of phonons is used in combination with DFT. The optimized molecular structure is then distorted in the range of 1000 times by applying small random perturbations of the atomic positions. The spin Hamiltonian coefficients of each distorted structure are determined with electronic structure methods and form the basis for the generation of a ML model. A part of this set is usually not used for training the ML model but just for testing purposes. The ML model used in Ref. [56] predicted the spin Hamiltonian coefficients by means of the knowledge

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Fig. 6.5 First-principles and ML approach to lattice and spin dynamics. a The 3 × 3 × 3 replica of the VO(acac)2 primitive cell used for the simulation of the crystal’s vibrational properties and the structure of the isolated molecular unit used to generate the training set for the ML algorithm. b The schematic structure of the ML algorithm used to predict the magnetic properties as a function of the general atomic displacements. Each atomic environment is converted into a vector of fingerprints that determine the atomic contributions to the A and g tensors. c The Fourier transform of the two-phonon correlation function integrated over the Brillouin zone. d Examples of ML predictions for the hyperfine and Lande’ tensors as function of the Vanadium displacements along x and z. Reprinted with permission from Ref. [56]. Copyright 2020 American Chemical Society All Rights Reserved

of each atom’s coordination environment, as graphically sketched in Fig. 6.5b. After a successful training of the ML, it is possible to very quickly scan the spin Hamiltonian coefficients along any pair of Cartesian directions [83]. Figure 6.5d reports just an example out the many 2D plots needed to computed all the derivatives needed in Eq. (6.178).

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Spin-Phonon Coupling Translational Invariance Similarly to what noted for the calculation of force constants, also spin-phonon coupling coefficients need to fulfil certain sum rules that arise from the symmetries of the problem. For instance, let us assume that the D tensor of a molecule depends linearly with respect to atomic positions Dαβ (X is ) = Dαβ +

∑ ( ∂ Dαβ ) is

∂ X is

0 ). (X is − X is

(6.179)

0 )= If the entire system is rigidly translated in space by an amount a Å, then (X is − X is a, leading to ∑ ( ∂ Dαβ ) . (6.180) Dαβ (X is ) = Dαβ + a ∂ X is is

Finally, since the value of D is independent on the particular position of the molecule 0 0 + a) = Dαβ (X is ). Therefore, we obtain the condition in space, Dαβ (X is ∑ ( ∂ Dαβ ) i

∂ X is

= 0, for each s.

(6.181)

Similarly to the method used to regularize the force constants in phonons calculations, translational invariance conditions can be enforced on the calculated derivatives of the spin Hamiltonian tensors by rescaling the values by a mean deviation of this condition. For instance, for the elements αβ of the tensor D, the correction for the s Cartesian direction reads (∂D ) αβ

∂ X is

=

(∂D ) αβ

∂ X is

− Devs , Devs =

Nat ( 1 ∑ ∂ Dαβ ) , Nat i ∂ X is

(6.182)

where Nat is the number of atoms in the molecule. An equivalent expression can be used for the components of the second-order derivatives that shares the same Cartesian degrees of freedom [56]. Spin-Phonon Relaxation Time The calculation of the Redfield matrices is now the last step required to compute the spin-phonon relaxation time. The calculation of the elements of Rab,cd is relatively straightforward once phonon frequencies and spin-phonon coupling coefficients are known, except for two important technical details: the integration of the Brillouion zone and dealing with the presence of a Dirac delta [55, 57]. These two aspects are strongly connected and need to be discussed simultaneously. Let us start by noting that the Dirac delta in not strictly speaking a function, but a distribution in the functional analysis sense, and that it only makes sense when appearing under the integral sign. This does not lead to any issue as all the expressions of Rab,cd contain a sum over q points, which for a solid-state

262

A. Lunghi

periodic system extend to an infinite summation, namely an integral. We thus have Rab,cd ∝

∑

{ δ(ω − ωq ) →

q

δ(ω − ωq )dq.

(6.183)

BZ

In fact, differently from the sparse spectra of an isolated cluster of atoms, the vibrational density of states P(ω) of a solid-state system is a continuous function of ω in virtue of an infinite number of degrees of system. Let us now discuss how to numerically implement an object such as the one in Eq. (6.183). The integration with respect to the q-points can be simply dealt with by discretizing the first Brillouin zone in a finite mesh of uniformly sampled q-points. The value of Rab,cd is then computed for each q-point, which are then summed up. By increasing the finesse of this mesh one eventually reaches convergence to the desired accuracy and the Redfield matrix will be numerically indistinguishable from the one obtained by integration. However, as already discussed, the Dirac delta loses sense when the sign of integral is removed, and we need a numerical way to address this issue. The common way to deal with the Dirac delta is to replace it with a real function that mimics its properties. A common choice is to use a Gaussian function with width σ in the limit of σ → 0. One can then substitute the Dirac delta functions with a Gaussian δ(x) ∼

1 2 2 √ e−x /σ , σ π

(6.184)

and recompute the Redfield matrix for decreasing values of σ until convergence is reached. To summarize, one needs to evaluate the Redfield matrix in the limit of infinite q-points and vanishing Gaussian smearing { Rab,cd ∝

δ(ω − ωq )dq = lim lim

σ →0 q→∞

BZ

∑ q

1 2 2 √ e−(ω−ωq ) /σ , σ π

(6.185)

One final important care in evaluating the limits of Eq. (6.185) is needed: the limit with respect to q must always be evaluated first than the one with respect to σ . This is required because in a simulation we are always dealing with a finite number of phonon states, regardless of the number of Nq , and if σ is reduced arbitrarily, the elements of R will invariably converge to zero. This is because exact degeneracy between two discrete spectra (the spin and the phonons ones) is never achieved, and without any smearing, the Dirac delta will always invariably result in a null value. However, we know that the discrete nature of the phonon spectrum is only due to the use of a finite mesh of q-points, and it is not intrinsic of a solid-state system. For this reason, σ can never be chosen to be much smaller than the finesse of the vibrational spectrum. To summarize, the correct numerical procedure to evaluate Eq. (6.185) is the following: (i) R is evaluated for a large smearing, e.g. σ = 10 cm−1 , and one qpoint, namely the ┌ point; (ii) keeping σ fixed, R is recalculated for increasing

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values of Nq until the desired level of convergence is reached; (iii) σ is reduced and Nq is increased further until new convergence is reached; (iv) step (iii) is repeated until convergence with respect to both σ and Nq is obtained. If this procedure is employed, the elements of the Redfield matrix converge to the right value without leading to any divergence of the relaxation time [55, 57]. Until now, we have considered a Gaussian smearing to reproduce the Dirac delta function but that is not the only option. Another common choice is the use of a Lorentzian profile, with line-width Δ. Both Gaussian and Lorentzian functions converge to the Dirac delta for their width tending to zero; however, the rate of convergence of the two functions is quite different. The tails of the Gaussian function indeed drop to zero exponentially, while those of a Lorentzian function they do so quadratically. Therefore, in order for a Gaussian and Lorentzian smearing to give results in agreement, the latter must be much smaller than the former. Finally, it is important to remark that in this discussion, we have assumed that smearing is only used to represent the Dirac delta, which correspond to a perfectly harmonic phonons bath. In Sect. 6.2, we have also contemplated the use of Lorentzian smering, but for a different reason. A Lorentzian line-width is in fact the natural line-shape of the Fourier transform of the phonon correlation function in the presence of phonon-phonon dissipation [79]. The use of Lorentzian smearing in these two circumstances should not be confused. In the former case, Δ must tend to zero in order to converge to a Dirac delta, while in the latter case Δ is set by the phonon bath properties and will in general have a finite value in the order of cm−1 or fractions. Once the Redfield matrix is computed, all which is left to do is to solve the equation of motion for the density matrix dρab (t) ∑ = Rab,cd ρcd (t). dt cd

(6.186)

In order to treat a mathematical object with four indexes, we introduce the Liuville space, L which is defined as the tensor product of the original Hilbert space, L = H ⊗ H. The space L is therefore spanned by vectors of size size(H)2 that lists all the possible pairs of vectors of the Hilbert space. What is expressed as a matrix in the Hilbert space then becomes a vector in the Liuville space. For instance, the density matrix ρab itself would read ρi , where i = ab. Accordingly, the Redfield matrix Rab,cd becomes Ri j , where i = ab and j = cd. An operator in matrix form in the Liuville space is called a super-operator and indicated with a double hat, e.g. ˆˆ to highlight that it acts on normal operators to give another operator. Finally, the R, equation of motion of the density matrix in the interaction picture reads dρi (t) ∑ = Ri j ρ j (t), or equivalently, dt j

dρˆ ˆˆ = Rˆ ρ(t) dt

(6.187)

This differential equation can now be solved with common numerical strategies. Once the Redfield matrix has been computed and diagonalized, one can construct the propagator as

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L i j (t) =

∑

Vik ei λk t Vk−1 j ,

(6.188)

k

where Vi j and λk are the eigenvectors and eigenvalues of Ri j , respectively, and Vi−1 j are the elements of the inverse matrix of the eigenvectors of Ri j . The evolution of the density matrix in the interaction picture therefore is ρi (t) =

∑

L i j (t)ρ j (t = 0).

(6.189)

j

Importantly, given the properties of R, λk should always have one null eigenvalue and all the others of negative sign. Moreover, the eigenvector associated with the null eigenvalue will correspond to the equilibrium distribution of spin states’ population. These properties of the propagator enforce that the time evolution of ρˆ in fact corresponds to a relaxation process, where out-of-diagonal elements go to zero and the diagonal terms tend to their equilibrium value for t → ∞. Once the evolution of ρˆ has been determined, it is possible to ( compute ) the time → → ˆ , where the evolution of the magnetization with the expression M(t) = Tr Sρ(t) trace is generally performed in the basis of the spin Hamiltonian eigenvectors. The → can be studied as commonly done in experiments, and in general, profile of M(t) ˆ = 0) and different orientations of the molecule in different initial conditions for ρ(t the external field will lead to different demagnetization profiles. A common choice is to orient the molecule with the easy/hard axis along the external field and initialize ρˆ in such a way that the molecule is fully magnetized along the same direction. In this scenario, for a single-spin system, an exponential decay of Mz (t) is generally observed. The latter can be fitted with the common expression ] [ Mz (t) = Mz (t = 0) − Mzeq e−t/T1 + Mzeq

(6.190)

eq

where Mz is the equilibrium value of the magnetization and T1 (or τ ) is the relaxation time. Under this conditions, 1/T1 coincide with the value of the second-smallest eigenvalue of Ri j (the first nonzero one).

6.5 The Origin of Spin-Phonon Coupling in Magnetic Molecules Now that we have established a robust computational protocol to implement the theory of spin-phonon relaxation, we are well positioned to discuss applications of this method for different molecular systems. In particular, we are interested in understanding how spin-phonon coupling is influenced by different molecular motions and by the chemical nature of the metal ions, ligands, and their bond. Normal Modes Composition

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Let us start our journey from a series of simple S = 1/2 mono-nuclear coordination compounds based on a V4+ metal centre. Vanadium in this oxidation state has a [Ar]d 1 electronic configuration, and for the common square pyramidal or distorted octahedral coordination geometries, the unpaired electron occupies a non-degenerate d orbital. This leads to a fully quenched angular momentum and a well-defined S = 1/2 ground state. The first electronic excited state in V+4 compounds is often found at energies above 5000 cm−1 [86], thus much higher energies than any vibrational frequency of molecular crystals. This electronic configuration makes it possible to use Vanadium complexes to study spin-phonon relaxation in a prototypical two-level systems. The spin Hamiltonian for an isolated V4+ molecule reads →·g·B → +S → · A · →I, Hˆ s = μB S

(6.191)

where the Lande’ tensor g and the hyperfine coupling tensor A determine the spin spectrum of the compound. Additionally, when the molecule is embedded in a crystal, the dynamics of the single spin is influenced by the dipolar interactions with all the surrounding molecules. This interaction is captured by a spin Hamiltonian of the form ∑ → j, → i · DDip · S (6.192) S Hˆ s = ij ij

where DDip depends on the molecule-molecule distance as r −3 and the magnitude and orientation of the molecular magnetic moments [1]. Besides shaping the static spin spectrum, the tensors g, A, and DDip are also the interactions that get modulated by molecular motion and that are at the origin of spin-phonon coupling. In Refs. [52, 55, 56, 72], the compound VO(acac)2 was studied following the methods detailed in the previous section, shedding light on the nature of molecular vibrations, and spin-phonon coupling in S = 1/2 molecular crystals. For this purpose, a 3 × 3 × 3 supercell of the VO(acac)2 crystal was optimized and used to sample the crystal phonons across the entire Brillouin zone [55]. The unit cell of the crystal is reported in the left panel of Fig. 6.6. Moreover, linear spin-phonon coupling coefficients were computed with Eq. (6.176) for all the interactions in Eqs. (6.191) and (6.192). Figure 6.6 reports the norm of spin-phonon coupling resolved as function of the phonon’s angular frequency and integrated over all the Brillouin zone. This analysis reveals that the modulation of dipolar interactions reaches a maximum around 50 cm−1 (Fig. 6.6), coincidentally very close to the energy of the first optical mode at the ┌-point. On the other hand, the contribution of hyperfine and Lande’ tensors to spin-phonon coupling (Fig. 6.6) maintains a finite value for all energies. This observation can be explained by considering the nature of the tensors g, A, and DDip . The latter only depends on the inter-molecular distance and is therefore most effectively modulated by rigid molecular translations. This sort of molecular displacements is expected to be associated with acoustic modes, in which density of states increases with ω but up to a finite cut-off value that depends on the intermolecular force constants and molecular mass. Figure 6.6 is compatible with such

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Fig. 6.6 Molecular structure and spin-phonon coupling in VO(acac)2 . The unit cell of VO(acac)2 is reported in the left panel. Vanadium atoms are in pink, carbon atoms in green, oxygen atoms in red, and hydrogen atoms in white. The distribution of spin-phonon coupling intensity as function of phonon frequencies for the g-tensor, the dipolar D-tensor, and the A-tensor is reported in the right panel, respectively [55]

an interpretation with the exception of the presence of a sharp cut-off frequency. On the other hand, hyperfine and Lande’ tensors represent interactions mediated by the electronic structure of the metal ion, and it is reasonable to expect them to be influenced by those optical modes that modulate the metal’s coordination shell. A full decomposition of the phonons density of states in terms of molecular motions was also performed in Ref. [55] and reported here in Fig. 6.7. The total vibrational density of states is decomposed in terms of rigid molecular translations, rotations, and intra-molecular distortions, revealing several important features of molecular crystals’ vibrational properties. Low-energy phonons are dominated by rigid translations or rotations up to frequencies of 100 cm−1 , after which optical modes are fully characterized by intra-molecular displacements. Not surprisingly, the dipolar contribution to spin-phonon coupling closely mimics the translation contribution to phonons which peaks at 50 cm−1 . Although intra-molecular contributions become dominant only at high frequencies, it is important to note that they are present at virtually any frequency. As shown by the inset of Fig. 6.7, a tiny amount of intramolecular contribution to the phonons is always present due to the finite rigidity of molecular structures. Since rigid translations cannot lead to the modulation of local molecular properties such as g and A, this intra-molecular contribution to acoustic phonons must be at the origin of spin-phonon coupling at frequencies much smaller than optical modes. The right panel of Fig. 6.7 provides an analysis of spin-phonon coupling in VO(acac)2 from a slightly different perspective. In this case, the spin-phonon coupling due to the modulation of the g-tensor along a path in the reciprocal space [52]. Figure 6.7 makes it possible to appreciate the three acoustic modes with vanishing frequencies for q → ┌. The same phonons also have a vanishing spin-phonon coupling, as acoustic modes at q = ┌ correspond to a collective translation of the entire crystal, which cannot lead to any coupling with the spin due to the isotropic nature

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Fig. 6.7 Mode composition and spin-phonon coupling in VO(acac)2 . The left panel reports the total density of states of VO(acac)2 (purple) and its decomposition into translational molecular contributions (red), molecular rotations (yellow), and intra-molecular distortions (green) [55]. The right panel reports the spin-phonon coupling intensity in arbitrary units due to the modulation of the g-tensor as function of phonon frequency and Brillouin zone vector q along the path X − ┌ [52]

of space. The spin-phonon coupling norm of acoustic modes reported in Fig. 6.7 increases linearly moving away from the ┌-point up to an energy of ∼ 2 meV (not shown explicitly) and abruptly changes for higher energies when the admixing with optical phonons becomes important. This is also highlighted by the presence of avoided crossing points in the phonon dispersions. Considering that the norm used √ the latter is then found to vary as q in Fig. 6.7 is the square of spin-phonon coupling, √ for small values of q, or equivalently as ω in virtue of a pseudo-linear dispersion relation. Although this behaviour is in agreement with the predictions of the Debye model, its origin is qualitatively different. The Debye model assumes that the magnetic properties of an atoms are directly influenced by the modulation of its distance with the atoms in the adjacent unit lattice cell as prescribed by acoustic phonons. As already discussed, the rigid translation of molecules in the crystal cannot lead to any sizeable spin-phonon coupling due to the fact that the number of atoms per unit cell is very large and that the magnetic properties of the metal ions are very well screened from inter-molecular interactions. However, since the molecules are not infinitely rigid, when molecules translate in space they also slightly deform, leading to a finite value of spin-phonon coupling. From the results of Ref. [52], we can deduce that the amount of this intra-molecular contribution to acoustic modes depends linearly with q (for small q), thus leading to results in qualitative agreement with the Debye model at frequencies much smaller that the optical modes. The study of linear spin-phonon coupling provides important insights into the origin of this interaction in terms of molecular displacements, and the same principles can expected to be applicable also to higher-order couplings [56]. A similar analysis to the one just presented has also been performed for transition metal coordination compounds with S > 1/2. Reference [71] reported a study on mono-nuclear compound based on a Fe2+ ion with a nominal S = 1 ground state,

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Fig. 6.8 Spin-phonon coupling in [(tpaPh)Fe]− . The top-left panel reports the relative spinphonon coupling intensity due to the modulation of the D by the displacement of the Cartesian coordinates of each atom. The molecular structure is reported for reference. The bottom-left panel reports the vibrational density of states due to the motion of the first coordination shell. The top-right panel reports the composition of the normal modes as function of frequency and the bottom-right panel the spin-phonon coupling intensity decomposed across translational, rotational, and intramolecular contributions. Reproduced from Ref. [71] with permission from the Royal Society of Chemistry All Rights Reserved

namely [(tpaPh)Fe]− (H3 tpaPh=tris((5-phenyl-1H-pyrrol-2-yl)methyl)amine). The molecular structure is reported in Fig. 6.8. The vibrations of this system were only studied at the level of the unit cell, therefore only giving access at the ┌-point vibrations. Despite this limitation, the study showed a consistent picture of spin-phonon coupling in molecular compounds to the one just presented. For instance, a comparison between the left and right panels of Fig. 6.8 shows that local distortions of the molecular units are responsible for the largest spin-phonon coupling and that the spin-phonon coupling due to the motion of atoms outside the first coordination shell rapidly decreases with the distance from the metal atom. Indeed, the intra-molecular distortions that were found to be strongly coupled to spin are those involving the first coordination shell. The right panel of Fig. 6.8 also reports the analysis of the molecular motion associated with each phonon and shows that optical modes are completely characterized by

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Fig. 6.9 Spin-phonon coupling in CoL2 . Panel a reports the spin-phonon coupling intensity for the high-energy window where vibrations are resonant with the excited KD energy. Panel b reports the low-energy window proper of acoustic and the first optical modes. Panels C-F depict the molecular vibrations with energy indicated by the white arrows in panels a and b. Reprinted from Ref. [57], with the permission of AIP Publishing All Rights Reserved

intra-molecular distortions for energies above 100–200 cm−1 , where the maximum of spin-phonon coupling is achieved. Finally, we present the case of the high-spin Co2+ complex [CoL2 ]2− (H2 L=1,2bis(methanesulfonamido)benzene) [87]. [CoL2 ]2− exhibit a tetrahedral coordination and a large uniaxial magnetic anisotropy with D ∼ 115 cm−1 . [CoL2 ]2− represents the only case other than VO(acac)2 where phonon dispersions and their role in spinphonon coupling have been studied with ab initio methods. Figure 6.9 reports the results of Ref. [57], where the spin-phonon coupling intensity was reported as function of frequency and q-vector along an arbitrary direction. Also in this case, optical modes are found to be significantly more coupled than acoustic phonons, except at border zone, where they have similar energies to the optical ones. The left panel of Fig. 6.9 reports the atomic displacements associated with some of the phonons and confirms that local distortions of the coordination shell strongly affect the magnetic degrees of freedom. Here we note a typical scenario, where low-energy optical modes are an admixture of rigid rotations and delocalized intra-molecular translations, while high-energy optical modes are strongly localized on specific intra-molecular degrees of freedom. When the latter involve the first coordination shell of the metal, they possess a large coupling with the spin.

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Finally, it is worth mentioning that the experimental estimation of spin-phonon coupling is also possible. Far infra-red or Raman spectroscopies recorded in the presence of external magnetic fields have been used to extract the spin-phonon coupling coefficients for the modes in resonance with spin transitions [45, 88–91]. Once combined with ab initio simulations, these experimental techniques provide one of the most direct insights on spin-phonon coupling. The Coordination Bond and Ligand Field Contributions The studies discussed so far highlight important features of spin-phonon coupling in crystals of magnetic molecules but do not address the critical point of how the chemistry of a compound influences this interaction and the nature of the lattice vibrations. Let us start once again from S = 1/2 systems. In the work of Albino et al. [86], four complexes were studied: two penta-coordinated vanadyl (VO2+ ) and two hexa-coordinated V4+ molecules, where the coordination is obtained by catecholate and dithiolate ligands, namely [VO(cat)2 ]2− , [V(cat)3 ]2− , [VO(dmit)2 ]2− , and [V(dmit)3 ]2− , where cat=catecholate and dmit=1,3-dithiole-2-thione-4,5-dithiolate. The analysis of spin-phonon coupling strength distribution as function of the frequency is reported in Fig. 6.10 for all four compounds. The notable result is that the molecules with cat ligands present higher frequencies of vibrations with respect to the corresponding dmit ligands, but at the same time overall larger spin-phonon coupling coefficients. The first observation can be traced back to the lower mass and harder nature of the oxygen donor atom of Cat with respect to the sulphur of dmit. To understand the effect ligand’s chemical nature on spin-phonon coupling coefficients, it was instead proposed a correlation with the static g-shifts, which depend on the position of the electronic excited states, in turn correlated to ligand’s donor power and covalency of the coordination bond. These observations also advance the hypothesis that the g-tensor could be used as a proxy for spin-phonon coupling, pointing to isotropic ones associated with low spin-phonon coupling and vice versa. These results for the spin-phonon coupling associated with the modulation of the g-tensor have also been formalized in the context of ligand field theory [92] and multi-reference perturbation theory [93]. Following the formalism of dynamical ligand field proposed by Mirzoyan et al. [94], we express the g-tensor in terms of second-order perturbation theory over d orbitals gi j = ge −

∑ v

ζ ⟨ψ0 |lˆi |ψv ⟩ ⟨ψv |lˆj |ψ0 ⟩ , SΔ0v

(6.193)

where lˆi is a component of the electronic angular momentum of the metal ion and Δ0v is the energy separation between the ground state and the v-excited state characterized by the wave functions ψ0 and ψv , respectively. Finally, ζ is the free-ion spin-orbit coupling constant and S is the value of the molecular ground-state spin. Considering once again the case of [V(cat)3 ]2− , where the V4+ ion experiences a distorted octahedral crystal field, the ground-state and the first excited-state wave

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Fig. 6.10 Spin-phonon coupling in V4+ complexes. The spin-phonon coupling density due to the g-tensor anisotropy is computed as function of vibrational modes’ frequency for [VO(cat)2 ]2− (1), [V(cat)3 ]2− (2), [VO(dmit)2 ]2− (3), and [V(dmit)3 ]2− (4). Reprinted with permission from Ref. [86]. Copyright 2019 American Chemical Society All Rights Reserved

functions ψ0 and ψ1 can be written as |ψ0 ⟩ = α|dz 2 ⟩ + α , |φ0 ⟩ |ψ1 ⟩ = β|d yz ⟩ + β , |φ1 ⟩

(6.194) (6.195)

where α and β describe the contribution of V4+ ’s d-like orbitals that participate to gx x , while α , and β , represent the contribution of the ligands’ orbitals. Assuming that only the portion of the orbitals localized on the V4+ centre can contribute to the integrals of Eq. (6.193) and substituting Eq. (6.195) into Eq. (6.193), one obtains that δgx x = gx x − ge = −6ζ

α2 β 2 = −ζeff Δ−1 01 , Δ01

(6.196)

It is then possible to derive an analytical expression for spin-phonon coupling by taking the derivative of Eq. (6.196) with respect to the atomic positions X is ( ∂g ) xx

∂ X is

0

( 2 2 ) α 2 β 2 ( ∂Δ01 ) −1 ∂(α β ) − 6ζ Δ 01 ∂ X is 0 Δ201 ∂ X is 0 ( ) ( ∂(α 2 β 2 ) ) δgx x ∂Δ01 − 6ζ Δ−1 . =− 01 ∂ X is 0 Δ01 ∂ X is 0 = 6ζ

(6.197) (6.198)

Equation (6.198) shows that there are two possible contributions to spin-phonon coupling due to the modulation of the g-tensor. The first one is due to the modulation of the first excited state’s energy, and the second one is due to the modulation of the

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metal-ligand hybridization due to the coordination bond. Both DFT and CASSCF results confirm that the modulation of covalency is not the dominant contribution to Eq. (6.198) [92, 93], thus confirming the important role of Δ01 . These observations also lead to the conclusion that spin-phonon coupling in S = 1/2 systems is proportional to the static g-tensor, which can be used as a simple rule-of-thumb to rank potential slow-relaxing molecules. This hypothesis has also been recently tested on a series of iso-structural S = 1/2 complexes with V4+ , Nb4+ , and Ta4+ , where it was possible to correlate relaxation time to the experimental g-shift [95]. These studies on the contributions to spin-phonon coupling also make it possible to rationalize a body of recent literature that proposes a variety of apparently orthogonal strategies to improve spin-lattice relaxation time in magnetic molecules. For instance, in the light of Eq. (6.198), several experimental works reporting, respectively, gtensor isotropy [19], large molecular covalency [77, 96], and molecular rigidity [16, 97] as primary ingredients for long spin-lattice relaxation times in molecular qubits can be reconciled by recognizing that they all point to different contributions to spin-phonon coupling and that they all must be considered at the same time in analysing spin-lattice relaxation times. This analysis holds for S = 1/2 in high field, where the g-tensor is likely to drive relaxation. The study of S > 1/2 transition metals or lanthanides needs to follow a different strategy. However, the analysis of g-tensor contributions to spin-phonon coupling brings forward an important general point: strong spin-phonon coupling arises from the modulation of those molecular degrees of freedom that more intensely contribute to the magnitude of the static spin Hamiltonian parameters. Interestingly, this analysis can also be extended to the discussion of molecular symmetry [98, 99]. Similarly to the case of g-tensor, magnetic anisotropy D in transition metals can be described within a ligand field formalism and second-order perturbation theory. Taking analytical derivatives of D, one obtains an expression equivalent to Eq. (6.196), pointing once again to the energy of excited electronic states as the key variable influencing spin-phonon coupling [100]. In the case of transition metals with very large anisotropy, electronic excited states are generally so close to the ground state that the perturbation theory used to describe D in the ligand field approach breaks down. This electronic configuration leads to a pseudo-degenerate ground state which maximize the effect of spin-orbit coupling beyond perturbative regime, similarly to what happens in lanthanides [101]. The molecular vibrations that contribute to the removal of such electronic degeneracy will then be those with large spin-phonon coupling. In the case of [(tpaPh)Fe]− , it was in fact observed that the vibrations that maximally couple to spin are those Jahn-Teller-like distortions that remove the C3v ͡ angles, thus quenching symmetry axis of the molecule by bending the in-plane NFeN the residual angular momentum [71]. The spin properties of single-ion Lanthanide complexes are largely dominated by electrostatic contributions, as discussed by Rinehart et al. [102]. Depending on the element of the Ln series, the electronic density associated with different M J states can either be prolate, oblate, or spherical. Crystal field interactions would then stabilize certain shapes of electronic density depending on the spatial orientation of

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Fig. 6.11 Spin-phonon coupling in Dy(acac). Panel a schematically shows two different types of vibrations similarly coupled to spin. Panel b reports the variation of local charge density on the Oxygen atoms during the molecular motion. Reprinted with permission from Ref. [107]. Copyright 2021 American Chemical Society All Rights Reserved

the ligands. For instance, the states with M J = ±15/2 and M J = ±1/2 of Dy3+ ions are associated with an oblate and prolate electronic density, respectively. A very axial distribution of ligands around the Dy ion would then stabilize in energy the M J = ±15/2 states with respect to M J = ±1/2, and vice versa for equatorial crystal fields [103–106]. As a consequence, it is natural to expect that vibrations able to break the axial symmetry of the Dy ion’s crystal field will be able to strongly couple to its magnetic moment. It has also been shown that a second coupling mechanism can arise [107]. This is due to the fact that the effective charge of the first coordination sphere, determining the crystal field felt by the ion, is also modulated during molecular motion. This adds a second channel of spin-phonon coupling, and in Ref. [107] it was shown that even vibrations that little affect the shape of the first coordination shell are able to strongly couple to spin if the second coordination shell motion is able to modulate the ligands’ local charges. Figure 6.11 provides a schematic representation of these two different spin-phonon coupling mechanisms for the molecule Dy(acac)3 (H2 O)2 [108]. Vib1, schematically represented in the left panel of Fig. 6.11, is a molecular vibration where the first coordination shell of Dy is little distorted, but at the same time the charges of the ligands show large fluctuations due to the vibrations involving the second coordination shell. Vice versa, Vib2 represents the typical case where local charges remain largely constant during vibration, but the first coordination shell gets significantly distorted. Both types of vibrations were found to be strongly coupled to spin.

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Fig. 6.12 Relaxation mechanism in VO(acac)2 . The spin levels are reported in the section a of the scheme. The part b of the scheme instead reports the vibrational levels involved in direct and Raman relaxation. Reprinted with permission from Ref. [56]. Copyright 2020 American Chemical Society All Rights Reserved

6.6 The Mechanism of Spin-Phonon Relaxation in Magnetic Molecules In this last section of the chapter, we will finally address the mechanism of spinphonon relaxation and the prediction of its rate. Direct and Raman Relaxation in S = 1/2 Systems The molecule VO(acac)2 will serve once again as benchmark system to present the main features of spin-phonon relaxation in S = 1/2 systems. In the case of a two-level system, one- and two-phonon transition rates are determined by secondorder perturbation theory and first- and second-order coupling strength, respectively. In Refs. [55, 56], the numerical strategies of Sect. 6.4 were used to compute the Redfield matrices of Eqs. (6.58) and (6.85) and determine the time evolution of the magnetization moment. The main results of that studies are reported in Figs. 6.12 and 6.13. Let us begin our analysis from direct spin relaxation. This relaxation mechanism involves transitions between spin states due to resonant one-phonon absorption or emission, as depicted in Fig. 6.12. Ab initio spin dynamics simulations were carried out as function of magnetic field intensity and showed that two possible relaxation regimes are possible: the first one is active at low field, where the modulation of hyperfine coupling drives relaxation, and the second one is active at high field,

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Fig. 6.13 Spin-phonon relaxation in VO(acac)2 . Panel a reports the simulated Raman relaxation time as function of field and temperature. Panel b and c reports both Raman (black) and direct (red) simulated relaxation time together with experimental values from AC measurements (blue) and inversion recovery on diluted sample of similar molecules (black star). Panel b refers to 20 K and Panel c to 5 T. Panel d reports the molecular displacements associated with the first two optical modes at the ┌-point. Reprinted with permission from Ref. [56]. Copyright 2020 American Chemical Society All Rights Reserved

where the modulation of the g-tensor is instead the main source of relaxation. For VO(acac)2 , the transition field between the two regimes was estimated to be around B ∗ ∼ 1 T. Importantly, the two regimes have a different field dependence, with the former going as τ ∼ B −2 and the latter as τ ∼ B −4 . Regarding the temperature dependence of relaxation, it was observed that τ ∼ T −1 at high T , and τ ∼ T 0 for T → 0. Interestingly, ab initio spin dynamics simulations reveal a picture of direct relaxation in agreement with the conclusions of Sect. 6.3. As anticipated in Sect. 6.5, this is because at very low vibrational frequencies typical of Zeeman/hyperfine splittings, the acoustic modes of a molecular crystal qualitatively behave as prescribed by the Debye model. Therefore, for Zeeman splittings much lower than the energy of the optical phonons, the qualitative τ versus B/T behaviour expected from classical models based on the Debye model holds. It is important to note, however, that the energies of VO(acac)2 ’s optical modes are on the high-end side of this class of compounds, with several examples known where the first optical vibrations fall as low as 10 cm−1 [86], as opposite to 40–50 cm−1 in VO(acac)2 [55]. In these compounds,

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severe deviations from the conclusions of the Debye model should be expected for fields as low as a few Tesla. Let us now turn to the Raman relaxation, which it was found to be driven by simultaneous absorption and emission of two phonons, as represented schematically in Fig. 6.12. Also in the case of Raman relaxation, it was shown that two field regimes are potentially active, with hyperfine driving relaxation as τ ∼ B 0 and g-tensor as τ ∼ B −2 [56]. Once again, this is in agreement with the conclusions reached in Sect. 6.3. The temperature dependence of Raman relaxation was found to behave as T ∼ T −2 for T > 20 K and then rapidly increase for lower T . In Sect. 6.3 we derived a similar behaviour of τ versus T coming from the acoustic phonons as accounted in the Debye model. However, as we have noted in the previous section, optical phonons are much more strongly coupled with spin, and since they fall at surprisingly low energies, it is natural to look at them as source of spin-phonon relaxation instead of acoustic ones. The results for VO(acac)2 were then interpreted by noticing that the Fourier transform of the two-phonon correlation function G 2−ph , reported in Fig. 6.5c for the process of simultaneous absorption and emission of two phonons at T = 20 K, shows a sharp maximum at some frequency close to the first ┌-point optical frequency. Since only the phonons in a small energy windows seem to contribute to relaxation, we can treat them as an effective localized mode. According to the structure of Eq. (6.85), a pair of quasi-degenerate phonons contributes to τ as τ −1 ∼

eβω (eβω − 1)2

(6.199)

which gives the correct simulated behaviour in the limit of kB T > ω. A relaxation process that follows a single contribution such as Eq. (6.199) is often associated with the local-mode relaxation mechanism [42]. The latter was originally proposed to origin in the presence of local low-energy optical modes in a magnetically doped crystal due to the mismatch in vibrational frequency between the magnetic impurities and the diamagnetic host [37]. However, in the case of magnetic molecules there is no reason to invoke such a mechanism as plenty of low-lying optical modes are always present, even in defect-free crystals [52]. The T −2 limit closely matches the experimental observation of relaxation following a low-exponent power law. However, deviations from T −2 are often observed in favour of power-law dependencies T −n with n > 2 [97]. Three possible explanations are likely to apply to this scenario. When several strongly coupled low-energy optical modes are present, multiple contributions in the form or Eq. (6.199) overlap, resulting in a more complex T dependency. On the other hand, when relaxation time is fast and rapidly exits the window of measurable values τ , it is hard to demonstrate that the high-T limit has been fully established. In this scenario, the profile of τ versus T might still be affected by the pseudo-exponential regime and therefore results might erroneously be fitted with a power law with exponent larger than n = 2. Finally, at this stage it is impossible to exclude the contribution of spin-phonon coupling strength beyond the quadratic order and that should be considered in future works. Despite the limitations of Eq. (6.199), we suggest that it should be preferred to power

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laws when fitting experimental data. The latter, unless supported by a physical picture as the one presented in Sect. 6.3, do not provide insights into the spin relaxation mechanism. On the other hand, Eq. (6.199) contains the frequency of the phonons contributing to relaxation and therefore is able to provide a clearer interpretation of relaxation experiments. Such approach has been successfully applied to a series of spin 1/2 systems [98, 99, 109, 110]. It is important to remark that the agreement between experiments and simulations becomes acceptable only in the regimes of high-T /high-B. This is due to the fact that in the opposite regime, dipolar spin-spin coupling, not included in the simulations, becomes the dominant source of relaxation. Indeed, when diluted samples are measured with EPR inversion recovery, a much better agreement is observed [58]. Ab initio spin dynamics simulations also make it possible to determine the phonons responsible for relaxation. In the case of direct relaxation, only phonons in resonance with the transition are able to contribute. As depicted in Fig. 6.7, at fields lower than 10 T only acoustic phonons are present, and their intra-molecular component is the main drive for relaxation. In the case of Raman relaxation instead, all the phonons in the spectrum are potentially able to contribute. However, only low-energy optical phonons significantly contribute to spin dynamics as they are at the same time thermally populated and strongly coupled. Indeed, the population of high-energy phonons diminish exponentially with their frequency, while the acoustic phonons, despite being thermally populated, have a low density of states. This can also be appreciated from Fig. 6.7, where the density of states rapidly drops below 40–50 cm−1 . Figure 6.13d shows typical molecular distortions associated with the first few optical modes at the ┌-point. Orbach and Raman relaxation in S > 1/2 systems with uniaxial anisotropy Our second case study is a high-spin Co2+ complex (S = 3/2) with large uniaxial magnetic anisotropy [87], namely with D = −115 cm−1 and vanishing rhombicity |E/D|. In the absence of an external field, the direct transitions within Kramers’ doublets M S = ±n/2 are prohibited by time-reversal symmetry, and one-phonon relaxation must occur through the Orbach mechanism. Interestingly, the same selection rule applies to two-phonon transitions due to second-order perturbation theory and quadratic spin-phonon coupling, as the matrix elements of Eqs. (6.58) and (6.85) have the same dependency on spin operators and Hˆ s eigenfunctions. The source of two-phonon relaxation is therefore to be looked for in Eq. (6.119), which uses fourthorder perturbation theory. The main results of Ref. [57] are presented in Fig. 6.14. At high temperature, the Orbach mechanisms become the most favourable relaxation pathway, and it is dominated by one-phonon absorption of phonons resonant with the M S = ±3/2 → M S = ±1/2 transition, as depicted in the right panel of Fig. 6.14. At low temperature, the Raman mechanism instead drives relaxation and shows a pseudo-exponential trend with respect to T . From the analysis of the transition rates computed with Eq. (6.119), it was possible to interpret the Raman relaxation as a direct spin flip M S = ±3/2 → M S = ∓3/2 due to the simultaneous absorption and emission of two degenerate phonons. This process is represented in the right panel of

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Fig. 6.14. Overall, the temperature behaviour of Orbach and Raman spin relaxation were found to follow the expression τ −1 = τ0−1 e−βUeff + V˜2−sph

eβ ω˜ , (eβ ω˜ − 1)2

(6.200)

In Eq. (6.200), Ueff is the energy of the excited Kramers doublet and ω˜ is a value of energy compatible with the lowest lying optical modes, computed around ∼ 20 cm−1 . Similarly to the case of S = 1/2, Raman relaxation receives contributions from all the vibrational spectrum. Among all the phonons, those contemporaneously populated and coupled to spin will drive relaxation. However, in the case of S = 3/2, the matrix element of Eq. (6.119) will also favour phonons in resonance with the spin excited states. In highly anisotropic compounds such as this, the excited Kramers doublet lies at a much higher energy than the first optical modes, suggesting that phonons well above the first optical transitions should be involved in Raman relaxation. However, it should be noted that the resonant condition with the excited states is only imposed at the power of two [see Eq. (6.119)], while the population of phonons decreases exponentially with their energy, as dictated by the Bose-Einstein population. At low temperature, the latter condition overcomes the former, leading to the conclusion that low-energy modes are the main drive for Raman relaxation in both S = 1/2 and S > 1/2. As discussed for S = 1/2, acoustic modes do not generally contribute significantly to Raman relaxation in molecules as they have a very low density of states. However, at very low temperature (T < 5 K), when even optical modes are strongly un-populated, then acoustic phonons might become relevant. Similarly to what discussed for relaxation in S = 1/2, the use of Eq. (6.200) should be preferred to power laws, as commonly done. As it can be appreciated from Fig. 6.200, the accuracy reached in Ref. [57] is not fully satisfactory and the error between experiments and predictions is around one and two orders of magnitude. As discussed in Ref. [58], this is almost entirely due to the misuse of the secular approximation, which does not totally decouple coherence and population terms of the density matrix for Kramers systems in zero external field. Once the secular approximation is correctly carried out, the agreement between experiments and theory becomes virtually exact [58]. Dy3+ complexes with axial symmetry have received a large attention from the molecular magnetism community in virtue of their slow spin relaxation [111]. Several molecules have then been studied by means of ab initio methods to elucidate the role of spin-phonon coupling and predict spin relaxation rates. Reference [80] represents the first attempt at computing the relaxation times in Ln complexes. The system ttt t + studied was [DyCpttt 2 ] (Cp = [C5 H5 Bu3 -1,2,4]), DyCp in short, i.e. the first singlemolecule magnet with a blocking temperature approaching nitrogen boiling point. DyCp exhibits a Dy3+ ion with a strongly anisotropic ground state J = 15/2 due to an almost perfectly axial coordination geometry. As mentioned in Sect. 6.5, this situation is ideal to create very large energy gaps between the KDs with M J = ±15/2 and M J = ±1/2, which is estimated to be around 1500 cm−1 in DyCp. The simulations presented in in Ref. [80] were able to capture the experimental results in the high-T

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Fig. 6.14 Orbach and Raman relaxation in a Co2+ complex. Left panel: dashed lines correspond to the measured relaxation time in zero external field. The red line and triangles are the computed Orbach relaxation, and the blue line and triangles are the computed Raman relaxation. Right panel: a The spin states of Co(pdms)2 are reported as function of their energy and the nominal Sz value. The red and blue arrows represent possible relaxation pathways. b An electronic excitation from the ground state to an excited state (red arrow in panel a) can be accompanied by the absorption of a phonon with energy in resonance with the spin transition (Orbach process). c The direct transition Sz =3/2 → Sz = −3/2 (blue arrow in panel a) can be induced by the simultaneous absorption and emission of two phonons with the same energy (Raman process). Reprinted from Ref. [57], with the permission of AIP Publishing All Rights Reserved Fig. 6.15 Orbach and Raman relaxation in Dy3+ complexes. Experimental relaxation time for Dy(acac) (black dots) together to the simulated Orbach (red line and symbols) and Raman (green line and symbols) ones. Adapted with permission from Ref. [107]. Copyright 2021 American Chemical Society All Rights Reserved

regime (T > 60 K) with only one order of magnitude of error. Simulations show that the modulation of the effective crystal field operator was responsible for transitions involving up to the sixth excited Kramers doublet, supporting the use of the first term of Eq. (6.200) to fit the Orbach relaxation regime and determining an effective anisotropy barrier Ueff ∼ 1225 cm−1 . The prediction of Orbach rates with ab initio spin dynamics has been further validated through the study of a series of Dy3+ molecules with long relaxation time, revealing interesting correlations between the molecular motion of high-energy modes in resonance with spin Kramers doublet

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and spin relaxation [112–114]. In particular, these studies showed that relaxation may occur though complex relaxation pathways, often involving several of the 16 states of the Dy3+ ground-state multiplet. Moreover, it was shown that, although the performance of slow-relaxing single-molecule magnets is primarily due to the axiality and strength of the crystal field splitting, among molecules showing similar values of Ueff , the energy alignment and the careful design of the vibrational density of states can play a fundamental role in bringing down relaxation rates even further [114]. These series of studies were performed including only gas-phase vibrations as well as considering only one-phonon processes. As a consequence, they were not able to predict spin relaxation at low temperature, where a second pseudo-exponential relaxation regime appears, similarly to what observed in Co2+ (see Fig. 6.14). The compound Dy[(acac)3 (H2 O)2 ], Dyacac in short, was also studied with ab initio spin dynamics, this time including all the phonons of the crystal’s unit cell and both one- and two-phonon transitions [107]. Similarly to what discussed for Co2+ , relaxation occurs through Raman mechanism at low temperature, while Orbach is the dominating mechanism at high temperature. Once again the relaxation time depends on temperature as reported in Eq. (6.200) and depicted in Fig. 6.15. Differently from DyCp, the Orbach relaxation in Dyacac was observed to be mediated by the first two excited Kramers doublets, due to the low symmetry of the coordination of the Dy ion and the consequently large admixing between different M J . Regarding the Raman mechanism, it was instead observed that only the first few optical modes at the ┌-point contribute to relaxation in virtue of a larger thermal population at low temperature. Simulations for DyCp were repeated using a similar approach to the one employed for Dyacac [58], confirming that a two-phonon Raman mechanism is able to explain spin relaxation at low-T . A similar interpretation to the dynamics in Dyacac also applies to DyCp. Despite the much higher energy of the spin transitions in DyCp respect to Dyacac ( the first spin excited state falls at ∼ 450 cm−1 in DyCp and ∼ 100 cm−1 in Dyacac), simulations confirm that low-energy vibrations up to a few THz are the only contribution to Raman spin-phonon relaxation. On the other hand, only high-energy optical phonons in resonance with spin transitions are responsible for Orbach relaxation. Importantly, the results on DyCp’s spin dynamics in zero-field showed a remarkable dependence on the inclusion of coherence terms in the spin reduced density matrix, with deviations of up to nine orders of magnitude from the experimental results when neglected. Vice versa, as observed for Co2+ system discussed above, once the secular approximation is correctly implemented, simulations become in perfect agreement with experiments over the entire range of temperature [58], therefore providing a conclusive proof that spin-phonon relaxation in magnetic molecules of Kramers ions presenting easy-axis magnetic anisotropy is fully understood. Outlook In this chapter, we have illustrated how the theory of open-quantum systems can be adapted to the problem of spin-phonon interaction and provide a quantitative ground for the study of spin dynamics. Despite the handful of systems studied so

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far, ab initio simulations have already provided deep insights into the nature of spinphonon coupling and spin relaxation in crystals of magnetic molecules. Here we will attempt to summarize the most important points. Most of our understanding of spin relaxation in solid-state comes from the application of the Debye model and perturbation theory. However, differently from the solid-state doped materials initially studied, molecular crystals possess many lowlying optical modes. Moreover, as one of the main goals pursued in the field of molecular magnetism is the enhancement of magnetic anisotropy, the energy splitting among spin states has been increasing by orders of magnitudes along the years [5, 115]. This situation made it so that the assumptions of Sect. 6.3 have lost their general validity. Ab initio simulations showed that for molecular materials, acoustic modes are not particularly relevant and that optical ones are instead responsible for relaxation. Moreover, it was found that optical modes appear at surprisingly low energies, so that even at very low temperature they remain sensibly populated. Thanks to ab initio spin dynamics, it has now been possible to demonstrate the nature of spin relaxation mechanism in Kramers systems. Direct and Orbach mechanisms are due to the absorption and emission of resonant phonons [55, 80]. The origin of Raman relaxation is attributed to different mechanisms for S = 1/2 and S > 1/2 systems. In the former case, quadratic spin-phonon coupling is responsible for two-phonon transitions [56], while in the latter case, the fourth-order perturbation theory is necessary to explain spin relaxation in zero external field [46, 57, 107, 116]. In both cases, Raman is mediated by low-energy optical phonons exhibiting local molecular rotations admixed to delocalized intra-molecular distortions. Importantly, ab initio simulations support the use of the expression τ −1 = V˜ 1−sph

1 (eβ ω˜ 1−ph − 1)

+

∑ i

2−ph

2−sph V˜i

eβ ω˜ i 2−ph

(eβ ω˜ i

− 1)2

,

(6.201)

to fit experimental trends of τ versus T , where possible. The first term of Eq. (6.201) either corresponds to Direct or Orbach relaxation depending on the energy of the phonon involved, hω˜ 1−ph , while the second term describes Raman relaxation as dictated by low-energy THz phonons with energy hω˜ 2−ph . Due to the non-resonant nature of Raman relaxation, a sum of multiple contributions is often needed to accurately describe the spin relaxation temperature profile [98, 99, 109]. This approach holds a good promise for giving more physical insights into the vibrations involved in Raman relaxation as compared to the use of less insightful power-law expressions. These findings have major consequences for molecular magnetism practices. In addition to well-established approaches to increase the axiality of the crystal field [102] and exchange coupling multiple ions [117], ab initio spin dynamics results point to the design of rigid molecular structures as key to reduce spin-phonon relaxation rates [71]. Indeed, by increasing molecular rigidity, optical modes are shifted up in energy with a twofold effect: (i) they become less admixed with acoustic modes [55], so reducing the rate of direct relaxation, and (ii) they become less populated so that Raman relaxation rate would also slow down [56, 57, 107]. A second approach sug-

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gested by ab initio spin dynamics involves the careful disalignment of spin and vibrational spectra in order to reduce the effect of resonant Orbach process [80, 113, 118]. Finally, although ab initio spin dynamics is not yet a fully quantitative computational approach, we have shown that it is rapidly maturing into one. Since its first proposals in 2017 [71, 78–80], a level of accuracy down to one order of magnitude has now been proved for different systems across different relaxation regimes [58]. Although further benchmarking is needed, this level of accuracy already hints at the possibility to rank the spin lifetime of different compounds in advance of experimental characterization [119]. This is an important milestone for the field of molecular magnetism, as it opens the gates to a long-sought fully in silico screening of new compounds. Let us now attempt to provide an overview of future developments as well as a brief discussion of the potential impact of ab initio spin dynamics for other adjacent fields. Ab initio theory of spin relaxation now includes all the terms up to two-phonon processes. This is essentially the same level of theory employed in the past to derive the canonical picture of spin-phonon relaxation, and we can therefore claim that the theoretical formulation of such theories in terms of ab initio models is available. However, this is far from claiming that no further work will be necessary to obtain a conclusive and universal picture of spin relaxation in solid state. On the contrary, this theoretical advancements form the stepping stone for an in-depth analysis of relaxation phenomenology, for instance going beyond the classical Born-Markov approximation and the inclusion of other decoherence channels such as magnetic dipolar interactions. These two avenues in fact represent some of the most urgent developments of the proposed theoretical framework and have the potential to account for phonon bottleneck [73, 120] and magnetization quantum tunnelling effects [44, 121, 122], respectively. These contributions to relaxation are indeed ubiquitous in experiments and must find their place into a quantitative theory of spin dynamics. Moreover, it is important to remark that so far ab initio spin dynamics has been applied to only a handful of systems, and there is large scope for investigating the large and varied phenomenology of molecular magnetism. Examples include the study of exchange-coupled poly-nuclear magnetic molecules [5], surface-adsorbed systems [14], and molecules in glassy solutions [43]. In order to reach a fully quantitative ground, ab initio spin dynamics now requires a careful testing of the electronic structure methods underlying it. It is well known that the inclusion of electronic correlation in the simulations is key to achieve accurate prediction of spin Hamiltonian parameters [48, 49], and the same effect can be expected for spin-phonon coupling coefficients. On the other hand, the simulation of phonon modes requires significant advancements. The introduction of sophisticated vdW corrections to DFT [123] and the inclusion of anharmonic lattice effects [70] stand out as two important avenues for future development. The development of efficient simulation strategies is another necessary step towards the widespread use of ab initio spin dynamics. On the one hand, the ab initio methods underlying spin relaxation theory are particularly heavy in computational terms. Magnetic molecules indeed often crystallize in large unit cells, and the simulation of phonons

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beyond the ┌-point is a daunting task. Similarly, the calculation of spin-phonon coupling coefficients requires the quantum chemistry simulations for many similarly distorted molecular geometries, rapidly leading to immense computational overheads. In Sect. 6.5, we have hinted to machine learning as a game-changer in this area. Machine learning, learning the underlying features of a distribution from examples drawn out it, makes it possible to interpolate complex functions such as the potential energy surface of a molecular crystal [57, 82] or the relation between spin Hamiltonian and molecular structure [83]. After the training stage, ML models can be used to make predictions at very little computational cost, therefore offering a significant speed-up of these simulations [124]. Further adaptation of machine learning schemes to this tasks is needed, but encouraging proof-of-concept applications have recently been presented [56, 57, 83, 85, 125–127]. Another interesting strategy involves the use of parametrized Hamiltonians to compute the spin-phonon coupling coefficients. When an accurate parametrization is possible, extensive speed-ups can be achieved [128]. Although this chapter has been focusing on crystals of magnetic molecules, many more systems are within the reach of the methods we have illustrated. Indeed, the only strong assumptions of ab initio spin dynamics are that the system must be well described with a spin Hamiltonian formalism and that it must exist an accurate ab initio method to predict lattice and spin properties. The computational scheme we have discussed should readily apply to any kind of system exhibiting localized magnetism. Indeed, solid-state defects or impurities in solid-state semiconductors are commonly described with the same theoretical tools used in the field of molecular magnetism, and DFT-based schemes for the simulation of their properties already exist [129]. Many of these systems are currently under scrutiny for quantum sensing technologies, and there is a large interest in their relaxation properties [130]. Another type of system that we believe is easily within the reach of the presented methods is nuclear spin. Indeed, electronic structure methods are routinely used to predict nuclear spin Hamiltonian’s parameters [131], and when such nuclear spins are part of a molecular system, there is virtually no difference from the treatment of electron spin dynamics. In conclusion, we have provided a comprehensive overview of the state of the art in ab initio spin dynamics as well as a practical guide to its application to crystals of magnetic molecules. We have shown that this novel computational method is rapidly maturing into a quantitative tool able to provide unique insights into spin relaxation, thus holding great promise for the field of molecular magnetism and beyond. As such, we hope that this work will serve as a starting point for the readers interested in adopting this strategy as well as a review of the strives so far for expert readers. Acknowledgements We acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. [948493]).

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Chapter 7

Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where to from Here? Sourav Dey, Tanu Sharma, Arup Sarkar, and Gopalan Rajaraman

Abstract Ab initio calculations have played an active role in the design and development of Lanthanide-based single-ion magnets (SIMs) for the last two decades or so. These methods not only offer insight into the molecules that are reported but also hold a significant predictive potential to take this area forward. In this chapter, we aim to give an overview of the electronic structure method (ab initio SA-CASSCF/ RASSI-SO/SINGLE_ANISO) to interpret, analyse and predict the magnetic properties of lanthanide-based SMMs. In the past few years, we have witnessed the evolution of blocking temperature (T B ) and blocking barrier (U eff ) of Dysprosium-based Lanthanide SIMs. Among other classes of molecules that have intriguing magnetic properties, a class of molecules with a higher-order Dnh symmetry (n = 4, 5, 6 and 8) were predicted to yield superior magnetic properties and were later synthesised and found to possess very large U eff values and decent blocking temperatures. Particularly, molecules with D5h symmetry played an important role in understanding various contributions to the barrier height, and more than fifty molecules are reported now, with several variations yielding the absence of SIM behaviour to some of the best. In this chapter, we aim to give an overview of the variation observed in this class and another new class of molecules, i.e. Ln-encapsulated fullerene as SMMs. Particularly, we aim to showcase the strength of these methods in understanding the mechanism of magnetisation relaxation and the role of symmetry in dictating the desired magnetic characteristics. Keywords SMM · Ab initio · DFT · Magnetic anisotropy · U eff

S. Dey · T. Sharma · A. Sarkar · G. Rajaraman (B) Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Rajaraman (ed.), Computational Modelling of Molecular Nanomagnets, Challenges and Advances in Computational Chemistry and Physics 34, https://doi.org/10.1007/978-3-031-31038-6_7

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7.1 Introduction Single-molecule magnets (SMMs) are a class of molecule-based magnets exhibiting blockade of permanent magnetisation below a certain temperature. This was discovered first by accident in a transition metal cluster popularly called {Mn12 Ac} by Sessoli and co-workers in 1993 [1–3]. The slow relaxation of magnetisation observed in this molecule led to several potential applications such as compact memory storage devices, Q-bits in quantum computing and molecular spintronics [4–12]. Each of the proposed potential applications demand certain criteria to be met to realise the end-user applications [13–18]. For example, to realise the molecule-based compact storage devices, it is important to have a very large blocking barrier for magnetisation reversal (U eff ) and blocking temperatures (T B ), defined as the temperature below which the magnetisation is fully frozen [19]. Over the years, several attempts have been made to increase both the U eff and T B values [20], while the initial efforts focussed on transition metal-based clusters to enhance the spin ground state S and zero-field splitting parameters (D) [21–28]. This has not yielded very attractive SMMs possessing larger T B values as controlling the magnetic anisotropy in larger polynuclear clusters is challenging, and S and D were found to be inversely proportional to each other [29, 30]. Furthermore, the magnetic coupling needs to be very high to ensure that excited states do not participate in the relaxation mechanism—a formidable task that demands exchange coupling of >300 cm−1 [10, 11, 31–36]. As controlling magnetic anisotropy is key to success, an alternative class of molecules suitable for exhibiting slow relaxation of magnetisation was explored. In this regard, the report of slow relaxation of magnetisation in [TbPc2 ]− (Pc = phthalocyanine) by Ishikawa et al. gained attention as this study elegantly exploits the inherently unquenched orbital angular momentum of the deeply buried 4f orbitals to build a new class of molecular magnets called Single-ion Magnets (SIMs) [10, 11, 31, 37]. Since then, there has been a tremendous interest in the geometry and coordination number in designing high-performance SIMs. In this regard, Long and co-workers have categorised the trivalent lanthanide ions as prolate or oblate as per their f-electron density of the ground |J, mJ ⟩ microstate [38]. This classification aids in designing suitable ligands for individual Ln(III) ions to exhibit SMM characteristics and led to the discovery of numerous high-barrier height lanthanide SMMs [14, 15, 18, 39–41]. Among all the Ln(III) SMMs reported, Dy(III) ions are the most prominent to exhibit SIM behavior due to their bistable ground state and even slightly strong axial ligand field compared to equatorial ligand field is sufficient to create anisotropy [38, 42]. It can also be ascribed to the large spin–orbit angular momentum (J = 15/2) in the axial ligand field, which creates a substantial energy gap between the successive ±mJ states. Furthermore, the choice of symmetry also plays a key role in dictating the above energy gap. It has been seen in many instances that the highly symmetric complexes possessing Dnh (n = 4–6) symmetry enjoy a very large barrier height for magnetisation reversal [43–49]. Achieving such high symmetry often demands low-coordinate Lanthanide complexes. There are three approaches to obtaining pseudo-low-coordinate lanthanide complexes (i) employ strong axial

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and weak equatorial ligands to mimic the two-coordinate environment. Using this strategy, several Dy(III) molecules possessing pseudo-Dnh symmetry were reported with very high blocking barriers and attractive blocking temperatures [17, 43, 44, 50–53]. The success here lies in how strong the axial and how weak the equatorial ligation is, and this has been chemist’s playground ground over the years that resulted in numerous air-stable high T B SMMs. (ii) Second approach employs a very large bulky ligand that can stabilise low coordination numbers. Using this strategy, Long, Mills, Layfield and co-workers came up with pseudo two-coordinate “Dysprocenium” complexes where the estimated T B values surpass the liquid nitrogen temperature [39, 40, 54, 55]. The discovery of these “Dysprocenium” complexes is an important breakthrough in lanthanide SIM. Although this class of molecules has very large U eff values, the T B value remains only a fraction. Further, these molecules are very fragile and do not sustain the harsh fabrication conditions required (iii) the high symmetry coupled with low coordination number is also attainable in endohedral lanthanofullerene molecules [56–58] these advantages led to the development of various high-performance lanthanide SIMs and SMMs such as LnSc2 N@C80 , Ln2 ScN@C80 and Ln2 @C79 N (Ln=Tb, Dy, Ho) [57–65]. In all three approaches, the anisotropy generated is governed by the crystal field of the coordinated ligands. Therefore, both the symmetry and ligand field play a key role in the magnetic anisotropy of lanthanide SIMs. A thorough understanding of the electronic structure is mandatory to control the factors that govern the magnetisation reversal. This not only helps to analyse and interpret the magnetic properties but also provides design clues to a new generation of molecular magnets. This chapter aims to discuss the electronic structure and magnetic anisotropy of various Dy(III) SIMs possessing a pseudo-Dnh/nd (n = 2, 3, 4, 5, 6, 8) symmetry using multireference SA-CASSCF /RASSI-SO/SINGLE_ANISO calculations. The SINGLE_ANISO routine employs ) ( 1 ˜ the pseudospin S = 2 Hamiltonian approach developed by Chibotaru and coworkers to extract magnetic anisotropy and other important parameters [42]. Later, we extended our studies to Dysprocenium SIMs and concluded the chapter with endohedral lanthanofullerene SIMs/SMMs.

7.2 Lanthanide-Based SMMs Prerequisites To begin with, a doubly degenerate ground state is required to attain magnetic bistability. This degeneracy is guaranteed for a system with a half-integer spin such as Dy(III) (referred to as Kramers ion) due to the time-reversal symmetry. Consequently, most of the reported lanthanide SIMs are found to be based on Dy(III) ions. The barrier height of magnetisation reversal is determined by the crystal field splitting of the ground spin–orbit state. For example, in the case of Dy(III) ion, the crystal field splits the ground 6 H15/2 state into sixteen |±mJ ⟩ states(±15/2, ±13/ 2, ±11/2, ±9/2, ±7/2, ±5/2, ±3/2 and ±1/2), termed as eight Kramers doublets (KDs). The significant separation between the KDs is the key to attaining substantial

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Ueff values. Furthermore, the larger the |±mJ ⟩ ground state, the greater will be its magnetic moment and, therefore, slower will be its relaxation. In simple terms, |±mJ ⟩ value resembles the spin ground state |±mS ⟩ value in transition metal clusters and the crystal field splitting of the |±mJ ⟩ states is similar to the zero-field splitting pattern, D. Furthermore, a large |±mJ ⟩ ground state is desired to reduce the QTM (quantum tunnelling of magnetisation) in the relaxation mechanism. The QTM is induced by the mixing of a ground |±mJ ⟩ with other excited |±mJ ⟩ states. The stabilisation of the largest mJ state as the ground state (such as mJ = ±15/2 for Dy(III)) depends on the surrounding ligand field and the nature of electron density (oblate /prolate) of a lanthanide (III) ion. The lanthanide (III) ions with oblate electron density such as Ho(III), Dy(III), Tb(III), Nd(III), Pr(III) and Ce(III) require a strong axial and weak equatorial crystal field to attain the largest ±mJ state as the ground state [14, 38]. On the other hand, a weak axial and strong equatorial crystal field is needed to stabilise the largest ±mJ as a ground state in lanthanide(III) ion with prolate electron density such as Yb(III), Tm(III), Er(III), Sm(III) and Pm(III) [14, 38]. Apart from the stronger axiality, the local symmetry around the coordination sphere also plays an important role in enhancing the U eff values. But which symmetry is well suited for the SIM behaviour of Dy(III) ions? To answer this question, we have studied a series of [Dy(OH)n ]3−n and [Er(OH)n ]3−n (n = 1 → 12) models [18]. Such models are later extended to [DyFn ]3−n (n = 1 → 12) systems with varying coordination numbers by Ungur and co-workers [14]. The ab initio calculations reveal that the Dy(III) models with point groups C ∞v , D∞h , S 8 /D4d , D4h , D5h and D6h /6d are well suited for improved SMM performance [29, 66, 67]. After these findings, concentrated efforts have been made to prepare a highly symmetric Dy(III) complex with a strong axial and weak equatorial ligand field. It leads to the generation of several D4h , D5h , D5d and D6h symmetric Dy(III) SIMs with a large blocking barrier and blocking temperatures. The unavoidable transverse magnetic field from the surrounding equatorial ligand always promotes the QTM. Therefore, minimising this transverse magnetic field is an essential criterion for designing a potential SMM. The transverse magnetic moment in ab initio calculations is represented by gxx and gyy values, while the axiality is represented by gzz value. The g values are estimated from the Zeeman interaction in the presence of a magnetic field (B). The total magnetic moment (μ) in a strong orbit coupling regime can be represented as μ = −μ B g S˜

(7.1)

Hˆ Z ee = −μ · B = μ B g S˜ B

(7.2)

μ B is called Bohr magneton. In the case of strong spin–orbit coupling, such as in lanthanides, the lowest electronic structure is described by pseudospin ( S˜ = 1/2) instead of true spin (S) [42]. In this regard, the derivation of pseudospin Hamiltonian by Chibotaru and co-workers and its one-to-one correspondence with multireference

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wave function from ab initio calculations enable to explain the magnetic properties in the weak, medium and strong spin–orbit regime as explained in one of the contributing chapter [42]. However, the gzz >> gxx /gyy criterion should be maintained in order to reduce the QTM in Kramers ion or tunnel splitting in the case of nonKramers ion. The tunnel splitting is the small energy difference between the |+mJ ⟩ and |−mJ ⟩ levels (such as between |+6⟩ and |−6⟩ for Tb(III)) or pseudo-Kramers doublets (pKDs) for a non-Kramers ion. Another way to represent the axiality is the use of the crystal field (CF) parameter from ab initio calculations. The CF parameters in the ab initio SA-CASSCF /RASSI-SO/SINGLE_ANISO calculations are estimated according to the Stevens Hamiltonian [42] Hˆ C F =

k ∑∑ k

Bk Oˆ k q

q

(7.3)

q=−k

q q where Bk and Oˆ k indicate the CF parameters and Stevens operator, respectively. Here, k is the rank tensor, and q can have values from −k to +k. The operator related to the even k values (k = 2, 4, 6) is accountable for the CF splitting and the operator related to odd k values (k = 1, 3, 5) is accountable for the intensity of electric dipole transitions, and hence, it does not associate with magnetism [68]. However, q Bk parameter with k = 2, 4, 6 and q = 0 denotes the axial crystal field while the q Bk parameters with k = 2, 4, 6 and q /= 0 represents the non-axial crystal field [68]. The value of the axial CF parameter should be larger compared to the non-axial CF parameter in order to generate a large blocking barrier for magnetisation reversal [68]. To summarise, in Lanthanide SIMs, a large |±mJ ⟩ as the ground state coupled with very large crystal field splitting among the |±mJ ⟩ states with no/substantially diminished QTM between the |±mJ ⟩ states is desired to obtain superior SMMs. The pseudo symmetry around the coordination sphere is desired as this is likely to diminish the QTMs for certain point groups.

7.3 Mechanism of Magnetisation Relaxation in SMMs/ SIMs After perceiving the Spin-Hamiltonian parameters, one can now qualitatively describe the mechanism of spin (or magnetisation) relaxation pathways through various spin–lattice vibrations [69]. In the practical application for memory storage in SMMs (or SIMs), one has to apply an external magnetic field to magnetise the molecule. As soon as these molecules get magnetised, one of the M J or M S components of the spin eigenstates remains overpopulated than the other opposite component. During the demagnetisation process, the slower the relaxation time better; it will behave like a memory device for data storage. As soon as the magnetic field is removed (demagnetisation process), the system tries to relax back to its original state

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via multiple relaxation pathways with changing Boltzmann populations. In general, spin or magnetisation relaxation is of two types—(i) spin–lattice relaxation (denoted as T 1 ) and (ii) spin–spin relaxation (denoted as T 2 ) [10, 11]. Spin–lattice or longitudinal relaxation of the magnetic moment takes place along the z-axis of the applied magnetic field, and the exchange of energy occurs via lattice vibrational states or phonons. In comparison, the spin–spin or transverse relaxation involves relaxation of the magnetic moment oriented in the perpendicular or XY plane of the applied magnetic field. The spin–spin relaxation is strongly influenced by the presence of surrounding spins (nuclear or electronic) and is an important factor in controlling quantum information processing applications [70]. Complexes having large magnetisation reversal barriers show spin–lattice relaxation through the Orbach mechanism at higher temperatures. They are thus crucial for the SMMs or SIMs to function as memory devices [71]. However, at low temperature, various other mechanisms such as direct, Raman and QTM effects can simultaneously operate to undercut the U eff values for SMMs/SIMs. Lanthanide coordination complexes, especially mononuclear Dy-SIMs, are a typical example that displays various spin–lattice relaxation mechanisms during the ac/dc magnetisation studies. Recent breakthroughs in the development of the Dy-SIMs possessing high blocking barrier and blocking temperature have come through the simultaneous progress of rational ligand design and ab initio analysis of spin dynamics. The relaxation rate of magnetisation in 4f-systems can be written as a sum of three processes [10]: ( ) ΔC F τ −1 = AH n1 T + C T n2 + τ0−1 exp − kB T Direct Raman Orbach

(7.4)

Here, A, C and τ 0 are the parameters that consist of the spin-phonon coupling matrix and the speed of sound. The first term of Eq. (7.4) represents a direct mechanism which is a simple two-level process and depends on the applied magnetic field H. However, for a 4f- or 5f-ion system, the relaxation occurs via two or more steps since there are many microstates or mJ states possible, and thus, the direct mechanism does not apply [10]. Generally, the released energy during magnetisation relaxation is taken up by the lattice phonons. When the system jumps to the respective excited KDs (or non-KDs) by acquiring thermal energy from the lattice, then it is known as the Orbach mechanism (black dotted arrow in Fig. 7.1) [71]. The molecule then can relax back to the ground state, accompanying the excitation of the phonon states (see Fig. 7.2). If the crystal field remains axial and high, the relaxation becomes slower, and changes in populations on these levels become slow. Orbach mechanism mainly occurs at high temperatures and follows the Arrhenius exponential rate law of magnetisation relaxation [72]. If the relaxation rate is defined by τ, then ln has a linear dependence with the inverse of temperature, i.e. 1/T in the Orbach regime. The spins can also tunnel through the opposite magnetisation or mJ (or mS ) states in the ground state, as shown in the red-dotted arrow in Fig. 7.1, via the Quantum Tunnelling process. This happens when the mixing coefficient of the opposite magnetisation becomes high. If the system has a half-integer S or J value, the Ms

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Fig. 7.1 Various pathways of magnetisation relaxation mechanisms through lower-energy Kramers Doublets (KDs). The arrows represent the probability of transition through the KDs

Fig. 7.2 Schematic view of various spin–lattice relaxation mechanism occurring in single-molecule magnets. The blue line represents phonon states, while red line indicates crystal field energy of the KDs. The Debye frequency is the maximum acoustic phonon frequency of a crystal. Optical phonons occur at discrete frequencies and remains in the higher energy region. Reprinted from Liddle et al. [10] with permission from the Royal Society of Chemistry

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or MJ states will have zero mixing coefficient for a free ion state due to the Kramers theorem of degeneracy. However, non-axial crystal field and non-zero nuclear spin (I /= 0) introduce considerable QTM in the M J states. One way to remove or suppress the QTM is to apply a small amount of dc magnetic field. In this regard, crystal field symmetry of the ligand geometry plays an important role. If the system has an integer spin or integer J ground state, then Kramers theorem does not hold, and strong mixing will result, and splitting of the non-Kramers states will take place. In the latter case, the term QTM is not appropriate; instead, tunnel splitting energy should be used in describing the relaxation behaviour. The larger the tunnel spitting or QTM, stronger will be the mixing, and hence, the system will relax faster. It can also happen that the system is excited to a particular microstate and then tunnel through the opposite magnetised state; this is called thermally assisted QTM (TA-QTM). If the released amount of energy is taken up by a superposition of two lattice waves in which one of the excited phonon states is a virtual intermediate state, then it is known as the first-order Raman process (see Fig. 7.2). In the second-order Raman mechanism, both the phonon states as well as the KDs undergo a two-step transition via a virtual intermediate state (olive green dotted arrow in Fig. 7.1) [73]. One system can have multiple relaxation pathways in real examples and might show various mechanisms at different temperatures. In addition to this, there is dipolar (intra- or inter-molecular) spin–spin relaxation (T 2 ) and electron-nuclear hyperfine coupling that could also help the system relax. During ab initio calculations on discrete molecular systems, the energy of respective KDs (or non-KDs) with respect to their effective magnetic moments and the various relaxation mechanisms are approximately derived from the mixing of the M J levels with the formula a|μ|b where a and b are the two initial and final M J wavefunctions within which the transition is occurring (see Fig. 7.1) [74]. Explicit consideration of the lattice or phonon states requires the inclusion of unit cell vibrations and density of state calculations of phonons. These spin relaxation mechanisms or dynamics also play a very important role in determining the effective (or experimental) energy barrier and relaxation rate of single-molecule magnets. It is noteworthy to mention that the ab initio calculated energy barrier (Ucal ) is quite often overestimated compared to the experimental one (Ueff ). While the inclusion of dynamic correlation and expansion of active space can reduce the difference, sometimes the discrepancy seems to be inherent. The difference between U cal and U eff can be ascribed to various factors: (i) although the ab initio calculations estimate the value QTM/TA-QTM and transition matrix element for Orbach, it does not give the probability of occurrence for any process, (ii) while the intermolecular interaction is present in an X-ray crystal structure, the same is not captured in the calculations often, (iii) the hyperfine interaction between the metal and ligand sometime facilitates the under-barrier relaxation and (iv) the presence of under-barrier relaxation process such as Raman process.

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7.4 Accessing the Electronic Structure and Spin-Hamiltonian Parameters of Reported Dy(III) Single-Ion Magnets In this section, we have classified the reported Dy(III) SIMs into seven categories depending on the point group symmetry. We will begin our discussion with some high symmetric models and ab initio prediction of the maximum barrier height that can be achieved. This includes (i) Dy(III) models with C ∞v point group and (ii) Dy(III) models with D∞h point group. This will be followed up by other SIMs where ab initio calculations were performed to understand the relaxation mechanism. This includes (iii) Dy(III) SIMs with D4h point group, (iv) Dy(III) SIMs with D5h point group, (v) Dy(III) SIMs with D6h point group, (vi) Dy(III) SIMs with D4d point group and (vii) Dy(III) SIMs with D5d point group. Each section deals with the ab initio study of molecules reported with associated symmetry.

7.4.1 Dy(III) SIM with C∞v Point Group The perfect axial symmetry for SIMs can be attained using a simple diatomic model [DyO]+ (model 1, Fig. 7.3), where the metal centre possesses a C ∞v point group. A decade ago, in 2011, Ungur et al. thought about this hypothetical model and performed ab initio calculations to unveil its magnetic anisotropy [13]. The calculations reveal an increasing order of energy with the mJ = |±15/2⟩ being the ground state (KD1). This is followed by |±13/2⟩, |±11/2⟩, |±9/2⟩, |±7/2⟩, |±5/2⟩, |±1/2⟩ and |±3/2⟩ for KD2, KD3, KD4, KD5, KD6, KD7 and KD8, respectively (Fig. 7.3), lying higher in energy. The computed gxx and gyy values are found to be negligible for KD1-KD7, and therefore, QTM/TA-QTM were found to be insignificant for magnetisation relaxation. On the other hand, the large transverse anisotropy (gxx ~ gyy ~ 10) in KD8 yields a significant TA-QTM value for magnetisation relaxation. Therefore, model 1 has to follow a long relaxation pathway from KD1 to KD8 to reverse its magnetisation, which places the U cal value in the order of 2132 cm−1 (Fig. 7.3). From this section, we will denote the experimental blocking barrier for magnetisation reversal as U eff and ab initio calculated blocking barrier as U cal . Although this model is very attractive, chemically, it is unrealistic as monocoordinate Ln(III) complexes are non-existent. This is due to relatively large atomic radii and larger cationic charge demands ligands with stronger electron donor capability. Having said this, the [DyO]+ fragments can be stabilised in a cage environment, such as in endohedral fullerene [56] or in Metal Organic-Frameworks (MOFs) [75, 76].

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Fig. 7.3 The relaxation pathway of magnetisation reversal of [DyO]+ which is indicated by the arrows. The horizontal black line denotes the energy of the KD as a function of magnetic moment. Reprinted from Ungur et al. [13] with permission from the Royal Society of Chemistry

7.4.2 Dy(III) SIM with D∞h Point Group To probe the role of geometry and coordination number on magnetic anisotropy, our group has explored various models of [Dy(OH)n ]3−n (n = 1→12) in 2014 [18]. This was followed by Chibotaru and co-workers on [DyFn ]3−n (n = 1→12) models in 2016 [14]. The calculations suggest the linear [F–Dy–F]+ or [HO–Dy–OH]+ model has the great capability of yielding the largest crystal field splitting of eight ground KDs. To validate these models, Chilton, along with others, has studied a hypothetical complex [(i Pr3 Si)2 N–Dy–N(Sii Pr3 )2 ]+ (2) which has been modelled from a reported Sm(II) analogue [(i Pr3 Si)2 N–Sm–N(Sii Pr3 )2 ] (see Fig. 7.4a) [16]. The metal centre in model 2 is found to possess a pseudo D∞h symmetry (N–Dy–N angle of 175.5°). The ab initio SA-CASSCF /RASSI-SO/SINGLE_ANISO calculations on 2 reveal a large span in energy (1861 cm−1 ) of eight ground KDs generated from 6 H15/2 . This is due to the strong ligand field from –N(Sii Pr3 )2 groups with a short Dy-N bond length of 2.48 Å. The computed axial B20 crystal field parameter is found to be two orders of magnitude larger than the non-axial crystal field parameter, which suggests a strong axiality in this complex. The six KDs of 2 are found to possess a significant contribution from mJ = |±15/2⟩, |±13/2⟩, |±11/2⟩, |±9/2⟩, |±7/ 2⟩ and |±5/2⟩ state, respectively. The smaller gxx and gyy values of KD1-KD6 lead to insignificant QTM/TA-QTM value for these states. On the other hand, the matrix elements for Orbach process are also found to be minimal. Furthermore, axes of all these KDs are strongly aligned to each other along N-Dy-N axis, which hampers the magnetisation relaxation. The strong mixing of the mJ levels KD7-KD8 (mJ = 0.64|±3/2⟩ + 0.26|±1/2⟩ in KD7, mJ = 0.68|±1/2⟩ + 0.31|±3/2⟩ in KD8) favours the magnetisation relaxation via this state. The computed U cal value (ca. 1800 cm−1 ) is found to be larger than the best SIMs reported till date. Since the T B value of the 1 th of the U eff value, one can safely assume a T B value of reported SIMs are ~ 20 >90 K for this complex if synthesised.

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Fig. 7.4 a The molecular structure of [Sm{C(SiMe3 )3 }2 ]. b The blocking barrier with the bending angle (θ), averaged all over ϕ values. Inset is the structure of the model complex. Error bars are the standard deviation from the mean of the torsion angles φ. c The mechanism of magnetic relaxation of [Dy{N(SiH3 )2 }2 ]+ with the variation of θ from 90° to 180° (the ϕ value is fixed at 90°). The transparency of each arrow is proportional to the normalised transition probability. Reprinted with permission from Chilton et al. [15] Copyright@2015 American Chemical Society

To further explore the effect of N-Dy-N bending angle (θ) and N–Dy–N–C torsional angle (ϕ) into the magnetic anisotropy, a magneto-structural correlation has been performed with θ (90° ≤ θ ≤ 180°) and ϕ (0° ≤ ϕ ≤ 90°) in a simplified model [Dy(L1 )2 ]+ (L1 =N(SiH3 )2 ) where Dy-N bond length fixed at 2.5 Å) as the bulky substituents do not alter the magnetic anisotropy (see Fig. 7.4b, c) [15, 16]. The ab initio calculations unveil a linear drop in the U cal value with the decrease in θ (Fig. 7.4b, c). The variation of ϕ with each θ value does not lead to a significant change in the U cal values (Fig. 7.4b, c). Nevertheless, strong axiality is maintained all over the bending angle, which implies that just a two-coordinate complex rather than θ = 180° is required to attain a strong magnetic anisotropy. To confirm whether this is valid for other ligands, the magneto-structural correlation has been performed with two other model systems [Dy(L2 )2 ]+ (L2 =C(SiH3 )3 ) and [Dy(L3 )2 ]+ (L3 =CH(SiH3 )2 ). Both of them show a similar trend with θ value as found with L1 (Fig. 7.4b). This strengthens the hypothesis that bending angle is not a major factor in a two-coordinate system. To further verify this hypothesis, ab initio calculations were performed with another model system [Dy{C(SiMe3 )3 }2 ]+ with the θ value of 137.0° (modelled from reported [Yb{C(SiMe3 )3 }2 ]+ ) and 143.4° (modelled from reported [Sm{C(SiMe3 )3 }2 ]) [77, 78]. The calculations reveal magnetisation relaxation via fifth or sixth excited KD with a U cal value of 1247 and 1484 cm−1 for the θ = 137.0

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and 143.4°, respectively, despite strong Dy•••H agostic interaction in the equatorial plane. The lowering of U cal compared to 2 suggests that the nitrogen donor ligands are more efficient for SIMs compared to carbon donor ligands. While many groups pursue a vigorous search for truly two-coordinate Dy(III) SIMs, this effort is so far unsuccessful as such compounds are very unstable and undergo decomposition or transformation very easily.

7.4.3 Six Coordinate Dy(III) SIM with Octahedral Geometry 7.4.3.1

Magnetic Anisotropy in {DyC2 N4 } Core

According to the ab initio study on various model systems, two-coordinate geometry with a strong ligand field (D∞h ) should be ideal for Dy(III) SIM since it lacks equatorial ligation. But there are synthetic challenges in preparing the two-coordinate SIMs. However, to mimic this two coordination environment, Liddle and co-workers have prepared a novel SIM, [Dy(BIPMTMS )2 ][K(18C6)(THF)2 ] (3, BIPMTMS = {C(PPh2 NSiMe3 )2 }2− , (Fig. 7.5a) with FC/ZFC (field cooled/zero-field cooled) T B value of 10 K (Table 7.1) [17]. The metal centre in 3 has a pseudo D4h symmetry with four equatorial nitrogen donors and two axial C donor centres (Table 7.1). The longer bond length of Dy-N compared to Dy = C bonds lead to the substantial charge accretion creating a pseudo linear C=Dy=C-like environment (∠C=Dy=C = 176.6°). The ab initio calculations on 3 reveal mJ = |±15/2⟩ as the ground state, as expected from the strong axial ligand field generated by methanediide centres (Table 7.1). The ground state gzz axis is oriented along the C=Dy=C axis to minimise the electrostatic repulsion from ground oblate electron density. This leads to a negligible QTM effect which is also reflected in the gxx, gyy (gxx ~ gyy ~ 0.0) and gzz (gzz ~ 20) values (Table 7.1). It is noteworthy to mention that for a system with a strong axiality and negligible equatorial interaction; one can expect gxx = gyy = 0 and gzz = 20 for the ground state. The first and second excited states of this molecule are found to be |±13/ 2⟩ and |±11/2⟩ mJ states, respectively. The smaller gxx and gyy values (KD2: gzz = 17.19, KD3: gzz = 14.27) in these states yield negligible TA-QTM for magnetisation relaxation. Furthermore, the gzz axes are also found to be aligned along with the C = Dy = C bond, which hinders the magnetisation relaxation via these states. However, the strong mixing of mJ occurs in the third excited KD, and this promotes a significant TA-QTM for magnetisation relaxation. The gzz axis of this KD4 is titled by ~90˚ with respect to the ground state gzz axis favouring relaxation via this state. The computed U cal1 value of 516 cm−1 , considering the relaxation via the third excited state, shows good agreement with the U eff1 of 493 cm−1 (Table 7.1). As complex 3 follows two thermally activated relaxation processes, another blocking barrier (U eff2 = 548 cm−1 ) can be rationalised considering the relaxation via the fourth excited KD (U cal2 = 563 cm−1 ). Further, the computed magnetic susceptibility shows a good agreement with the experiments.

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Fig. 7.5 Molecular structure of a 3, b 3a. Reprinted from Gregson et al. [17] with permission from the Royal Society of Chemistry

However, the precursor of complex 3, [Dy(BIPMTMS )(BIPMTMS H)] (3a, Fig. 7.5b) also displays the slow relaxation of magnetisation in zero field [17]. The ab initio calculations on 3a reveal a lowering in the energy of eight KDs (from 6 H15/2 ) compared to 3. This can be ascribed to the reduction in axiality from C=Dy–CH (one mono and one dianionic centre instead of two dianionic centres as in 3) along with the lowering of ∠C = Dy–CH to 158.3°. The ground KD of 3a is found to contain mJ = |±15/2⟩ state, while the first excited state possesses a dominant (81%) contribution from mJ = |±13/2⟩ state (Table 7.1). The small transverse anisotropy on the ground and first excited KD yield QTM and TA-QTM, respectively. Furthermore, the gzz axis of the first excited KD is found to be nearly aligned (deviation is 15°) with the ground state, which also supports the magnetisation relaxation via higher excited states. The substantial mixing of mJ levels in the second excited KD favours the magnetisation relaxation. This is further corroborated by the tilting of the gzz axis (~88°) with the ground state. This yields an U cal value of 170 cm−1 , which is in close agreement with the experimental one (177 cm−1 , Table 7.1).

7.4.3.2

Magnetic Anisotropy in {DyO3 X3 } (X=–Cl, –Br, –I –O or –N) Core

Most of the reported lanthanide SIMs lack air stability. To circumvent this issue, Chilton and co-workers have studied an air-stable octahedral Dy(III) SIM, [Dy(DiMeQ)2 Cl3 (H2 O)] (4, DiMeQ = 5,7-dimethyl-8-oxoquinolinium, Fig. 7.6a) [79]. The metal centre in 4 is six coordinated from two trans DiMeQ ligands, three meridional-Cl ions and one water molecule. Therefore, the Dy(III) ion resides in an {O3 Cl3 } core with one H2 O and Cl− ion trans to each other, which gives rise to the strong hydrogen-bonded network with other molecules in the crystal. The continuous shape measurement (CShM) analysis reveals near octahedral geometry in 4b (Table 7.1). The shorter Dy–O (DiMeQ, 2.15 Å) bond length compared to Dy–O (H2 O, 2.31 Å) and Dy-Cl (2.68–2.90 Å) creates significant magnetic axiality in complex 4. It is confirmed by the electrostatic potential calculations, where the metal centre is found to possess an oblate electron density (Fig. 7.6). Here, it should also be noted that the electrostatic factors do not have a significant influence on the

0.642

Oh Oh Oh Oh Oh D4h D4h D4h

[Dy(OAr*)Cl2 (THF)3 ] (6)

[DyCl3 (OPPh3 )2 (THF)]·THF (7)

[DyBr3 (OPPh3 )2 (THF)]·THF (8)

[Dy(Cy3 PO)3 I3 ]·2THF (9)

[Dy(Cy3 PO)2 I3 CH3 CN] (10)

[Dy(OCPh3 )2 (THF)4 ][BPh4 ] (11)

[Dy{OB(NArCH)2 }2 (THF)4 ] [BPh4 ] (12)

[Dy(Ot Bu)2 (4-phenylpyridine)4 ]+ (13)

[Dy(Ot Bu)2 (1–4-piperidin-1-ylpyridine)4 ]+ (14) D4h

[Dy(Ot Bu)2 (4-pyrrolidin-1-ylpyridine)4 ]+ (15)

*

0.756

Oh

[Dy(ImDipp NH)Cl2 (THF)3 ] (5)

–

0.337

2.567

2.845

1.400

0.850

0.967

0.960

1.725

5

4

5

–

7

–

–

–

–

–

–

< 1.8

10

8

8

5

7

1.8

9

4

–

–

–

–

–

–

–

0.02

0.02

0.02

0.02

0.019

0.018

0.018

–

–

–

–

0.015

–

–

1258

1311

1442

1088

1385

738

48

49

34

37

558

771

177

548 493

TH corresponds to the temperature below which the opening of magnetic hysteresis is observed

D4h

0.693

Oh

[Dy(DiMeQ )2 Cl3 (H2 O)] (4)

–

Oh

[Dy(BIPMTMS )(BIPMTMS H)] (3a)

–

D4h

1218

1300

1404

1068

1420

738

283

235

147

291

470–528

761

170

563 516

2nd 2nd 2nd 2nd

−3.19 19.81 −1.20 19.76 −2.22 19.92 −3.44 19.86

4th 4th 4th

−9.50 19.98 −8.61 19.88 −7.91 19.89

4th 4th

19.87 19.86

–

–

5th

4th

−3.67 19.86

19.88

4th

1.08

2nd

4th

19.79

19.88

[85]

[85]

[85]

[43]

[84]

[83]

[83]

[82]

[82]

[81]

[81]

[79]

[17]

[17]

gzz (KD1) Relaxation Refs via KD

−5.00 19.86

–

–

Point CShM FC/ T H (K)* Sweep U eff U cal (cm−1 ) B20 group ZFC rate (cm−1 ) T B (K) (Tesla/ min)

[Dy(BIPMTMS )2 ] [K(18C6)(THF)2 ] (3)

Complex

Table 7.1 The summary of the computed and experimental figure of merit of illustrated octahedral SIMs with pseudo point group symmetry

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determination of the main magnetic axis in low symmetry complexes, whereas these work well for high symmetry complexes [80]. Furthermore, the larger values of the axial B20 , B40 , B60 CF parameter compared to non-axial CF parameter also suggests strong axiality in this complex (Table 7.1). The ab initio calculations on 4 reveal mJ = |±15/2⟩ ground state as expected (Table 7.1). The gzz axis is oriented along the highest order C4 axis (along with the (C)OvDy–O(C) bond, see Fig. 7.6). The first (mJ = |±13/2⟩, 388 cm−1 ) and second exited KD (mJ = |±11/2⟩, 656 cm−1 ) possess strongly axial principal g values, suggesting negligible TA-QTM. Furthermore, the gzz axis of these KDs shows minimal deviation from the ground state. The rhombic g values (gxx = 10.57, gyy = 6.45) in the third excited KD (mJ = 0.41|±1/2⟩ + 0.25|±9/ 2⟩ + 0.11|±11/2⟩ + 0.10|±5/2⟩) unveil the magnetisation relaxation via this state. The computed U cal value (761 cm−1 ) shows excellent agreement with the U eff of 771 cm−1 (Table 7.1). But the computed temperature-dependent magnetic susceptibility (χT) is overestimated compared to the experiments. This discrepancy can be attributed to the strong hydrogen bonding network with other molecules, which has not been included in the calculations. However, complex 4 could not retain its magnetisation even at 2 K, although its doped sample with Y reveals retention of magnetisation. The effect of hyperfine interactions due to H nuclei in magnetic properties has been ruled out as the deuterated sample did not alter the magnetic anisotropy. The blocking barrier is considered as a figure of merit of SIM, as most often, the T B value is proportionate with this parameter. The enhancement of U eff requires a strong axial ligand. From this viewpoint Zheng and co-workers reported an imido ligand (ImDipp NH, 1,3-bis(2,6-diisopropylphenyl)imidazolin-2-imine) which forms

Fig. 7.6 a Electrostatic potential at the Dycentre. Scale with blue at −0.07 to red +0.02 V. Water molecules are described with a −2/3 charge for O and +1/3 charge for H. b The main magnetic axis of 4. Colour code: Dy-violet, Cl—green, O—red, N—green, C—grey, H—white. Reprinted from Giansiracusa et al. [79] with permission from the Royal Society of Chemistry

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a short Dy–N bond in [Dy(ImDipp NH) Cl2 (THF)3 ] (5, Fig. 7.7) [81]. The formation of a short Dy–N bond (2.12 Å) is ascribed to the σ + π donor-like cyclopentadienyl (Cp) ligand (Fig. 7.7). Furthermore, DFT (density functional theory) studies indicated a Dy–N bond order of 2.23, also suggesting a strong Dy–N bond. The metal centre in 5 possesses a pseudo-octahedral geometry with two THF and two chloride molecules in the equatorial and one ImDipp NH and one THF in the axial position (Fig. 7.7a). The ab initio calculations reveal the ground state gzz axis points towards the Dy– ImDipp NH bond due to the strong ligand field from the imido ligand (Fig. 7.7a). The ground state has an axial set of g tensors (gxx = gyy = 0.0, gzz = 19.86) as expected for a mJ = |±15/2⟩ KD (Table 7.1). While the first excited KD (mJ = |±13/2⟩) also shows an axial g tensor, the low symmetry components of the structure cause a significant transverse anisotropy in the second (gxx = 2.2, gyy = 5.8, gzz = 10.57) and third excited (gxx = 8.6, gyy = 7.2, gzz = 0.50) KDs. It is further corroborated by the strong mixing of |±mJ ⟩ states along with the significant deviation of the gzz axis with respect to the ground state. This facilitates the magnetisation relaxation via these states and yields an U cal value in the range of 470–528 cm−1 , which is lower than the experimental estimate of 558 cm−1 (Table 7.1). The underestimation of the blocking barrier suggests that there is a significant TA-QTM in the second and third excited KD; the relaxation via higher excited states is viable. However, the computed χT data shows a good agreement with the experiment. To compare the axial ligand strength of the imido ligand in 5 with an alkoxide ligand, Zheng and co-workers have studied another analogue octahedron complex [Dy(OAr*)Cl2 (THF)3 ] (6, Fig. 7.7b) [81]. The gzz axis is again found to be along the Dy-O(OAr*) bond due to the strong ligand field from the alkoxide ligand (Fig. 7.7b). The ab initio calculations reveal stabilisation of mJ = |±15/2⟩ in the ground state along with axial g tensor (Table 7.1). But the strong transverse anisotropy (gxx = 0.9, gyy = 1.8, gzz = 15.85) in the first excited KD, along with significant mixing of mJ states (0.86|±13/2⟩), promotes significant TA-QTM for magnetisation relaxation. It leads to a U cal value of 291 cm−1 , which does not match the experimental U eff values (37 cm−1 , Table 7.1). This discrepancy has been attributed to the relaxation mechanism via the barrierless Raman process along with QTM. The lowering of both

Fig. 7.7 The main magnetic axis of a 5, b 6. Colour code: Dy—gold, Cl—green, O—red, N—blue, C—grey, H—white. c, d The isosurface of metal–ligand π bonding HOMO orbitals in 5. Reprinted from Liu et al. [81] with permission from the Royal Society of Chemistry (Here, the Z-axis is chosen as the highest order symmetry axis, while the choice of the X and Y axes is arbitrary)

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U cal /Ueff in 6 compared to 5 can also be rationalised with the computed CF parameter. The axial CF parameter is larger than the non-axial crystal field parameter in 5, while they are comparable in 6. Further, DFT studies reveal that Dy-N has a bond order of 2.23 (one σ + one π, Fig. 7.7c, d) while the Dy-O in 6 possesses a bond order of 1.03 (one σ). It suggests a stronger axial crystal field in 5 compared to 6, which explains the trend in the blocking barrier. The chloride ions in 5 create a strong equatorial ligand which restricts it from substantial magnetic anisotropy despite having a gigantic axial ligand field from the imido ligand. In this regard, Zheng and co-workers performed ab initio calculations on the model [Dy(THF)3 (ImDipp N)]2+ (5a), where the chloride ions in the equatorial plane were removed [81]. The calculation on 5a reveals ~ 1.5 times increment in the energy splitting of eight KDs compared to 5. This is corroborated by the increase in axial B20 CF parameter (B20 = −3.67 and −5.95 in 5 and 5a, respectively). Furthermore, the QTM/TA-QTM is reduced, which leads to the relaxation via third excited KD. This yield an U cal value of 858 cm−1 , which is more than 300 cm−1 larger than 5. On the other hand, to fully quench the equatorial ligand field, further ab initio calculations were performed on the model [Dy(ImDipp N)2 ]+ (5b). Quite interestingly, the KD1-KD5 on 5b is found to contain almost pure mJ = |±15/2⟩, |±13/2⟩, |±11/ 2⟩, |±9/2⟩, |±7/2⟩ state, respectively, which quenches the QTM/TA-QTM. In this model, the relaxation occurs via the fifth excited KD due to the strong mixing of mJ states (0.78|±5/2⟩ + 0.15|±1/2⟩). This state gzz axis is titled by ~ 90˚ with respect to the ground state gzz axis. The computed U cal value in this model is 2733 cm−1 , which is more than two times larger than the “Dysprocenium” complexes due to multiple bonds between Dy and the N of the imido ligand. This paves the way for the chemical design of high-performance SIM. In 2020, Dunbar and co-workers synthesised and characterised two Dy mononuclear complexes with octahedral (Oh ) geometry having the formula of [DyX3 (OPPh3 )2 (THF)]·THF (X = Cl (7) and Br(8)) (Ph3 PO = triphenylphosphine oxide, THF = tetrahydrofuran) [82]. Both the complexes are isostructural, and the Dy ion is six-coordinate with three chlorides or bromides, one THF in the equatorial position, and two OPPh3 ligands in the axial position. The molecular structure of 8, along with ab initio computed magnetic anisotropy, is shown in Fig. 7.8a. The ac susceptibility measurement suggests complexes 7 and 8 as SIMs with energy barriers of 34.1 cm−1 and 49.3 cm−1 , respectively, under a small dc field. The ab initio calculations reveal that the ground state KD of Dy ions in both the complexes has small transverse components (gxx , gyy ), and the gzz value is ~ 20 expected for pure Ising |mJ = ± 15/2⟩ KD; thus, the observation of small QTM values. The enhanced transverse components at first excited KD lead to large TA-QTM processes (0.8 and 0.2μB for 7 and 8, respectively). This situation allows for magnetic relaxation via the first excited states with an energy barrier of 147.3 cm−1 for 7 and 234.9 cm−1 for 8 (Fig. 7.8a, b). All the examples stated above have a significant equatorial ligand field, which decreases the magnetic axiality of the complexes. To reduce the equatorial ligand field, Gao and co-workers have employed iodide ligand with a long Dy-I bond length of ~ 3.0 Å in two six coordinated Dy(III) SIMs, [Dy(Cy3 PO)3 I3 ]·2THF (9, Cy3 PO =

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Fig. 7.8 a The molecular structure of complex 8 along with the orientation of main magnetic (gzz ) axis. The solvent and H atoms are omitted for clarity. Colour code: Dy-purple, P-pink, O-red, Cblack. b Magnetisation blocking barrier for the Dy site in 7 (c) 8. The thick black line indicates the KDs as a function of computed magnetic moment. The green/blue arrows show the possible pathway through Orbach/Raman relaxation. The dotted red lines represent the presence of QTM/ TA-QTM between the connecting pairs. The numbers provided at each arrow are the mean absolute values for the corresponding matrix element of the transition magnetic moment. Reprinted from Vignesh et al. [82] with permission from the Royal Society of Chemistry

tricyclohexylphosphine oxide, Fig. 7.9) and [Dy(Cy3 PO)2 I3 CH3 CN] (10, Fig. 7.9) [83]. The large size of iodine helps delocalize the negative charge over a larger domain space and reduces the equatorial ligand field. The CShM analysis with {DyO3 I3 } and {DyO2 NI3 } cores for 9 and 10 reveals octahedral geometry around metal ions, where more deviation from the corresponding geometry is found in the former (Table 7.1). In complex 9, two Cy3 PO groups lie in the axial position, while three iodides and one Cy3 PO group lie in the equatorial position. In complex 10, one Cy3 PO group in the equatorial plane is replaced by CH3 CN. The FC/ZFC magnetic data for 9 and 10 shows clear divergence at 4 K and 7.4 K, respectively. The ab initio calculations on 9 and 10 reveal pure mJ = |±15/2⟩ ground state for both the complexes. This is further corroborated by the axial g factors in the ground state (gxx = 0.002, gyy = 0.003, gzz = 19.854 and gxx = 0.001, gyy = 0.001 and gzz = 19.876 for 9 and 10, respectively, Table 7.1). The ground magnetic gzz axis of both the complexes is found to orient along the axial (Cy3 PO)O-Dy-O(OPCy3 ) axis to minimise the electrostatic repulsion with the oblate electron density of mJ = |±15/2⟩ state. For 10, the first and second excited KDs contains mJ = |±13/2⟩ and |±11/2⟩, respectively, with negligible TA-QTM (TA-QTM = 9.28 × 10–3 and 0.22 μB for KD2 and KD3, respectively). Further, the main magnetic axes of KD1, KD2 and KD3 are almost collinear (maximum deviation is 2.6°) and block the relaxation via these states. On the other hand, the substantial mJ mixing (0.28|±9/2⟩ + 0.65|±1/2⟩) in the fourth excited KD favours the magnetisation relaxation. This leads to the U cal value of 732.8 cm−1 , which is in excellent agreement with the experimental estimate of 738.1 cm−1 (for a diluted sample, Table 7.1). In the case of complex 9, the presence of one Cy3 PO group in the equatorial significantly reduces the axiality compared to 10. | position | The computed axial | B20 | CF parameter is found to be 3.44 (7) and 10.79 (10),

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309

Fig. 7.9 The molecular structure of a 9, b 10. Colour code: Dy—yellow, O—red, N—blue, P— green, C—grey, I—violet. Hydrogens are omitted for clarity. The dashed line denotes the ab initio computed main magnetic axis of ground and excited KDs. Reprinted with permission from Li et al. [83] Copyright@2015 American Chemical Society

which also explains the lowering of magnetic axiality of 9 compared to 10 (Table 7.1). However, the first excited KD of 9 shows moderate mJ mixing (0.92|±13/2⟩), and a significant deviation (~6°) of the gzz axis with the ground state is enough to facilitate the magnetisation relaxation via this state. Considering this, the estimated U cal value of 9 is 292.8 cm−1 , which is not in agreement with the U eff of 48 cm−1 in the experiments (Table 7.1). Even the inclusion of dynamic correlation does not improve the results (XMS-CASPT2 U cal value is 314.3 cm−1 ). Therefore, the discrepancy has been attributed to (Table 7.1) to barrierless Raman and QTM process of magnetisation relaxation, which was further confirmed by the experiments. Hence, this comparative study offers to find the role of equatorial ligand in magnetic anisotropy. Complexes 9 and 10 possess significant equatorial ligand fields from the OPCy3 and -NCCH3 groups, respectively, which limits their blocking barrier below 800 cm−1 . With an aim to further increase the blocking barrier, Trifonov and co-workers have replaced the all equatorial ligands of 9 and 10 with four THF molecules and axial ligands by -OCPh3 groups, resulting in the synthesis of [Dy(OCPh3 )2 (THF)4 ][BPh4 ] (11, Fig. 7.10a) [84]. The CShM analysis with {DyO6 } core reveals an octahedron geometry (D4h symmetry) around the metal centre (Table 7.1). The shorter Dy-O(OCPh3 ) bond length of 2.103 Å (the lowest among all [DyX2 L4 ] complexes) compared to the Dy-O(THF) bond length of 2.370–2.398 Å suggests significant magnetic axiality in the complex. This is confirmed by the magnetic measurement, which reveals a massive blocking barrier of 1385 cm−1 for this complex (Table 7.1). Additionally, a significant FC/ZFC T B value of 7 K was estimated for this complex (Table 7.1). To get a further insight into the magnetisation relaxation, ab initio calculations were performed. The calculations reveal a 1384.5 cm−1 splitting of eight KDs from the 6 H15/2 state of complex 11 due to the

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axially compressed structure. The main magnetic axis of 11 is oriented along the pseudo-C 4 axis due to the strong ligand field from the -OCPh3 group (Fig. 7.10a). The ground state (mJ = |±15/2⟩) of 11 is found to be Ising in nature (gxx , gyy = 0.000, gzz = 19.866, Table 7.1), which is reflected in the negligible QTM value. The first and second excited KD at 557.9 and 930.5 cm−1 are well described by mJ = |±13/2⟩ and |±11/2⟩ states, respectively. The very small transverse anisotropy in these states (KD2: gxx = 0.048, gyy = 0.051, gzz = 16.820; KD3: gxx = 0.350, gyy = 0.746, gzz = 13.213) renders little TA-QTM (1.6 × 10−2 μB (KD2) and 0.19 and μB (KD3)) for magnetisation relaxation. On the other hand, a strong mixing of mJ = |±9/2⟩ (33.7%) with mJ = |±1/2⟩ (46.1%) in the third excited KD promotes significant QTM. Further near perpendicular orientation of the gzz axis with the ground state favours magnetisation relaxation via third excited KD. This leads to the U cal value of 1084.4 cm−1 , which is underestimated compared to the U eff . This can be ascribed to the lack of dynamic correlation in the calculations. It is noteworthy to mention that the dynamic correlation is important in 11 due to the very short axial Dy-O(OCPh3 ) bond length, which induces metal–ligand covalency. To include this dynamic correlation ab initio calculations were performed on a model complex [Dy(OCH3 )2 (OH2 )4 ] (11a ) to reduce the computational cost. Then, the CASPT2 corrected CF on the full structure was obtained by the addition of SA-CASSCF calculated CF on 11 with the dynamic electron correlation obtained as a difference of CF from a CASSCF and CASPT2 calculations for the 11a [84]. Quite interestingly, the CASPT2 calculation reveals an enhancement in the energy splitting compared to CASSCF. The ground, first and second excited KD is found to contain almost pure mJ = |±15/2⟩, |±13/2⟩ and |±11/ 2⟩ state, respectively. The magnetisation relaxation via the first (725.6 cm−1 ) and second excited state (at 1202.0 cm−1 ) is blocked due to their large energy separation, which is inaccessible to the available phonons. However, the magnetisation relaxation occurs via the third excited KD at 1419.7 cm−1 due to the large TA-QTM (2.9 μB ) in this state. This leads to the U cal value of 1419.7 cm−1 , which agrees well with the U eff value. This massive energy barrier is obtained due to the lack of available high-energy phonons despite the low axiality of excited KDs. This explains the low blocking temperature of this complex, although it possesses a huge blocking barrier. The QTM/TA-QTM values are larger in 11 compared to best lanthanide SIMs due to the significant transverse anisotropy originating from the THF ligands. Therefore, the removal of THF ligands with nitrogen donor ligands such as pyridine can further improve the blocking barrier and blocking temperature. To unravel the effect of linearity and role of coordinated solvents in magnetic anisotropy, theoretical ab initio calculations were performed by Chilton on a series of [Dy(L)2 (THF)4 ]+ (L = monoanionic ligand such as N(SiH3 )2 , CH(SiH3)2 and C(SiH3)3 ) model complexes. The computed U cal value is found to be in the range of ca. 695 cm−1 (for L = N(SiH3 )2 , CH(SiH3)2 ) and ca. 1112 cm−1 (for L = C(SiH3 )3 ) [15]. Inspired by the results, Liddle and co-workers have investigated a pseudo-D4h symmetric Dy(III) SIM, [Dy{OB(NArCH)2 }2 (THF)4 ][BPh4 ] (12, Fig. 7.10c) with an isoelectronic boryloxide analogue (OB(NArCH)2 ) of imido ligand (NC(NArCH)2 ) of 5 to compare the donor strength ability of axial ligand [43]. Complex 12 possesses pseudo-octahedral geometry with two axial boryloxide and four equatorial THF

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where …

311

Fig. 7.10 a Molecular structure of 11. Colour code: Dy—orange, O—red, C—grey. Hydrogens are omitted for clarity b The mechanism of magnetic relaxation of 11. Reprinted from Long et al. [84] with permission from the Royal Society of Chemistry. c The X-ray structure of 12. Reprinted from Thomas-Hargreaves et al. [43] with permission from the Royal Society of Chemistry

ligands (Table 7.1). The (B)O-Dy-O(B) angle is found to be 175.9° in 12. The shorter (more than 0.3 Å) axial Dy-O(B) bond length compared to equatorial Dy-O(THF) bond length hints at the strong magnetic axiality of this complex. Complex 12 shows retention of magnetisation up to 7 K (Table 7.1). The ab initio calculations on 12 reveals a mJ = |±15/2⟩ (97%) ground state with negligible transverse anisotropy (gxx = 0.00, gyy = 0.00, gzz = 19.86, Table 7.1). The first excited KD resides at 530 cm−1 , contains mJ = |±13/2⟩ (98%), and yields also a very small transverse anisotropy (gxx = 0.05, gyy = 0.06, gzz = 16.83). The transverse anisotropy (gxx = 0.63, gyy = 1.08, gzz = 13.15) started to increase in the second excited KD (mJ = 0.87|±11/2⟩) at 910 cm−1 . However, the gzz axis of the ground, first and second excited KDs are almost collinear to each other, which does not favour the magnetisation relaxation via these states. On the other hand, a significant mJ mixing (mJ = 0.54|±1/2⟩ + 0.29|±9/2⟩) occurs at third excited KD at 1068 cm−1 and leads to the significant transverse anisotropy (gxx = 3.53, gyy = 6.79, gzz = 11.18) that causes relaxation of magnetisation. Further, its gzz axis is nearly orthogonal (84.1°) to the ground state gzz axis, substantiating the magnetic relaxation via the third excited KD. Considering this, the estimated U cal value (1068 cm−1 ) shows excellent agreement with the U eff of 1088 cm−1 (Table 7.1). The huge U eff value of 12 also validates the ab initio calculated U cal values based on [Dy(L)2 (THF)4 ]+ series.

7.4.3.3

Magnetic Anisotropy in {DyO2 N4 } Core

Although the above discussed few pseudo D4h Dy(III) SIMs shows great potential for a high blocking barrier, the U eff values are found to be less than 1100 cm−1 in all the examples studied. To increase the U eff values further, Zheng and co-workers have studied three D4h symmetric Dy(III) SIMs, [Dy(Ot Bu)2 (L)4 ]+ (L = 4-phenylpyridine

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(13), 1–4-piperidin-1-ylpyridine (14), 4-pyrrolidin-1-ylpyridine (15)) with two tertbutoxide groups in the axial position and four bulky substituent pyridine derivatives in the equatorial position [85]. The CShM analysis with {DyO2 N4 } core reveals octahedral geometry in all the complexes. The deviation from the corresponding geometry is found to be in the order of 14 > 13 > 15 (Table 7.1). Ac magnetic data unveils the third height blocking barrier to date in all the three complexes (Ueff = 1442 (13), 1311 (14) and 1258 (15) cm−1 , Table 7.1). The large barrier height can be ascribed to the shorter axial Dy-O bond length (ca. 2.1 Å) and longer equatorial Dy-N bond length (ca. 2.5 Å) along with ca. 180° O-Dy-O angle. This is further corroborated with more than ca. 2.5 times larger Loprop charge of axial donor atoms compared to the equatorial donor atoms. Furthermore, the T B (FC/ZFC)values are reported to be 5, 4 and 5 K for 13, 14 and 15, respectively (Table 7.1). To get a clear understanding of the magnetisation dynamics, ab initio calculations were carried out on the three complexes. The calculation reveals an energy span of ca. 1668 cm−1 of eight ground KDs in the three complexes. The ground state contains mJ = |±15/2⟩ with highly axial g factors (gzz = 19.98 (13), 19.88 (14) and 19.89 (15), Fig. 7.11a). The ground state gzz axis is oriented along the pseudo-C 4 axis (along the O-Dy-O axis). The first excited KD (mJ = |±13/2⟩) at 644 (13)/605 (14)/576 (15) cm−1 and second excited KD (mJ = |±11/2) at 1113 (13)/1041 (14)/986 (15) cm−1 are found to have small transverse anisotropy which quenches the TA-QTM (Fig. 7.11a). On the other hand, significant mJ mixing on (13: 0.83|±9/2⟩ + 0.10|±1/2⟩, 14: 0.76|±9/2⟩ + 0.12|±1/2⟩, 15: 0.77|±9/2⟩ + 0.20|±1/2⟩) the third excited KD creates significant transverse anisotropy (13: gxx = 2.25, gyy = 2.37, gzz = 11.02; 14: gxx = 3.32, gyy = 3.49, gzz = 10.08; 15: gxx = 4.43, gyy = 4.71, gzz = 9.12, Table 7.1 and Fig. 7.11a). This yields substantial TA-QTM values (3.7, 4.7 and 3.6 μB for 13, 14 and 15, respectively) for magnetisation relaxation. The computed U cal values are 1404, 1300 and 1218 cm−1 for 13, 14 and 15, respectively, which agree with the experimental ones (Table 7.1). The computed B20 axial CF parameter is −9.50 (13), −8.61 (14) and −7.91 (15), reflecting same trend as blocking barrier (Table 7.1). The larger axiality of 13–15 compared to 12 can be rationalised with the computed Loprop charges. The computed Loprop charges of Dy, O and N are 2.47−2.50, −1.09−1.12, −0.39 to −0.40 in 13–15, while they are 2.57, −1.02 and −0.56 for Dy, O(B) and O(THF), respectively in 12. These data suggest that alkoxide ligands are stronger donors than boryloxide, and pyridine is weaker than THF. While the bond lengths guide the SIM behaviour, the ligand donor strength is also a key aspect in dictating the magnetic anisotropy. To further gain the insight into axial Dy-O bond distance (R) into magnetic anisotropy, a magneto-structural correlation has been performed by symmetrically varying the R from 1.9 to 3.0 Å in a simplified model system [Dy(OMe)2 (NH3 )4 ]+ to reduce the computational cost (Fig. 7.11). The correlation reveals mJ = |±15/2⟩ ground state with highly axial g tensor (gxx ~ 0.0, gyy ~ 0.0, gzz ~ 20.0) with the Rvalue of 1.9 to 2.6 Å. In this regime, the energy of KD2-KD8 increases linearly with the decrease in Dy-O bond length. At R = 2.7 Å, the ground KD becomes isotropic (gxx ~ gyy ~ gzz ~ 6), and the CF splitting of the eight ground KDs is found to be the lowest. Further increase in the R-value creates an easy plane magnetic anisotropy

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where …

313

Fig. 7.11 a The mechanism of magnetic relaxation of 13. b The energy splitting of the eight ground KDs as function of Dy-O bond length. The figure in the right side is the structure of model complex [Dy(OMe)2 (NH3 )4 ]+ . The pink-shaded region denotes the KDs with negligible transverse anisotropy. Reprinted from Ding et al. [85] with permission from John Wiley and Sons

with mJ = |±1/2⟩ as the ground state. Therefore, one can expect a zero blocking barrier with an R-value greater than 2.6 Å. Overall, the correlation suggests that the D4h geometry is also an attractive point group to attain a large blocking barrier.

7.4.4 Seven Coordinate Dy(III) SIM with Pentagonal Bipyramidal Geometry We have so far discussed how to design the strong axial ligand field to obtain a large barrier height for magnetisation reversal, but how the variation of point group symmetry alters the barrier height for the same molecule is not established. In this regard, Tong and co-workers have also investigated a [Zn-Dy-Zn] complex ([Zn2 DyL2 (MeOH)]NO3 .3MeOH.H2 O (16), L = 2,2' ,2'' -(((nitrilotris(ethane2,1-diyl))tris(azanediyl))tris(methylene))tris-(4-bromophenol)), which changes its symmetry from quasi-D5h to quasi-Oh (pentagonal bipyramidal to octahedron) by single-crystal to single-crystal transformation (Fig. 7.12a) [86]. The metal centre in D5h symmetry is bound to seven oxygen atoms, among which four equatorial oxygens are bridged to Zn(II), and the remaining equatorial oxygen comes from the methanol. The phenoxyl oxygens occupy the axial position. Upon methanol removal, the bridging oxygens get closer to the Dy(III), resulting in a quasi-Oh geometry (16a, Fig. 7.12b). The CShM analysis reveals a significant deviation from idealised symmetry for both the geometries (the deviation is 0.610 and 1.879 for D5h and Oh symmetry, respectively, Table 7.2). It changes the blocking barrier from 305 cm−1 (D5h ) to a negligible value (Oh ). The ab initio calculations reveal negligible transverse anisotropy (gxx = 0.000, gyy = 0.000, gzz = 19.870) in the ground state of 16 but a significant transverse anisotropy (gxx = 0.012, gyy = 0.013, gzz = 19.763) in

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16a makes it a non SIM. It is further corroborated with the experiments, in which the octahedral complex does not show any signal for ac magnetic susceptibility in zero fields. The computed axial crystal field parameters become comparable to non-axial crystal field parameters in 16a. In contrast, the non-axial crystal field parameters are much lower than the axial crystal field parameter 16, suggesting a strong axiality of 16. This is further supported by the shorter axial Dy-O and longer equatorial Dy-O bond length in 16. However, a significant transverse anisotropy in the first excited KD (gxx = 0.382, gyy = 0.822, gzz = 18.053) of 16 favours magnetisation relaxation via this state. This results in the U cal value of 290 cm−1 , which is slightly underestimated compared to the experimental estimate of 305 cm−1 (Table 7.1). To understand whether the difference in magnetic anisotropy with geometry originates from the deviations from the idealised symmetry, ab initio calculations were performed on two model complexes [Dy(OCH)7 ]4− (D5h ) and [Dy(OCH)6 ]3− (Oh ). The calculations on [Dy(OCH)7 ]4− reveal an axial set of g factors (gxx ~ gyy ~ 0 and gzz ~ 20 in KD1) in the ground state, suggesting quenching of QTM. On the other hand, isotropic g factors (gxx = gyy = gzz = 6.58) were found in the model complex [Dy(OCH)6 ]3− which suggests a lack of SIM behaviour. But the octahedral isomer of 16 shows field-induced SIM behaviour due to the deviation from the idealised octahedral symmetry, which creates relatively small magnetic anisotropy. Although the barrier height of D4h symmetric SIMs has reached up to 1400 cm−1 , the same is not reflected in the T B value, which has not reached beyond 10 K. Furthermore, most of these molecules are not air-stable. To address this issue, in 2016, our group along with Murugavel and co-workers studied an air-stable Dy(III) SIM, [L2 Dy(H2 O)5 ][I]3 .L2 .H2 O (17, L = t BuPO(NHi Pr)2 ) which possesses recordbreaking T B value (FC/ZFC) of 12 K (Fig. 7.12c and Table 7.2) [50]. The metal centre binds with five water molecules in the equatorial plane and two phosphonic diamide ligands in the axial position. The CShM analysis with {DyO7 } motif unveils a pentagonal bipyramidal geometry (pseudo D5h symmetry) around the metal ion (Table 7.2). The ab initio electronic structure calculations on 17 reveal an mJ = |±15/2⟩ ground state with very small transverse anisotropy (gxx ~ gyy = 0.4*10–4 , gzz = 19.86), which leads to the negligible QTM (Table 7.2). The ground gzz axis is found to be along the pseudo C 5 axis (along with the axial (P)O-Dy-O(P) axis, Fig. 7.12c). The first excited KD at 321 cm−1 is also found to contain very small transverse anisotropy (gxx , gyy = 0.02, gzz = 17.08) with mJ = |±13/2⟩ , quenches the TA-QTM for magnetisation relaxation. Furthermore, the gzz axis of this state shows a small deviation (6.0°) from the ground state. On the other hand, the stabilisation of mJ = |±1/2⟩ in the second excited state at 478 cm−1 promotes significant TA-QTM (gxx = 0.58, gyy = 3.13, gzz = 16.53) for magnetisation relaxation (Fig. 7.12d). Further, the gzz axis of this state is oriented nearly perpendicular (~94°) to the ground state. It leads to the U cal value of 478 cm−1 , which is slightly overestimated than the U eff of 453 cm−1 (Table 7.2). But the U eff value of 511 cm−1 of the diluted sample suggests significant intermolecular interaction in the crystal. The large barrier height of 17 can be rationalised with the computed NPA charges, where the charges of axial oxygen atoms become ~ 0.3e− higher than the equatorial oxygen atoms. The large U cal value is corroborated

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where …

315

Fig. 7.12 The main magnetic axis of a 16, b 16a and c)17. Colour code: Dy—violet/green, Zn— indigo, Br—dark yellow, C—grey, P—pink, O—red, N—blue, H—white/black. d The mechanism of magnetisation of 17. e The comparison of experimental magnetic susceptibility (red) with the computed line (indicated by solid line). Reprinted from Liu et al. [86] with permission from the Royal Society of Chemistry. Reprinted from Gupta et al. [50] with permission from the Royal Society of Chemistry

with the larger value of B20 , B40 and B60 compared to non-axial crystal field parameters. The computed temperature-dependent magnetic susceptibility shows excellent agreement with experiments, suggesting the robustness of the ab initio approach (Fig. 7.12e). Furthermore, ab initio calculations have been carried out on three model structures [L' 2 Ln(H2 O)5 ] (17a, L' 2 = MePO(NHMe)2 ) and [L2 Ln]3+ (17b) to find out the role of solvent and water molecules in the magnetic anisotropy. The model 17a also relaxes via second excited KD at 757 cm−1 . The removal of iodides in 17a diminishes the hydrogen bonding in the equatorial plane, resulting in a decrease in the NPA charges of the equatorial oxygens. This leads to the enhancement of the blocking barrier compared to 17 and unveils the importance of the secondary coordination sphere in magnetic anisotropy. On the other hand, model 17b relaxes via fifth excited KDs and raises the U cal value to 2072 cm−1 . This is due to the removal of the equatorial ligand field from water molecules, which suggests that the 2-coordinate geometry of Dy(III) is best suited for SIM.

0.131 0.639 2.048

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

[L2 Ln(H2 O)5 ][I]3 .L2 .H2 O (17)

[Dy(Cy3 PO)2 (H2 O)5 ]Cl3 · (Cy3 PO)·H2 O·EtOH (18)

[Dy(Cy3 PO)2 (H2 O)5 ]Br3 · 2(Cy3 PO)·2H2 O·2EtOH (19)

[Dy(CyPh2 PO)2 (H2 O)5 ]Br3 · 2(CyPh2 PO)·EtOH· 3H2 O (20)

[Dy(H2 O)5 (HMPA)2 ]Cl3 HMPA.H2 O (21)

[Dy(H2 O)5 (HMPA)2 ]I3 .2HMPA (22)

[Dy(OPCy3 )2 (H2 O)5 ](CF3 SO3 )3 ·2OPCy3 (23) D5h

D5h

[Zn2 DyL2 (MeOH)]NO3 3MeOH.H2 O (16)

[Dy(bbpen)Cl] (24)

[Dy(bbpen)Br] (25)

[Dy(bbpen-CH3 )Cl] (26)

[Dy(bbpen-CH3 )Cl] (27)

[Dy(bbpen-F)Cl] (28)

[Dy(bbpen-F)Br] (29)

2.142

1.796

2.111

1.824

2.327

0.154

0.174

0.142

0.239

2.677

0.660

Symmetry CShM

Complex

–

–

12.1

–

9.5

7.5

8.5

7

7

10.5

11

8

12

–

30

20

15

9

14

8

2

9

6

19

20

11

14

11

– 0.02

– 0.02

0.2 0.02

0.02

0.02

0.02

0.018

0.004

0.004

0.02

0.02

0.02

0.02

0.02

800

582

808

502

712

492

391

417

320

353

375

328

453

305

770

611

820

634

721

586

509

445

410

206 207a

378

299

478

290

19.97 3rd 19.88 3rd 19.88 4th

−3.13 −4.00 −4.40

–

–

–

19.99 4th

19.98 3rd

19.88 4th

19.88 3rd

19.97 3rd

−2.58

–

19.99 3rd

19.88 4th

−1.85

−2.43

19.86 3rd

−1.80

19.88 2nd

19.86 3rd

−2.82

–

19.87 2nd

(continued)

[92]

[92]

[91]

[91]

[90]

[90]

[46]

[89]

[89]

[88]

[87]

[87]

[50]

[86]

gzz of Relaxation Refs. KD1 via KD

−1.72

FC/ T H (K)* Sweep U eff (cm−1 ) U cal (cm−1 ) B20 ZFC rate TB (K) (T/ min)

Table 7.2 The summary of the computed and experimental parameters that are relevant to the performance of PBP SIMs

316 S. Dey et al.

D5h

D5h

D5h

D5h

D5h

D5h

C 5v

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

D5h

C 5v

C 5v

C 5v

[Dy(Bpen)(Cl)3 ] ( 30 )

[Dy(Bpen)Cl(OPhCl 2 NO2 )2 ] ( 31 )

[Dy(Bpen)(OPhCl 2 NO2)3 ] ( 32 )

[Dy(Bpen)(OPhNO2 )3 ] ( 33 )

[Dy(Ot Bu)2 (py)5 ][BPh4 ] (34)

[Dy(L)2 (py)5 ][BPh4 ] (35)

[Dy(t BuO)Cl(THF)5 ][BPh4 ]·2THF (36)

[DyCl2 (THF)5 ][BPh4 ] ( 37 )

[Dy(OCMe3 )Cl(THF)5 ] [BPh4 ](38)

[Dy(OSiMe3 )Cl(THF)5 ][BPh4 ](39)

[Dy(OCMe3 )Br(THF)5 ][BPh4 ](40)

[Dy(OSiMe3 )Br(THF)5 ] [BPh4 ](41)

[Dy(OPh)Cl(THF)5 ] [BPh4 ] (42),

[Na(THF)5 ][Dy(OPh)2 (THF)5 ][BPh4 ]2 (43)

[Dy(OPh)2 (py)5 ][BPh4 ](44)

[Dy(OCMe3 )2 (py)5 ][BPh4 ] (45)

[Dy(OSiMe3 )2 (py)5 ][BPh4 ] (46)

[Dy(OCMe3 )2 (4-Mepy)5 ][BPh4 ] (47)

[Dy(tfpz)Cl(THF)5 ][BPh4 ] (48)

[Dy(Mepz)Cl(THF)5 ][BPh4 ] (49)

[Dy(Iprpz)Cl(THF)5 ][BPh4 ] (50)

14

–

–

–

–

–

–

–

0.618

0.683

0.801

0.700

0.556

0.456

0.483

0.386

0.352

0.293

0.196

–

–

–

–

16.0

15.0

14.0

13.0

12.0

6.8

4.5

4.5

4.5

7.0

–

–

–

–

–

23.0

22.0

25.0

16.0

18.0

9.0

9.0

9.0

9.0

11.0

–

–

22

4

–

–

–

–

19

24

60

16

54

660

–

–

–

218

143

–

0.0015 1041

0.0015 1109

0.0015 1255

0.0015 832–905

0.0015 924

0.0015 512

0.0015 509

0.0015 569

0.0015 557

0.0015 636–652

–

–

0.2 1130 0.0010

0.0012 1262

–

–

–

–

186

150

74

1072

1101

1183

826

853

406

473

558

561

562

42

562

1086–1109

1183

234

218

307

37

19.89 4th 19.44 2nd 19.87 2nd 19.88 3rd

−0.62 −0.80 −1.01

19.89 4th

19.89 5th

19.88 4th

19.88 4th

19.88 3rd

19.88 3rd

19.88 3rd

19.88 3rd

19.88 3rd

14.83 1st

–

–

–

–

–

–

–

–

–

–

–

19.88 3rd

19.89 4th

−6.26 −6.45 –

19.89 5th

19.77 2nd

19.77 2nd

19.83 2nd

17.29 2nd

(continued)

[96]

[96]

[96]

[52]

[52]

[52]

[52]

[52]

[52]

[52]

[52]

[52]

[52]

[52]

[95]

[53]

[44]

[93]

[93]

[93]

[93]

gzz of Relaxation Refs. KD1 via KD

–

–

–

–

–

FC/ T H (K)* Sweep U eff (cm−1 ) U cal (cm−1 ) B20 ZFC rate TB (K) (T/ min)

0.759–0.787 23

0.801

1.355

1.780

1.904

1.271

Symmetry CShM

Complex

Table 7.2 (continued)

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where … 317

C 5v

C 5v

C 5v

C 5v

D5h

D5h

D5h

D5h

[Dy(tfpz)2 (THF)5 ][BPh4 ] (54)

[Dy(pz)2 (THF)5 ][BPh4 ] (55)

[Dy(pz)2 (py)5 ][BPh4 ]·2py (56)

[Dy(pz)2 (NS)5 ][BPh4 ] (57)

(Et 3 NH)[(H 2 L)DyCl 2 ] (58)

[(L)Dy(Cy3 PO)Cl] (59)

[(L)Dy(Ph3 PO)Cl] (60)

[Dy(LN5 )(Ph

–

–

–

3

3.5

4

–

–

–

–

temperature below which the opening of magnetic hysteresis is observed

14

–

–

–

5

5

5

–

–

–

–

264

327

188

250

0.01

–

–

–

771

168

145

49

0.0012 309

0.0012 327

0.0012 362

–

–

–

–

723–731

155

158

147

257

288

318

222

334

180

216 19.87 3rd 19.91 4th 19.79 2nd 19.82 3rd 19.81 3rd 19.78 3rd 19.74 2nd 19.74 2nd 19.73 2nd

−1.91 −0.91 −1.97 −1.28 −1.37 −2.80 −2.18 −2.32

(−4.96) 19.98 3rd −(−5.20)

19.88 3rd

−0.97

[98]

[48]

[48]

[48, 97]

[96]

[96]

[96]

[96]

[96]

[96]

[96]

gzz of Relaxation Refs. KD1 via KD

−1.21

FC/ T H (K)* Sweep U eff (cm−1 ) U cal (cm−1 ) B20 ZFC rate TB (K) (T/ min)

1.293–1.681 5

1.505

1.446

1.210

–

–

–

–

–

–

The Italics Emphasised coded texts represent field-induced SIMs

H corresponds to the a XMS-CASPT2 values

*T

C 5v

[Dy(pz)Cl(THF)5 ][BPh4 ] (53)

(61)

C 5v

[Dy(Ipr2 pz)Cl(THF)5 ][BPh4 ] (52)

3 SiO)2 ](BPh4 ).CH2 Cl2

C 5v

[Dy(Me2 pz)Cl(THF)5 ][BPh4 ] (51)

–

Symmetry CShM

Complex

Table 7.2 (continued)

318 S. Dey et al.

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To explore the effect of (P)O-Dy-O(P) angle and Dy-O bond distance on magnetic properties, Overgaard and co-workers have applied hydrostatic pressure from 0.0 to 3.6 GPa on 17 [47]. The CShM analysis with the crystal structures in hydrostatic pressure reveals irregular deviation from the idealised D5h symmetry with pressure; the structure at 1.32 GPa shows the lowest distortion compared to other structures. To gain a deeper insight into the magnetic properties with pressure, ab initio calculations have been carried out on X-ray structures at different pressure points. The calculation reveals an mJ = |±15/2⟩ ground state in all the pressure regimes with the orientation of the gzz axis along the pseudo C 5 axis. Furthermore, an oblate ground electron density is found in all the pressure points. The magnetisation relaxation occurs via the second excited KDs in all the pressure points except at 1.62 GPa, where a strong transverse anisotropy in the first excited KD (gxx = 0.658, gyy = 1.172, gzz = 15.741) at 284.6 cm−1 reinforces for magnetisation relaxation via this state. The computed U cal value is found to be 326.5, 513.4, 422.6, 476.0, 284.6, 390.7, 598.4, 567.7, 491.6 and 631.4 cm−1 at 0.00, 0.53, 0.90, 1.32, 1.62, 2.05, 2.21, 3.02, 3.26 and 3.59 GPa, respectively (Fig. 7.13a). The computed U cal values can be corroborated with computed Loprop charges and crystal field parameters. The computed U cal value becomes highest at 3.59 GPa and lowest at 1.62 GPa. However, computed U cal values unveil uneven “up” and “down” with the pressure (Fig. 7.13a). The CShM values at the different pressure points do not correlate with the U cal values. In this regard, we have derived a new structural parameter, R, considering the (P)O-Dy-O(P) angle and Dy-O bond length. R=

( ) axialO − Dy − O (◦ ) + Av.(equatorial − axial)Dy − O bond length Å 1000 (7.5)

The computed U cal values at different pressure points are found to exhibit an excellent correlation with the R values (Fig. 7.13b). Consideration of intermolecular

Fig. 7.13 a The energy splitting of eight KDs of 17 with pressure b The variation of structure R parameter with pressure. Reprinted with permission from Norre et al. [47] Copyright@2019 American Chemical Society

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interaction at ambient pressure and 3.59 GPa reveals a significant increase (>40 cm−1 ) in the U cal values, again reiterating the fact that intermolecular interactions are of utmost importance in these pseudo D5h complexes. While the discovery of 17 with a record-breaking blocking temperature made a breakthrough in lanthanide SIM, at the same time (in 2016), Tong and co-workers have reported magnetic properties of two similar pentagonal bipyramidal Dy(III) complexes, [Dy(Cy3 PO)2 (H2 O)5 ]Cl3 ·(Cy3 PO)·H2 O·EtOH (18, Cy3 PO = tri cyclohexyl phosphine oxide, Fig. 7.14a) and [Dy(Cy3 PO)2 (H2 O)5 ]Br3 ·2(Cy3 PO)·2H2 O·2EtOH (19, Fig. 7.14b) [87]. Complexes 18 and 19 differ from 17 with respect to the axial ligand field and the nature of the surrounding counteranions. The axial (P)O-Dy-O(P) bond angle is found to be 175.6º and 179.0° for 18 and 19, respectively. The complexes 18 and 19 can retain the magnetisation up to 11 and 20 K, respectively, although the FC/ZFC T B (8 K (18) and 11 K (19)) and U eff values (328 (18) and 375 (19) cm−1 ) are found to be smaller than 17 (Table 7.2). This suggests stronger axial donation from t BuPO(NHi Pr)2 groups in 16 compared to Cy3 PO groups in 18 and 19. The ab initio calculations reveal negligible transverse anisotropy, rendering very small QTM in the ground state of both the complexes. The computed g tensors (18: gxx = 0.001, gyy = 0.001, gzz = 19.863; 19: gxx = 0.000, gyy = 0.000, gzz = 19.876, Table 7.2) reveals larger axiality of 19 compared to 18. The larger negative B20 value (−1.80 (18) and −1.85 (19)) of 19 compared to 18 also supports the higher axiality of the former (Table 7.2). The computed χT value of both the complexes shows excellent agreement with the experiments. The first excited KD at 267 and 250 cm−1 was also found to be axial in nature, rendering a very small TA-QTM (Fig. 7.14d). The magnetisation relaxation occurs via 2nd excited KD in 18 due to the large transverse anisotropy (gxx = 1.0420, gyy = 1.4306, gzz = 16.7249) and large deviation (88.1°) of gzz axis with the ground state (Fig. 7.14a, d). On the other hand, complex 19 relaxes via the third excited KD due to the large TA-QTM and near perpendicular orientation of the corresponding gzz axis with respect to the ground state gzz axis (Fig. 7.14e). This results in the U cal value of 299 and 378 cm−1 for 18 and 19, respectively. Although the U cal value obtained for 19 is in close agreement with the experiment, the same is underestimated in 18 (Table 7.2). The discrepancy in 18 can be ascribed to the poor choice of the basis set. However, the larger U cal value of 19 compared to 18 can be attributed to the various structural aspects; (i) a larger (P)O-Dy-O(P) angle of 19 compared to 18 and (ii) shorter axial Dy-O(P) distance in 19 compared to 18. Furthermore, the higher U cal value in 19 compared to 18 can be rationalised by the stronger equatorial donation to the metal centre from chloride (18) compared to bromide (19) ions. Based on the earlier study, one year later, the same group, replacing two cyclohexyl groups of 19 by phenyl groups, reported a new pentagonal bipyramidal Dy(III) SIM, [Dy(CyPh2 PO)2 (H2 O)5 ]Br3 ·2(CyPh2 PO)·EtOH·3H2 O (20, Fig. 7.14c) [88]. The CShM analysis with {DyO7 } core of 20 reveals a pseudo D5h symmetry around the metal centre with a slightly larger deviation compared to 19 (Table 7.2). The experimental magnetic characterisation reveals a slightly lower U eff and FC/ZFC T B value of 20 compared to 19 (Table 7.2). Calculations reveal more than ~100 cm−1

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Fig. 7.14 a The main magnetic axis of KD1-KD3 in 18. b The main magnetic axis of KD1-KD3 in 19. c The main magnetic axis of KD1 in 20. Counter ions and hydrogens are omitted for clarity. Colour code: Dy—cyan, P—purple, O—red, C—grey. The mechanism of magnetisation relaxation in d 18 e 19 f 20. Reprinted with permission from Chen et al. [87] Copyright@2016 American Chemical Society. Reprinted from Chen et al. [88] with permission from John Wiley and Sons

smaller crystal field splitting of eight ground KDs (6 H15/2 ) in 20 compared to 19. This is due to the reduction in the axial crystal field with cyclohexyl substitution by phenyl, which is less electron-donating in nature. The ground state of 20 contains a dominant mJ = |±15/2⟩ contribution, which reduces the QTM (KD1: gxx = 0.000, gyy = 0.000, gzz = 19.881). While the magnetisation relaxation occurs via the second excited KD in 19, for 20, it relaxes via the first excited KD due to the large transverse anisotropy (gxx = 0.957, gyy = 3.427, gzz = 16.696, Fig. 7.14f). This leads to the U cal value of 206 cm−1 , which is highly underestimated compared to the U eff value (Table 7.2). This discrepancy may be due to the lack of dynamic correlation in the calculations. To include this dynamic correlation, high-level XMS CASPT2 calculations were performed on a modelled structure of 20 (Cy and Ph groups were replaced by Me) to reduce the computational cost. The computed U cal value using XMS-CASPT2 (crystal field correction approach due to the modelling) is found to be 297 cm− 1 , greatly improved compared to CASSCF but still underestimated compared to U eff . The enlargement of the active space by including the metal’s ligand orbitals and 5d virtual orbitals could close the gap. Inspired by the earlier works on 17–20, Murrie and co-workers have reported two other pentagonal bipyramidal Dy(III) SIMs, [Dy(H2 O)5 (HMPA)2 ]Cl3 .HMPA.H2 O

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(21, HMPA = [(CH3 )2 N]3 PO) and [Dy(H2 O)5 (HMPA)2 ]I3 .2HMPA (22) with the axial ligand field to hexamethylphosphoramide (Fig. 7.15) [89]. The change in the axial ligand field leads to a slight increase in the U eff value (320 and 417 cm−1 for 21 and 22, respectively). The CShM analysis shows a minimal deviation from both complexes’ idealised D5h symmetry (Table 7.2). The yttrium-doped samples of 21 and 22 were found to retain the magnetisation up to 9 and 10 K, respectively. The ab initio calculations reveal a large span in energy (882.1 (919.0) cm−1 for 21 (22)) of the eight KDs from the 6 H15/2 state. The ground, first excited and second excited KD contain a dominant contribution from mJ = |±15/2⟩, |±13/2⟩ and |±1/2⟩ , respectively, in both the complexes. At the same time, ground and first excited states are purely Ising in nature (gxx ~ gyy ~ 0.0, gzz ~ 19.97 (ground state), 17.10 (first excited state)), and the significant transverse anisotropy in the second excited KD renders sufficient TA-QTM (~0.5(3.5) μB for 21(22)) for magnetisation relaxation via this state. Furthermore, the near perpendicular orientation of the gzz axis (89.4°(91.8°) for 21(22)) of second excited KDs with respect to the ground gzz axis facilitates the magnetisation relaxation. Considering these, the U cal value is estimated to be 409.5 (444.7) cm−1 for 21(22). While the U cal and U eff show good agreement in 22, the computed U cal value in 21 is overestimated by ~90 cm−1 . This can be ascribed to the larger QTM/TA-QTM value in 21 compared to 22. The larger U cal of 22 compared to 21 can be explained with Loprop charges along with structural aspects. The Loprop charges of the Cl− ions in the equatorial position of 21 are found to be larger than I− ions in 22, suggesting a stronger equatorial ligand field in the former, which reduces the axiality. The larger axiality of 22 is further corroborated by the computed CF parameters, where the non-axial terms are far smaller than 21. Ab initio calculations have been performed with step-by-step elimination of each fragment in the secondary coordination sphere (Fig. 7.15). The calculations unveil an increment in the U cal value with the removal of each fragment. It is due to the removal of hydrogen bonds with equatorial water molecules, which consequently lowers the charges on the water oxygen atoms. Therefore, weakening the equatorial ligand field enhances the blocking barrier. As the replacement of chloride with larger halide ions like I− leads to an increase in the barrier height, a further model with larger counter anions like PF6 − could potentially help in enhancing the blocking barrier. To assess the importance of the flexibility of counter anions in magnetisation relaxation, Colacio and co-workers have reported another pentagonal bipyramidal Dy(III) SIM, [Dy(OPCy3 )2 (H2 O)5 ](CF3 SO3 )3 ·2OPCy3 (23, Fig. 7.16a) [46]. Complex 23 consists of large triflate counter anions instead of halide counter anions as in 17–22, which provides additional rigidity. A comparison of the electronic structures of 23 with 17–22 unveils the effect of flexible counter anions in magnetic anisotropy. The CShM analysis with {DyO7 } core of 23 reveals a significant deviation (0.639) from the ideal D5h symmetry compared to 17–22. Complex 23 is reported to have an FC/ ZFC T B value of 8.5 K. It seems like the large triflate counter anions help to increase the blocking barrier compared to 17–22. The ab initio calculations reveal an Ising (gxx = 0.000, gyy = 0.000, gzz = 19.974) mJ = |±15/2⟩ ground state, where the gzz axis is lying along the pseudo C 5 axis (along with the (P)O-Dy-O(P) axis). The

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where …

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Fig. 7.15 The U cal (K) values of the models (21) with the stepwise removal of secondary coordination sphere and equatorial water molecules. Colour code: Dy—gold, P—purple, O—red, N—blue, P—pink, Cl—green. Hydrogens are omitted for clarity. Reprinted from Canaj et al. [89] with permission from the Royal Society of Chemistry

first excited KD at 341.7 cm−1 has dominant mJ = |±13/2⟩ state with negligible transverse anisotropy (gxx = 0.017, gyy = 0.024, gzz = 17.086) rendering very small TA-QTM. On the other hand, substantial mJ mixing (0.54|±1/2⟩ + 0.28|±3/2⟩ + 0.10|±11/2⟩) in the second excited KD promotes significant TA-QTM (1.58 μB ) for magnetisation relaxation. This results in the U cal value of 508.5 cm−1 , which is more than 100 cm−1 higher than the experimental U eff of 391 cm−1 . This discrepancy may be due to the intermolecular interactions and other factors that are not considered in the ab initio calculations. Furthermore, the stepwise removal of counter anions and solvent molecules in 23 also leads to the enhancement in the U cal values due to the weakening of the equatorial ligand field. The larger U cal value of 23, compared to 17–22, is intriguing, as the latter is close to the ideal D5h polyhedron. This suggests that there are several factors other than structural aspects which are dictating the U cal value, one of them is the Loprop charge. The average computed Loprop charges on the axial O(P) atoms of {DyO7 } core is found to be ~1.128e− in 23, while it is ~1.115e− in 18–19. On the other hand, the average computed Loprop charge on the equatorial water oxygens is found to be 0.74e− in 23, which is slightly smaller than ~0.76e− in 18–19. The difference between the average axial and equatorial Loprop charges becomes higher in 22, compared to 18–19, which rationalises the larger U cal value of 23 compared to 18–19. To check

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Fig. 7.16 a The crystal structure of 23. Colour code: Dy—green, O—red, F—blue, S—orange, C—grey. Hydrogens are omitted for clarity. b The correlation plot of computed U cal value with Δ of 17–23. Reprinted from Díaz-Ortega et al. [46] with permission from the Royal Society of Chemistry

whether the difference between axial and equatorial Loprop charges (Δ) correlates with the U cal value in {[Dy(L)2 (H2 O)5 ]X3 ·2L} family, such as 18–19, the Loprop charges have been computed for all of them. Quite interestingly, all the complexes show an excellent correlation of the U cal values Δ (Fig. 7.16b). Despite a similar structural framework, the larger U cal value of 23 compared to 18–19 can be explained by the charges in the secondary coordination sphere. The negative charges on the oxygen atom of the triflate group are delocalised over the entire molecule. Further, the electron-withdrawing CF3 group reduces the charge on the oxygen atoms. This reduction in charges weakens the equatorial ligand field, which offers the larger U cal value in 23 compared to 18–19. However, the U eff value of 23 is smaller than 17–22, despite higher U cal values of the former. It is worth noting that the magnetisation relaxation depends on not only the electronic structure but also its coupling with molecular vibration, as has been seen in previous studies. The triflate ion is more flexible compared to halides due to its large size. This offers rapid magnetisation relaxation in 23, which effectively lowers the U eff value compared to 17–23. To obtain a large U eff value for magnetisation relaxation, Tong and coworkers have studied two other air-stable distorted pentagonal bipyramidal Dy(III) complexes; [Dy(bbpen)X] (X=Cl (24) and X=Br (25), H2 bbpen=N,N' -bis(2hydroxybenzyl)-N,N' -bis(2-methylpyridyl)ethylenediamine, Fig. 7.17a) [90]. The metal centre is bound to two phenyl oxygen atoms (from bbpen ligand) in the axial position and four-nitrogen atoms (from bbpen), and one Cl(Br) in the equatorial plane. The axial O–Dy–O bond angle is found to be 154.3° and 155.8° for 24 and 25, respectively. The CShM analysis with {DyO2 N4 Cl(Br)}core reveals a significant deviation (2.048(2.327) in 24(25)) from the ideal D5h symmetry in both the complexes. However, complexes 24 and 25 are reported to have FC/ZFC T B values of 7.5 and 9.5 K and U eff values of 492 and 712 cm−1 , respectively. Therefore, the weaker ligand field from the Br− ion in 25 leads to improved magnetic performance compared to 24. The calculations reveals a negligible transverse anisotropy in the

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ground (24: gxx = 0.001, gyy = 0.001, gzz = 19.875; 25: gxx = 0.001, gyy = 0.001, gzz = 19.881) and first excited KD (24: gxx = 0.112, gyy = 0.164, gzz = 16.912; 25: gxx = 0.062, gyy = 0.080, gzz = 16.985). The computed g factors reveal the larger axiality of 25 compared to 24. This is further corroborated by the axial B20 crystal field parameter (−0.27 × 10–2 and −0.44 × 10–2 in 24 and 25, respectively). Further, the larger values of the non-axial crystal field in 24 also suggest lower axiality than 25. The main magnetic axis is found to be close to the axial oxygen atoms in both complexes (Fig. 7.17a). However, a significant transverse anisotropy (gxx = 2.258, gyy = 5.362, gzz = 11.421) at second excited KD at 586 cm−1 of 24 favours the magnetisation relaxation, while transverse anisotropy of 25 is found to be minimal (gxx = 0.868, gyy = 1.159, gzz = 13.510) at second excited KD at 627 cm−1 (Fig. 7.17b). For 25, the third excited KD at 721 cm−1 possess significant transverse anisotropy (gxx = 4.742, gyy = 6.038, gzz = 9.820), that promotes sufficient TA-QTM for magnetisation relaxation (Fig. 7.17c). This results in the U cal value of 586 and 721 cm−1 for 24 and 25, respectively. These values are in excellent agreement with experiments for 25 but highly overestimated in 24. The discrepancy in 24 might be due to the higher probability of a barrierless relaxation process like QTM/Raman compared to 25. However, the larger U cal value 25 compared to 24 can be ascribed to the structural aspects, where 25 is found to possess a shorter axial Dy-O bond length and longer Dy-Br bond length compared to 24. A perfect agreement between the experimental and computed χT of 24 and 25 suggests the accuracy of the ab initio approach in both the complexes. It should be noted that the TA-QTM (0.4 μB ) in the second excited KD of 25 is sufficiently large, although it follows the relaxation path via the third excited KD due to the even higher TA-QTM of 2.5 μB (Fig. 7.17c). The rate of TAT (thermally −E −E assisted tunnelling transition) via a KD is proportional to e kT μ2 , where e kT (E = energy, k = Boltzmann constant, T = temperature) represents the thermal population of a KD and μ is the matrix element of the average magnetic moment of a KD. At 60 K (the maxima of the imaginary component of ac susceptibility of 25), the ratio

Fig. 7.17 a The main magnetic axis of 25. Color code: Dy—purple, Br—lime, O—red, N—blue, C—grey, H—white. The mechanism of magnetisation of b 24 c 25. The black line indicates the energy of KD as a function of magnetic moment. The red arrow represents the QTM/TA-QTM via ground and excited state. The blue arrow indicates possible pathway of magnetisation relaxation. Reprinted with permission from Liu et al. [90] Copyright@2016 American Chemical Society

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of TAT rates for the relaxation via third and second excited KD of 25 is found to be ~4. The ratio is a function of temperature, which should increase with the increase in temperature. This justifies the magnetisation relaxation of 25 via third excited KD at elevated temperatures. To further enhance the blocking barrier of complexes 24–25, Jiang et al. have introduced an electron-donating –CH3 group in the bbpen ligand called bbpen-CH3 , resulting in [Dy(bbpen-CH3 )Cl] (26) and [Dy(bbpen-CH3 )Br] (27, Fig. 7.18a–c) [91]. The increased electron density in the metal coordinating oxygen atoms leads to the enhancement in the U eff value to 502 (808) cm−1 in 26 (27) compared to 24–25. On the other hand, complexes 26 and 27 are reported to have FC/ZFC T B values (or T IRREV ) of 8.8 and 12.1 K, respectively, which is considerably higher than the bbpen analogues. The calculations reveal an easy-axis magnetic anisotropy which is reflected in the Ising type g values of the ground KDs (gzz is approaching ~20 g in 26 and 27, Table 7.2). The gxzzy (gx y is called average transverse g values) ratio is considered as a figure of merit of axiality of lanthanide SIMs, lower the values higher the axiality. Here, the above ratio is found to be in the order of 27 < 25 < 26 < 24, which is in accordance with their U eff values. Furthermore, the QTM in the ground state was also found to be in the order of 27 < 25 < 26 < 24, the same as the trend observed in U eff values. The complex 26 (27) relaxes via the second (third) excited KD due to the large TA-QTM value of 0.931(1.93) μB . This leads to the U cal of 634 (820) cm−1 for 26 (27), which is slightly overestimated compared to the U eff . However, the axiality of the complexes 24–27 can also be rationalised with respect to electrostatic potential (ESP) felt by a

Fig. 7.18 The molecular structure of a 27, b 28, c 29. d The energy of the four low-lying KDs of P(eq) 26 (1), 24 (1' ), 27 (2) and 25 (2' ) with EE SS P(ax) . . The mechanism of magnetisation relaxation of e 28 f 29. Reprinted from Jiang et al. [91] with permission from the Royal Society of Chemistry. Reprinted with permission from Zhu et al. [92] Copyright@2016 American Chemical Society

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metal ion. Ideally, for Dy(III) ion, the electrostatic potential should be larger in the axial position compared to the equatorial position for its improved performance as P(eq) ratio should indicate the performance of Dy(III) SIM, i.e. the SIM. Therefore, EE SS P(ax) lower the ratio, the better the SIM performance. The ratio is found to be 0.843, 0.834, 0.807 and 0.789 for 24, 26, 25 and 27, respectively, suggesting sufficient axiality in all the complexes (Fig. 7.18d). The lower ratio of bbpen-CH3 complexes (26 and 27) compared to their bbpen analogue (24 and 25) suggests increasing axiality with the methyl substitution. Another way to increase the axiality is to introduce the electron-withdrawing groups in the equatorial position. From this viewpoint, Zhu et al. has reported two analogues Dy(III) SIMs, [Dy(bbpen-F)Cl] (28) and [Dy(bbpen-F)Br] (29) [92]. These complexes have the same ligand as in complexes 24–25, with two electron-withdrawing fluoride groups added to the pyridine ring of the bbpen ligand (Fig. 7.18b, c). The structural parameters of 28–29 are found to be similar to 24–25. This substitution leads to the enhancement of the U eff values to 582 (800) cm−1 for 28 (29) compared to 24–25. Quite interestingly, the U eff value of 28 becomes larger than analogues 26 with methyl substitution, although the U eff of 27 and 29 remain similar in magnitude. Complexes 28 and 29 can retain magnetisation up to 20 and 30 K, respectively, which is considerably higher than 24–27. Calculations on 28–29 reveal an Ising ground state (gzz = 19.98 (19.99) in 28(29)) this lead to a negligible QTM (0.23 × 10–2 (0.98 × 10–3 ) μB in 28 29 (see Fig. 7.18e–f) for magnetisation relaxation. The larger computed QTM for 28 suggests lower axiality compared to 29, and this is due to the stronger ligand field from chloride ion compared to the bromide ion in the equatorial position. Further, calculations reveal the relaxation of magnetisation occurs via second and third excited KD for 28 and 29, respectively, due to the significant TA-QTM in these states (3.00 (1.64) μB 28 (29), Fig. 7.18e– f). This leads to the computed U cal value of 610.7 and 769.6 cm−1 for 28 and 29, respectively, which is slightly overestimated in 28 and slightly underestimated in P(eq) ratio of 28 and 29 is calculated to 29 compared to the U eff values. The EE SS P(ax) be 0.824 and 0.778, respectively, which is lower than 24–25, suggesting increasing axiality with fluoride substitution in the equatorial position. This is further corroborated by the average lower Loprop charge of the equatorial nitrogen atoms in 28–29 compared to 24–25, while the Loprop | charge of the axial | oxygen atoms | 2 | remains | | B2 | CF parameter | ratio such as similar in both sets of complexes. The | non−axial | B 0 | becomes axial CF parameter | 2 0.409(0.407) in 24(28), and 0.241(0.332) in 27(29) also suggests fluoride substitution in the equatorial position of the bbpen ligand of 24–25 increases the axiality. On the other hand, using slightly modified ligand of bbpen in 24–25, one year later, Chen and co-workers have reported a series of pentagonal bipyramidal Dy(III) complexes (Fig. 7.19); [Dy(Bpen)(Cl)3 ] (30, Bpen = N,N’-bis(2-methylenepyridinyl)ethylenediamine), [Dy(Bpen)Cl(OPhCl2 NO2 )2 ] (31), [Dy(Bpen)(OPhCl2 NO2 )3 ] (32) and [Dy(Bpen)(OPhNO2 )3 ] (33) [93]. Bpen is a tetradentate ligand with four-nitrogen “pockets” (U-type) which produces a weak ligand field in the equatorial plane from nitrogen donor centres. As Bpen ligand coordinates with the metal centre in the equatorial plane, the remaining sites (one

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equatorial and two axial) are occupied by 3Cl− ions in 30, one Cl− (equatorial) and two phenoxide oxygens (axial) in 31, and three phenoxide oxygens in 32 and 33. The CShM analysis reveals a pseudo D5h symmetry around the metal centre, where the complex 30 shows the smallest deviation from the ideal polyhedron. This value becomes the largest in 31 among all four complexes. The average equatorial Dy–N, Dy–Cl and axial Dy–O distances are found to be 2.5, 2.6 and 2.2 Å, respectively, indicating the compressed electronic structure of 31–33. Further, the axial O–Dy–O/ Cl–Dy–Cl bond angle is found to be 168.52°, 165.6°, 166.06°, 164.45° in 30, 31, 32 and 33, respectively. The blocking barrier of magnetisation reversal is estimated to be 15.6, 59.6, 23.8 and 18.6 cm−1 for 30–33, respectively, in the external dc field. The smallest U eff value of 30 can be ascribed to the two Cl− ions in the axial sites, which creates a weak axial ligand field. Why do complexes 30–33 show fieldinduced SIM behaviour with very small U eff values despite having similar geometry with other high-performance pentagonal bipyramidal Dy(III) SIMs? Particularly, 31 and 24 (U eff = 492 cm−1 ) have similar coordination environment and CShM values. The ab initio calculations reveal an axial ground state (gzz approaching 20) in all the complexes except 30, where a strong transverse anisotropy is found in the ground state (gxx = 0.601, gyy = 2.944, gzz = 17.293). This rules out the zero-field SIM behaviour of 30. The ground state gzz axis of these complexes is found to be oriented along with the axial coordinating atoms with a small deviation from the ground state gzz axis (Fig. 7.19). The ground state QTM values of 31, 32 and 33 are estimated to be 0.591 × 10–2 , 0.213 × 10–1 and 0.312 × 10−1 μB, respectively. These values are ~2 orders of magnitude higher than the estimated one in 24, suggesting fast relaxation in the absence of an external dc field and hindering them from behaving as zero-field SIMs. In comparison, an external dc field quenches the QTM, and the energy of the first excited state is found to be 36.5(30), 307.3(31), 218.1(32) and 233.6(33) cm−1 , much higher than the U eff . This may be ascribed to under-barrier relaxation caused by the direct and Raman processes. The direct process takes place in the external dc field when the degeneracy of the KDs is removed, which is the case here. Raman process takes place when higher excited KDs become mixed in character (mixing between various |±mJ ⟩ levels) and non-collinear with the ground states; eventually, that is the case for 30–33.

Fig. 7.19 The ab initio computed main magnetic axis of a 30, b 31, c 32, d 33. The only core structure of all complexes is represented here. Reprinted from Li et al. [93] with permission from John Wiley and Sons

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The magnetic axiality of 24 and 30–33 can also be compared with the electrostatic potential (ESP) generated by the ligand atoms. The axial ESP should be larger than P(eq) the equatorial in order to have significant magnetic anisotropy for SIM. The EE SS P(ax) is found to be 1.365, 0.920, 1.160, 0.985 and 0.472 for 30, 31, 32, 32 and 24, respectively. The lower value indicates the larger axiality. Therefore, the magnetic axiality is in the order of 30 < 32 < 33 < 31 < 24, which is in accordance with the U eff /U cal values. The magnetic anisotropy of 30–33 could further rationalise with structural aspects, where complex 31 has a more compressed structure compared to other complexes. Further, two equatorial nitrogen atoms adopt sp2 hybridisation in 30–33 while they are in sp3 hybridisation in 24, which induces more electron density in the equatorial plane of 30–33 compared to 24. This might be the reason for lower ESP in 24 compared to 33, despite having similar structural parameters. Additionally, other functional groups could also affect the magnetic properties. The ab initio study on complexes 17–23 with [DyX2 L5 ]+ (X = anionic ligand, L = neutral ligand) suggests that a large energy barrier for magnetisation relaxation is obtained when the negatively charged ligands lie along the axial direction. From this viewpoint, Zheng and co-workers have studied another pentagonal bipyramidal Dy(III) SIM, [Dy(Ot Bu)2 (py)5 ][BPh4 ] (34, Fig. 7.20a), which is reported to have a massive U eff (highest to date among reported pseudo D5h SIM) of 1262 cm−1 , along with a very high FC/ZFC T B value of 14 K [44]. The large U eff value is ascribed to the very short axial Dy-O bond (~2.11 Å) and long equatorial Dy–N bonds (2.53– 2.58 Å). The axial (C)O–Dy–O(C) is found to be almost linear (angle is 178.9°). The ground, first, second and third excited KDs are found to contain almost pure mJ = |±15/2⟩, |±13/2⟩, |±11/2⟩, |±9/2⟩ states at 0.0, 564, 940 and 1141 cm−1 , respectively (Fig. 7.20b). The Ising nature of these KDs (KD1: gxx = 0.00, gyy = 0.00, gzz = 19.89; KD2: gxx = 0.00, gyy = 0.00, gzz = 16.97; KD3: gxx = 0.04, gyy = 0.04, gzz = 14.28; KD4: gxx = 0.04, gyy = 0.29, gzz = 11.39) does not promote significant QTM/TA-QTM for magnetisation relaxation. Further, the gzz axis of these KDs lies along the O-Dy-O axis with a small deviation ( −0.66(40). The replacement of THF with pyridine in the equatorial position leads to a less prominent effect in the Loprop charge (−0.366(43) vs. −0.553 (45)). This suggests that the magnetic axiality and crystal field in pentagonal bipyramidal geometry is dictated by the electronegativity of the axial donor centres. This is further supported by the lower magnetic axiality of 37, where the Loprop charge of the Cl− (axial) and O(equatorial) becomes ~ − 0.83 and −0.55, respectively, which are pretty close. The formal charges seem less important in magnetic axiality since they are (−1) and 0 for the axial and equatorial ligands, respectively, in all the complexes. The previous discussions suggest a relation of U eff with Dy-X bond length. In this regard, Chilton et al. correlate the U eff values of complexes 24–25 and 38–47 with a structural parameter 2r Dy . This structural parameter is defined as (average Dy–X bond length)-(radii of Dy + radii of axial coordinating atoms such as X1 /X2 ). For Dy(III) ion and single atom donors such Cl− /Br− , the ionic radii were used. On the other hand, the covalent radii of oxygen were used for polyatomic ligands such as aryloxide/siloxide/alkoxide. An excellent linear correlation (R2 = 0.9226) between the two (U eff and 2r Dy ) has been obtained with U eff = −5032 × 2r Dy + 16,171 (Fig. 7.23a). Considering ten more examples from 17–30, the R2 value to 0.8465 (U eff = −4147 × 2r Dy + 13,125) remain similar. The inclusion of other parameters such as X1 –Dy–X2 angle in the correlation also gives similar results. For doping samples as well a similar R2 values are obtained (U eff = −3737 × 2r Dy + 12,190 R2 = 0.9432) (Fig. 7.23b). This correlation is purely empirical, does not consider the covalent contribution of the ligands, and limits its applicability. Inspired by the above correlation, Zheng and co-workers have further reported a series of pentagonal bipyramidal Dy(III) SIMs (Fig. 7.24) which can be classified into two categories, one with mono-pyrazole ligand in the axial position and have a general

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formula [DyX1 Cl(THF)5 ][BPh4 ]; (48: X1 = 3-(trifluoromethyl)pyrazole (tfpz); 49: X1 = 3-methylpyrazole (Mepz); 50: X1 = 3-isopropyl-1H-pyrazole (Iprpz); 51: X1 = 3,5-dimethylpyrazole(Me2 pz); 52: X1 = 3,5-diisopropylpyrazole (Ipr2 pz); and 53: X1 = pyrazole (pz)) [96]. The other complexes are based on pyrazole ring on two axial sites; [Dy(tfpz)2 (THF)5 ][BPh4 ] (54), [Dy(pz)2 (THF)5 ][BPh4 ] (55), [Dy(pz)2 (py)5 ][BPh4 ]·2py (56) and [Dy(pz)2 (NS)5 ][BPh4 ] (57) [96]. The coordination geometry around the metal centre can be described as a “destroyed” pentagonalbipyramid (DPB) with pseudo C 5v symmetry. All the complexes possess one N–N bond (two in 54–57) and one Cl− ion in the axial position and five THF/py/thiazole in the equatorial position. The pyrazolate ring offers “end-on” η2 binding to the metal site, which enhances the rigidity of the molecular structure and increases the magnetic axiality. The complexes 48–53 with the general formula [DyX1 Cl(THF)5 ][BPh4 ] are found to possess lower U eff compared to 54–57 with two pyrazole rings on two axial sites. However, the U eff of these complexes (48–57) is much lower than the prototype pentagonal bipyramidal Dy(III) SIMs due to severe distortion from the ideal D5h polyhedron, which reduces the magnetic axiality. The calculations on 48–53 reveal the eight low-lying KDs span below 432 cm−1 . The gzz axis of these complexes is pointed towards the centroid of the pyrazole ligand in the axial position, and it is collinear with the pseudo C 5 axis (Fig. 7.24). A strong mixing of mJ levels (0.10|±15/2⟩ + 0.12|±11/2⟩ + 0.18|±9/2⟩ + 0.18|±7/ 2⟩ + 0.15|±5/2⟩ + 0.10 ± |3/2⟩) is observed in the ground KD of 48, which renders significant QTM and rules out the possibility of zero-field SIM behaviour of this complex. It is due to the electron-withdrawing -CF3 group of the pyrazole ligand in the axial position, which increases the Dy–cen(N–N) distance and reduces the axial crystal field. For 49–53 (the electron-donating effect is observed due to the alkyl substituents in the pyrazolate ring), the ground KD is found to have a dominant (~90%) contribution from mJ = |±15/2⟩, although mixing with mJ = |±13/2⟩ is observed. It unveils the relaxation via higher excited states. For example, in 49, the

Fig. 7.24 The ground state main magnetic axis of 48–57. Colour code: Dy—green, O—red, N— blue, S—yellow, C—grey. Hydrogens are omitted for clarity. Reprinted from Li et al. [96] with permission from the Royal Society of Chemistry

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ground KD is composed of mJ = 0.89|±15/2⟩ + 0.10|±13/2⟩, leading to the Ising ground state (gzz = 19.87). The strong mixing of mJ levels (0.19|±13/2⟩ + 0.13|±5/ 2⟩ + 0.39|±3/2⟩ + 0.19|±1/2⟩) in the first excited KD at 150 cm−1 results in a stronger QTM probability (3.8 μB ), facilitating the magnetisation relaxation. The first excited KD of complexes 50–53 is found to have a dominating contribution (70–80%) from mJ = |±13/2⟩ state, leading to the TA-QTM process. Apart from 53, the magnetisation relaxation of these complexes occurs via second excited KD due to the stabilisation of mJ = |±3/2⟩ and |±1/2⟩, resulting in the U cal value of 186, 216 and 180 cm−1 for 50, 51 and 52, respectively. The second excited KD of 53 contains a dominating (70%) contribution from mJ = |±11/2⟩, which leads to a weak TA-QTM via the third excited KD at 334 cm−1 . This is due to the strong transverse anisotropy and near perpendicular orientation of the principal axis with respect to the ground gzz axis. This results in the U cal value of 334 cm−1 for 53. On the other hand, the ab initio computed gzz axis of 54–57 is pointing along the cen(N–N)–Dy–cen(N–N) direction (Fig. 7.24). For 54, the ground state (mJ = 0.94|±15/2⟩) is found to be Ising in nature (gzz = 19.79). But, the weak ligand field from the tfpz ligand (due to the electron-withdrawing -CF3 group in the pyrazolate ring) leads to the strong mixing of mJ levels (0.10|±13/2⟩ + 0.15|±9/2⟩ + 0.27|±7/2⟩ + 0.34|±5/2⟩) in the first excited state, along with misalignment of gzz with respect of the ground state, forces the magnetisation relaxation via this state. It results in the U cal value of 222 cm−1 . Complexes 55–57 possess identical axial ligands but differ in the equatorial ligand. For 55, the contribution of mJ = |±13/2⟩ in the first excited state is significantly enhanced (49%) leading to a very small transverse anisotropy (gxx = 0.20, gyy = 0.99, gzz = 13.61). Further, the principal axis of this state is nearly collinear with the ground state gzz axis. The relaxation occurs via the second excited KD due to a significant transverse anisotropy (gxx = 1.55, gyy = 2.95, gzz = 9.15), resulting in the values 318 cm−1 . In 56 and 57, the relaxation process is similar to 55. Calculations yield an U cal values of 288 and 257 cm−1 for 56 and 57, respectively. The lower U cal value of 56 and 57 compared to 55 can be attributed to a stronger equatorial ligand field with the removal of THF molecules in the former. However, magnetic measurements suggest the highest U eff value (362 cm−1 ) in 55 among all the complexes (43–52). This U eff value is ~4 times lower than the U eff obtained in [Dy(Ot Bu)2 (py)5 ][BPh4 ] (34). It can be ascribed to the reduced symmetry from D5h in 55 due to the bidentate chelating ligand in the axial position. Furthermore, the -Ot Bu ligand in the axial position in 34 creates a much stronger ligand field compared to the pyrazolate ligand, which leads to the almost ~2.5 times larger energy splitting of eight KDs compared to 55. Further, the highest FC/ZFC blocking temperature of 4 K is attained in 55 compared to the complexes with THF equatorial ligand. It can be rationalised by the presence of intra/intermolecular π•••π interaction between the pyridine and the pyrazolate ring, which increases the stiffness of the complex. Although a very high U eff value has been obtained in pentagonal bipyramidal Dy(III) SIMs, the coordination geometry is not rigid enough; hence, even a slight chemical alternation around the coordination environment destroys its symmetry/

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geometry. To make a rigid coordination environment, Chandrasekhar and coworkers have introduced a chelating ligand in the equatorial plane and kinetically labile ligands in the axial position (Fig. 7.25), resulting in the formation of (Et3 NH)[(H2 L)DyCl2 ] (58, H4 L = 2,6-diacetylpyridine bis-salicylhydrazone), [(L)Dy(Cy3 PO)Cl] (59, H2 L = 2,6-diacetylpyridine bis-benzoylhydrazone) and [(L)Dy(Ph3 PO)Cl] (60) complexes [48, 97]. All three complexes possess two oxygen and three nitrogen donors in the equatorial plane. They differ only in the axial ligands, i.e. 58 has two Cl− ions in the axial position, 59 has one Cl− and one Cy3 PO group in the axial position, and 60 has one Cl− and one Ph3 PO group in the axial position. The CShM analysis for 58 with {DyO2 N3 Cl2 } core, for 59–60 with {DyO3 N3 Cl} core reveals PBP geometry around the metal centre with pseudo D5h symmetry (Table 7.2). The experimental studies reveal a U eff value of 49, 145 and 168 cm−1 for 58, 59 and 60, respectively. The calculations reveal an Ising ground state (gxx = gyy ~ 0, gzz ~ 20) in all the complexes where the gzz axis is oriented along the axial Cl–Dy–Cl or O–Dy–Cl direction (Fig. 7.25). The ground state is found to contain a dominant contribution (~98%) from mJ = |±15/2⟩ state (Fig. 7.25d). The mixing of mJ levels (dominant mJ = |±13/2⟩) leads to a large TA-QTM (0.24, 0.64 and 0.41 μB for 58, 59 and 60, respectively) in the first excited state, favouring the magnetisation relaxation (Fig. 7.25d). It results in the U cal values 147 (58), 158 (59) and 155 (60) cm−1 . Although the U cal values of 59 and 60 show closer agreement with the U eff value, 58, it is overestimated. The computed average Loprop charges of the axial donor centres are larger than equatorial donor centres, suggesting significant axiality in all the complexes. However, the Loprop charges of the axial Cl− ions and equatorial oxygens are the same in 58, indicating lower axiality than 59 and 60. Furthermore, the non-axial B22 CF parameters are comparable to axial B20 CF parameters in 58. This also suggests lower axiality for 58 compared to 59 and 60. Among 59 and 60, the axial B20 CF parameters are larger in 59, and this indicates a larger axiality of 59 compared to 60, which is in accordance with the U cal /U eff values. On the other hand, Murrie and co-workers have introduced a macrocyclic LN5 ligand in the equatorial plane to increase the rigidity, resulting in another pentagonal bipyramidal Dy(III) SIM; [Dy(LN5 )(Ph3 SiO)2 ](BPh4 ).CH2 Cl2 (61, Fig. 7.26) [98]. This complex possesses two asymmetric units in the crystal lattice. The CShM analysis with {DyN5 O2 } core reveals pseudo D5h symmetry around the metal centre. The axially compressed structure (Dy–O(Ph3 SiO) = 2.13–2.16 Å, Dy-N = 2.40–2.56 Å) indicates a strong axiality of this complex. This is supported by magnetic measurements (U eff = 771 cm−1 ) and FC/ZFC T B values (5 K). The ground state gzz axis of this complex is oriented along the axial (Si)O–Dy–O(Si) bond due to the strong ligand field from the Ph3 SiO ligand. While the ground state is a pure mJ = |±15/ 2⟩ state, the first (at ~417 cm−1 ) and second excited KD (at ~723 cm−1 ) is found to have a dominant contribution from mJ = |±13/2⟩ and |±11/2⟩ level, respectively. The magnetisation relaxation occurs via 2nd excited KD due to the strong TA-QTM, thanks to the strong mixing of mJ = |±11/2⟩ with other states (Fig. 7.26b). This results in the U cal value of 723 cm−1 , which is slightly underestimated compared to the U eff value (771 cm− 1 ). Further, the SA-CASSCF calculations also reveal

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Fig. 7.25 The main magnetic axis of a 58, b 59, c 60. Colour code: Dy—gold, O—red, N—blue, C—grey. Hydrogens are omitted for clarity. d The mechanism of magnetic relaxation of 58. The ground state beta electron density of e 59 f 60

that the –NH– groups of LN5 ligand (61-N5) possess similar Loprop charges on the axial oxygen atoms. This suggests the substitution of the -NH- groups with fewer electron-donating groups would enhance the blocking barrier. From this viewpoint, ab initio calculations were performed on an in silico model 61-O2N3 where the -NHgroups have been substituted by oxygen atoms. The calculations predict an increase in blocking barrier to 1000 cm−1 in 61-O2N3, where the relaxation of magnetisation takes place via third excited KD due to significant mixing of mJ = |±9/2⟩ with |±3/ 2⟩ state. It is further corroborated by the ~50% lowering of the Loprop charges on the equatorial oxygen atoms compared to –NH– groups. Based on these results, a crown ether ligand has been introduced in the equatorial plane (results in model 61-O5)

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due to its longer metal–ligand distance (Fig. 7.26c). The ab initio calculations on 61-O5 reveal a negligible QTM/TA-QTM in the ground, first and second excited KD (Fig. 7.26d). The mixing of mJ = |±9/2⟩ (92%) with |±3/2⟩ (4%) in the third excited KD promotes significant QTM (0.5 μB ) for magnetisation relaxation (Fig. 7.26d). This results in the massive U cal value of 1243 cm−1 . This is further corroborated by the lowering of Loprop charges in the equatorial oxygen atoms in 61-O5 compared to 61 and 61-O2N3. The ratio of the axial B20 CF parameters with the average non-axial CF parameters (B2−1 , B21 , B2−2 , B2−2 ) yield the following trend 61-O5 > 61-O2N3 > 61, and this trend rationalises the estimated U cal values.

Fig. 7.26 a LoProp charges of the atoms attached to the Dy(III) centre for Dy1A in complex 61. b The mechanism of magnetisation relaxation of 61. c The main magnetic axis of model 61-O5. d The mechanism of magnetisation relaxation of 61-O5. Colour code: Dy—gold, N—blue, O— red, Si—light blue, C—grey. Hydrogens and counter anions are omitted for clarity. Reprinted from Canaj et al. [98] with permission from the Royal Society of Chemistry

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7.4.5 Eight Coordinate Dy(III) SIM with Pseudo D6h Point Group It is evident from previous studies that the pseudo D4h and D5h complexes are well suited for attaining a large blocking barrier for magnetisation reversal. But barring a few, most lack ambient stability. However, it is clear that obtaining a large barrier height requires controlling the crystal field and symmetry around the metal centre. Apart from D5h symmetry, other possible higher coordination symmetry should also be of interest. As higher coordination numbers tend to offer greater stability, Murrie and our groups jointly reported the first example of D6h symmetric SIMs [49]. In this report, there were three air-stable hexagonal bipyramidal Dy(III) SIMs (Fig. 7.27); [Dy(LN6 )(2,4-di-t Bu-PhO)2 ](PF6 ) (62,), [Dy(LN6 )(Ph3 SiO)2 ](PF6 ) (63) and [Dy(LN6 )(Ph3 SiO)2 ](BPh4 ) (64) [49]. The alternation of the counter anion from 63 to 64 leads to the change in axial O-Dy-O angle from 179.8° to 176.1;° further, the axial Dy-O bond length reduces by ~0.1 Å in the latter. All three complexes possess a weak equatorial ligand field from a rigid macrocyclic ligand and a strong axial ligand field from Ph3 SiO and 2,4-di-t Bu-PhO groups. This leads to a very high U eff of 676, 751 and 781 cm−1 for 62, 63 and 64, respectively. The calculations reveal the eight KDs from the 6 H15/2 state span up to 1134, 1192 and 1260 cm−1 for 62, 63 and 64, respectively. The ground state KD is strongly anisotropic (gzz = 19.978 (62), 19.992 (63), 19.979 (64)) and is found to contain a dominant (99%) contribution from mJ = |±15/2⟩ level. The ground gzz axis is oriented along with the axial Dy–O bond or pseudo C 6 axis in all the complexes due to the stronger ligand field from Ph3 SiO groups compared to the equatorial LN6 ligand (Fig. 7.27). This is further corroborated by the ~1/4th of Loprop charge on the equatorial nitrogen donor atoms compared to the axial oxygen donors. The first (at 431, 446 and 467 cm−1 for 62, 63 and 64, respectively) and second excited KD (at 747, 791 and 840 cm−1 for 62, 63 and 64, respectively) is well described by mJ = |±13/2⟩ and |±11/2⟩, respectively. Although the first excited state is found to be axial in nature, a strong transverse anisotropy (62: gxx = 0.677, gyy = 1.923, gzz = 12.687; 63: gxx = 0.614, gyy = 1.695, gzz = 12.766; 64: gxx = 0.117, gyy = 0.731, gzz = 13.342) in the second excited KD favours the magnetisation relaxation via this state (Fig. 7.27). This leads to the U cal value of 747, 791 and 840 cm−1 for 62, 63 and 64, respectively, which is in close agreement with the experiment. A small discrepancy between U eff and U cal in 62 can be ascribed to the quite large QTM (Fig. 7.27). The larger U cal value of 63 and 64 compared to 62 can be explained by the Loprop charge of the axial oxygen donor centres, which is found to be ~0.3 higher in 63 and 64 compared to 62. The U cal value is also correlated to the B20 axial CF parameters were estimated to be −5.53, −5.93 and −6.45 in 62, 63 and 64, respectively. To check the effect of robust and rigid macrocyclic ligand in the magnetic anisotropy of 62–64, the equatorial LN6 ligand has been replaced by six -NH3 groups carving models 62a-64a. The ab initio calculations on models 62a-64a reveal relaxation of magnetisation via first excited KD, yielding an U cal value of 421, 437 and 482 cm−1 for 62a, 63a and 64a, respectively. The lower U cal value of these models

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Fig. 7.27 The molecular structure of a 62, b 63, c 64, d 65, e 66, f 67. Colour code: Dy—gold/ green, O—red, N—blue, Si—light turquoise, P—purple, C—grey, B—dark yellow. Hydrogens are omitted for clarity. Reprinted from Canaj et al. [49] with permission from John Wiley and Sons. Reprinted from Li et al. [99] with permission from John Wiley and Sons

compared to 62–64 is due to the stronger equatorial donation from the –NH3 groups than the LN6 ligand, creating larger transverse anisotropy. This unveils the importance of the macrocyclic ligand in weakening the equatorial ligand field, which in turn generates a huge magnetic anisotropy. Furthermore, the calculations on 62b64b models by removing the counter anions reveal similar U cal values compared to 62–64. This also indicates the role of macrocyclic ligand in minimising the effect of counter anions in magnetic anisotropy. Further, to investigate the effect of bulky axial ligand in magnetic anisotropy, the axial ligand of 64 has been replaced by F− ion, resulting in 64c. The calculations on 64c reveal relaxation of magnetisation via 2nd excited KD with the U cal value of 830 cm−1 . The similar U cal value of 64 and 64c unveils the crucial role of bulky axial ligand in magnetic anisotropy. The LN6 ligand in 62–64 is not rigid enough and is also non-planar in the equatorial plane. The non-planar nature of the LN6 ligand contributes to the transverse anisotropy in 62–64. To increase the rigidity, Zheng and co-workers used a π conjugated moiety (LE ) has reported three other air-stable hexagonal bipyramidal Dy(III) SIMs (Fig. 7.27), [Dy(LE )(PhO)2 ](BPh4 )·4THF (65), [Dy(LE )(4-MeOPhO)2 ](BPh4 )·3THF (66) and [Dy(LE )(naPhO)2 ](BPh4 )·4THF (67) [99]. The metal centre in all the complexes is found to coordinate with the six nitrogen atoms in the equatorial plane and two phenolic oxygen atoms in the axial position. The Dy– O bond lengths in 65–67 range from 2.09 to 2.11 Å, the shortest to date for any Dy(III) complexes with a coordination number greater than 7. The complexes 65, 66 and 67 possess nearly perfect hexagonal planes. This is further supported by

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the CShM analysis, which reveals symmetry to be closer to the D6h point group in 65–67 (complex 66 has a smaller CShM value) compared to 62–64 (Table 7.2). Further, the intramolecular (LE )C-H···π (axial aromatic ring) interactions have been detected in all the complexes. Magnetic studies reveal a large U eff of 765, 930 and 852 cm−1 for 65, 66 and 67, respectively. While complex 65 can retain magnetisation up to 2 K, the same is slightly higher for 66 and 67 (6 K). The improved magnetic behaviour of 65–67 compared to 62–64 can be ascribed to the former’s planarity of the conjugated equatorial ligand. The calculations reveal an Ising nature of anisotropy (gzz ~ 20) for the ground KD in all the complexes, and the computed χT perfectly matches with experiments. The gzz axis of 65–67 is directed via the pseudo C 6 axis due to the strong axial ligand field from phenolic oxygen and weak equatorial ligand field from LE (Fig. 7.27) ligand framework. The ground, first (at 409(65), 485(66) and 464(67) cm−1 ) and second excited KD (at 698(65), 844(66) and 804(67) cm−1 ) of 65–67 can be well described by almost pure mJ = |±15/2⟩, |±13/ 2⟩ and |±11/2⟩ states, respectively. The first and second excited state of 65–67 is also found to be Ising in nature (KD2: gzz = 17.27(65)17.31(66)17.30(67); KD3: gzz = 13.74(65)14.38(66)14.30(67)). Further, the deviation of the gzz axis (of first and second excited state) from the ground state is found to be 30°) of the gzz axis of the third excited state with respect to the ground state also favours the magnetisation relaxation via this state. This leads to the U cal value of 827, 1051 and 992 cm−1 for 65, 66 and 67, respectively and these values are slightly overestimated compared to the U eff values and were attributed to anharmonic phonons that promote the under-barrier relaxation process to lower the U eff values. The larger U cal value of 66 compared to 65 and 67 can be explained from structural aspects. The shorter Dy–O bond length (2.108 and 2.088 Å in 65 and 66, respectively) helps increase the axial ligand field, which generates the larger U cal value. Although axial Dy–O bond length is found to be similar in 66 and 67, the lower O–Dy–O angle (180° and 168.9° in 66 and 67, respectively) in 66 explains axial the lower U cal value estimated for 67 compared to 66. Further, the average equatorial Loprop charges are found to be in the order of 65 < 67 < 66, which also explains the trend observed in the U cal value from 65 < 67 < 66. The T B value of complexes 62–64 is found to be ~4 K as the molecular structure is severally distorted from D6h symmetry. On the other hand, complexes 65–67 also possess a very low T B of 6 K despite having a larger U eff /U cal value. It has been seen the flexible nature of equatorial LN6 ligands in 62–64 creates a distorted skeleton in these complexes. To circumvent this, Tang and co-workers have reported two optically pure air-stable hexagonal bipyramidal Dy(III) SIMs with more rigid macrocyclic ligand in equatorial position (Fig. 7.29a–b), namely RRRR-Dy-D6h F12 (68) and SSSS-Dy-D6h F12 (69) [100]. The synthetic procedure of the two enantiomers is similar to 64. The only change is in the slight modification of the LN6 ligand. The

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Fig. 7.28 The mechanism of magnetisation relaxation of complexes a 64, b 66

axial ligand and counter anions remain the same as in 64 and 68–69. The introduction of electron-withdrawing fluoride groups in the equatorial macrocyclic ligand not only reduces the equatorial ligand field but also increases the thermal stability of the complexes 68–69. The CShM analysis shows a slightly larger deviation from the ideal D6h polyhedron compared to 65–67. The complexes are found to have a relatively flat equatorial skeleton. The shorter axial Dy-O bond (2.122–2.147 Å) compared to the equatorial Dy–N bond (2.658–2.744 Å) indicates a strong axiality of this complex. This was confirmed by the magnetic measurements, which reveal a massive U eff value of 1274 and 1264 cm−1 for 68 and 69, respectively. Therefore, both the enantiomers reveal a similar magnetic anisotropy. Not only that, both of them show opening up of the hysteresis loop up to 20 K, the highest for a D6h symmetric SIM. Furthermore, a relaxation time up to 2497 s at 2.0 K was detected for these complexes, the highest known for any air-stable SIM. Replacing the fluoride ion by a bromide ion does not significantly change the structural parameters, hence the magnetic anisotropy. The ground state gzz axis is oriented along with the axial O–Dy–O bond due to the stronger ligand field from Ph3 SiO groups in the axial position (Fig. 7.29). The gzz axis of the first and second excited KD deviates by 2.2° and 4.9° from the ground state gzz axis. The calculated Loprop charge of the equatorial nitrogen atoms is found to be ~1/4th times lower than the axial oxygen atoms suggesting a strong axiality in these complexes. More importantly, the Loprop charges on the equatorial nitrogen atoms are found to be significantly smaller in 68– 69 compared to 64 due to the presence of electron-withdrawing fluoride ions in the former. The estimated B20 and B40 CF parameters are negative, and both of them are larger compared to the non-axial CF parameters. The ground state KD is perfectly axial (gxx = 0.000, gyy = 0.000, gzz = 19.899) and composed of 100% mJ = |±15/2⟩. The first KD at 551.7 (548.2) cm−1 for 68 (69) and the second excited KD at 1014.5 (1007.3) cm−1 for 68 (69) are also found to be axial (gzz = 16.96 and 14.02 in KD2 and KD3, respectively) in nature and contains a dominant (99% for KD2 and 97% for KD3) contribution from mJ = |±13/2⟩ and |±11/2⟩, respectively. The substantial mixing of mJ = |±9/2⟩ (~81%) with mJ = |±3/2⟩ (~13%) in the third excited KD

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where …

345

leads to a significant transverse anisotropy (68: gxx = 0.729, gyy = 1.148, gzz = 9.608, 69: gxx = 0.866, gyy = 1.202, gzz = 9.600) for magnetisation relaxation. This is confirmed by the significant TA-QTM of 0.32 μB in the third excited KD and also >10° deviation of the corresponding gzz axis with the ground state (Fig. 7.30a). Thus, the computed U cal value is estimated to be 1316.8 and 1306.3 cm−1 for 68 and 69, respectively, and these values are in agreement with U eff values. The slightly larger U cal value of 68 compared to 69 can be ascribed to the lower average equatorial Loprop charges (−0.325 vs. −0.327 in 68 vs. 69) in the former. Furthermore, the high thermal stability of these complexes hints at the tailoring of the apical ligand with surface binding groups, and this approach will provide a new avenue for the deposition of SIMs on surfaces. An ideal hexagonal bipyramidal geometry should reduce the QTM as the non-axial q 6 crystal field parameters Bk (k = 2,4,6; q /= ( 0) apart from ) B6 vanishes in D6h symmetry ( Hˆ C F = B20 Oˆ 20 + B40 Oˆ 40 + B66 Oˆ 66 + B66 Oˆ 66 + Oˆ 6−6 . From this viewpoint, Dunbar and co-workers have studied two pseudo hexagonal bipyramidal Dy(III) SIMs (Fig. 7.29c–d), [Dy(t Bu3 PO)2 (NO3 )3 ] (70) and [Dy(t Bu3 PO)2 (NO3 )3 ]0.0.5CH3 CN

Fig. 7.29 The main magnetic axis of a 68, b 69, c 70, d 71A. Colour code: Dy—olive green/ cyan, O—red, N—blue, C—black/grey, F—green, Si—orange, P—pink. Hydrogens are omitted for clarity. Reprinted with permission from Zhu et al. [100] Copyright@2021 American Chemical Society. Reprinted with permission from Li et al. [101] Copyright@2019 American Chemical Society

Fig. 7.30 The mechanism of magnetisation relaxation of a 68, b 70, c 71A

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S. Dey et al.

(71) which differs by surrounding solvent molecules [101]. Complex 71 possesses two independent molecules in the asymmetric unit called 71A and 71B. The metal centres in 70 and 71 are found to coordinate with t Bu3 PO groups in the axial position and three chelating nitrate groups in the equatorial position. The CShM analysis with {DyO8 } core reveals a larger deviation (0.636 (70), 1.108(71A), 1.237(71B)) in 71 compared to 70 from the ideal D6h polyhedron. The ~0.2 Å shorter axial Dy-O distance compared to the equatorial one indicates a compressed structure for magnetic axiality. The axial O–Dy–O angle is found to be larger in 71 (178.8°(71A), 176.8°(71B) compared to 70 (172.6°). The magnetic studies reveal a blocking barrier of 25.6–26.7 cm−1 in 70 and 30.2–33.6 cm−1 in 71 in an external field. The calculations reveal a well-separated ground state from the first excited state at 162.9(168.8) cm−1 in 71A(71B). The ground KD in 71 is found to be highly axial in nature (gxy ~ 10–3 ) and composed of mJ = |±15/2⟩. The magnetic axiality reduces in 70, where the ground KD possess strong transverse anisotropy (gxx = 0.455, gyy = 0.703, gzz = 17.900), resulting in the QTM of 0.19 μB (Fig. 7.30b). This rules out the possibility of observing a zero-field SIM behaviour, as evident from the experiments. The ground state gzz axis of 70 is oriented in the equatorial plane, while in 71, it is found to align along the axial O–Dy–O bond axis (Fig. 7.29c, d). With an external dc field, the QTM observed in 70 can be quenched, leading to a possible relaxation via the first excited KD (mJ = |±15/2⟩ →|±5/2⟩). This will result in the U cal value of 72.3 cm−1 , and this is ~50 cm−1 larger compared to the U eff value. For 71A and 71B, a significant TAQTM (3.3μB , Fig. 7.30) in the first exited KD (mJ = |±7/2⟩ (71A), |±9/2⟩ (71B)) favours the magnetisation relaxation and results in the U cal value of 162.9(168.8) cm−1 in 71A(71B). The overestimation of the U cal compared to the U eff value in 70–71 suggests an under-barrier relaxation process via low-energy local vibrations. The computed axial B20 (−1.56) and non-axial B22 (1.50) crystal field parameters are similar to 55. On the other hand, the B20 parameters are ~10 times larger than the B22 in 71, and this indicates the uniaxial anisotropy. To verify whether the distortion from the ideal D6h polyhedron leads to poor magnetic behaviour in 70–71, ab initio calculations have been performed on models wherein the equatorial nitrate groups are replaced with 18-crown-6 ligand and the axial t Bu3 PO group with Me3 PO group. The calculations reveal almost pure stabilisation of mJ = |±15/2⟩, |±13/2⟩ and |±11/2⟩ in the ground, first (400 cm−1 ) and second excited KD (at 707.9 cm−1 ), respectively. All these states possess very small transverse anisotropy and suggest magnetisation relaxation via other higher excited states. This finding unveils that D6h symmetry is a viable one for targeting the high-performance SIM (Table 7.3).

7.4.6 Eight Coordinate Dy(III) SIM with Pseudo D4d Point Group Compared to D4h , D4d symmetric SIMs have been less explored due to the small axial anisotropy originates from the diminished axial B20 CF parameters. The first D4d SIM, [DyPc2 ]− (72) (Pc = phthalocyanine) discovered by Ishikawa et al. in 2003, shows an mJ = |±13/2⟩ ground state despite possessing a uniaxial crystal

D6h

D6h

D6h

D6h

[Dy(LN6 )(Ph3 SiO)2 ](PF6 ) (63)

[Dy(LN6 )(Ph3 SiO)2 ](BPh4 ) (64)

[Dy(LE )(PhO)2 ](BPh4 )·4THF (65)

[Dy(LE )(4-MeO-PhO)2 ](BPh4 )·3THF

1.332

D6h

D6h

D6h

SSSS-[Dy(F12 LN6 )(Ph3 SiO)2 ][BPh4 ] (69)

[Dy(t Bu3 PO)2 (NO3 )3 ] ( 70 )

[Dy(t Bu3 PO)2 (NO3 )3 ]0.0.5CH 3 CN (71) 1.108–1.237 –

–

–

–

–

–

–

–

–

–

–

–

20

20

6

6

2

–

–

–

–

–

0.02

0.02

0.012

0.012

0.012

–

–

–

30–34

26–27

1264

1274

852

930

765

781

751

676

163–168

72

1306

1317

992

1051

827

840

791

747

[101] [101]

19.90 4th 19.90 4th 17.90 2nd

−8.56 −1.56

−1.01-(−1.10) 19.60 2nd

[100]

[100]

19.99 4th −19.75

[99]

[99]

[99]

[49]

[49]

[49]

–

19.99 4th

−6.54

19.98 3rd

−6.45

19.99 4th

19.99 3rd

−5.93 –

19.98 3rd

gzz of Relaxation Refs. KD1 via KD

−5.53

FC/ T H (K) Sweep U eff (cm−1 ) U cal (cm−1 ) B20 ZFC rate TB (K) (T/ min)

H corresponds to the temperature below which the opening of magnetic hysteresis is observed The Italic Emphasised coded texts represent field-induced SIMs

*T

1.338

0.636

1.249

D6h

1.028

1.069

2.163

2.271

RRRR-[Dy(F12 LN6 )(Ph3 SiO)2 ][BPh4 ] D6h (68)

[Dy(LE )(naPhO)2 ](BPh4 )·4THF (67)

(66)

D6h

[Dy(LN6 )(2,4-di-t Bu-PhO)2 ](PF6 ) (62) 2.472

Point CShM Group

Complex

Table 7.3 The summary of the computed and experimental parameters that are relevant to the performance of hexagonal bipyramidal SIMs

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where … 347

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field [37]. This yields a very small U eff (28 cm−1 ) for magnetisation relaxation. After that, Murray and co-workers have assumed that a chelating ligand like βdiketone should be ideal for preparing this D4d symmetric complex due to the oxophilic nature of Dy(III) ions. Further, ionic radii of Dy(III) suits perfectly for the binding with β-diketone type ligands. Based on this idea, they have reported one D4d symmetric Dy(III) SIM; [Dy(paaH)2 (H2 O)4 ][Cl]3 .2H2 O (73, paaH = N-(2-pyridyl)-ketoacetamide, Fig. 7.31) [102]. The complex is reported to have slow relaxation of magnetisation with an U eff of 123 cm− 1 . The CShM analysis with {DyO8 } core reveals square antiprismatic geometry around the metal centre (Table 7.4). However, the metal centre binds with two zwitterionic paaH ligands in the axial position and four water molecules in the equatorial position. It results in strong magnetic anisotropy as the π electron cloud of the zwitterionic ligand above and below the XY plane, which creates a large electrostatic repulsion to stabilise the oblate electron density of Dy(III) ion (Fig. 7.31b). Calculations reveal that eight KDs span in the range of 801.6 cm−1 for 73. The ground state gzz axis of 73 is pointing towards the β-diketonate ligand in the axial position to minimise the repulsion with the oblate electron density (Fig. 7.31a). The ground KD of 73 is Ising in nature (gzz = 19.61) and composed of pure mJ = |±15/2⟩. The principal axis of the first three KDs of 73 are nearly co-linear, but the significant transverse anisotropy (gxx = 0.05, gyy = 0.07, gzz = 16.33) in the first excited KD at 249.2 cm−1 suggests magnetisation relaxation via this state. It results in the U cal value of 249.2 cm−1 for 73. Inspired by the above study, Pardo and co-workers have introduced electrondonating groups in the β-diketonate ligand (tmhd, 2,2,6,6-tetramethyl-3,5heptanedione) to increase the axial ligand strength [103]. With this substitution, they have reported four D4d symmetric Dy(III) SIMs (Fig. 7.32); [Dy(tmhd)3 (Br2 bpy)] (74, Br2 -bpy = 5,5’-dibromo-2,2’-bipyridine), [Dy(tmhd)3 (Br-bpy)] (75, Br-bpy = 5-bromo-2,2’-bipyridine), [Dy(tmhd)3 (dppz)] (76, dppz = dipyrido [3,2-a:2’,3’-c]phenazine) and [Dy(tmhd)3 (mcdpq)] (77, mcdpq = 2-methoxyl-3cyanodipyrido[3,2-f:2,3’-h]quinoxaline)) [103]. In all the complexes, the Dy(III)

Fig. 7.31 a The main magnetic axis of 73. Colour code: Dy—green, O—red, N—blue, C—grey, H—white, b The schematic representation of the oblate electron density of 73. Reprinted from Chilton et al. [102] with permission from the Royal Society of Chemistry

0.700 0.511

D4d D4d D4d D4d D4d

D4d D4d D4d D4d

[Dy(tmhd)3 (dppz)] ( 76 )

[Dy(tmhd)3 (mcdpq)] ( 77 )

[DyLz2 (o-vanilin)2 ].Br.methanol {methanol = solvent} (78)

[DyLz2 (o-vanilin)2 ].NO3 .methanol (79)

[DyLz2 (o-vanilin)2 ].CF3 SO3 .methanol(80) D4d D4d

[Dy(tmhd)3 (Br-bpy)] ( 75 )

[Dy(bpy)(tffb)3 ](C 4 H 8 O2 )1/3 ( 81 )

[Dy(Phen)(tffb)3 ] ( 82 )

[Dy(bpy)(tfmb)3 ] ( 83 )

[Dy(bpy)(tfmb)3 ]0.0.5C 4 H 8 O2 (84)

[Dy(bbpen)(tpo)2 ][BPh4 ] (85)

1.396

0.557

0.834

1.344

1.018

0.541

0.670

0.768

0.646

7

–

–

–

–

–

–

–

–

–

–

–

–

–

6

–

–

–

–

6

7

6

0.02

–

–

–

–

–

–

–

–

–

–

–

–

–

656

100

41

64

61

84

484

154

35

172

156

143

123

28

H corresponds to the temperature below which the opening up of magnetic hysteresis is observed The Italics Emphasised coded texts represent field-induced SIMs

*T

0.855

D4d

[Dy(tmhd)3 (Br 2 -bpy)] ( 74 )

0.338

D4d

–

D4d

[Dy(paaH)2 (H2 O)4 ][Cl]3 .2H2 O (73)

671

122

43

92

130

181

471

346

153

155

151

127

249

–

–

[105] [106]

[105]

[105]

[105]

[104]

[104]

[104]

[103]

[103]

[103]

[103]

[102]

[37]

19.51 2nd

18.32 2nd

19.03 2nd

19.38 2nd

19.68 2nd

19.80 3rd

19.72 3rd

19.40 2nd

19.68 2nd

19.54 2nd

19.26 2nd

19.68 2nd

–

gzz of Relaxation Refs. KD1 via KD

−10.32 19.83 4th

–

–

–

–

–

–

–

–

–

–

–

–

–

Symmetry CShM FC/ T H (K) Sweep U eff (cm−1 ) U cal (cm−1 ) B20 ZFC rate T B (K) (T/ min)

[DyPc2 ]− (72)

Complex

Table 7.4 The summary of the computed and experimental parameters that are relevant to the performance of square antiprismatic SIMs

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where … 349

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S. Dey et al.

ion coordinates with six oxygen atoms from the three tmhd ligands and two nitrogen atoms from the N-capped co-ligands. The average Dy–O and Dy–N bond lengths are found to be ca. 2.3 Å and 2.6 Å, respectively, in these complexes. The CShM analysis with {DyO6 N2 } core reveals square antiprism geometry around the metal centre in all the complexes. All the complexes show QTM in the zero-field but differ in terms of barrier height due to the different substitutions in the N-capping co-ligands. The magnetic measurement unveils slow relaxation of magnetisation with the U eff value of 29(74), 43(75), 54(76) and 2 (77) cm−1 obtained in the absence of static field and U eff of 143(74), 156(75), 172(76) and 35 (77) cm−1 are obtained in an optimum field of 1500 Oe (1200 Oe for 76 and 77). The calculated gzz axis of the ground state of 74–77 is nearly identical and oriented along the face of four oxygen atoms from two tmhd ligands above and below the xy plane (Fig. 7.32). The gzz value of KD1 is close to 20, suggesting the Ising nature and uniaxial anisotropy in these complexes. It is reflected in the computed QTM, which ranges in the order of 10–2 →10−3 μB in contrary to 10–4 →10−5 observed in zero-field SIM. The relatively larger value (ca. 10–1 μB ) in 77 indicates a larger possibility of QTM in this complex. The ground and first excited states in 74–77 are well described by mJ = |±15/2⟩ and |±13/2⟩ states, respectively. However, the strong TA-QTM in the first excited KD favours magnetisation relaxation via this state. It results in the U cal values of 127, 151, 155 and 153 cm−1 for 74–77, respectively, which agrees well with the U eff in the presence of dc field obtained for complex 77. Although the U cal value of 77 is large, a stronger QTM in the ground state hinders

Fig. 7.32 The ab initio computed main magnetic axis of a 74, b 75, c 76, d 77. Colour code: Dy—yellow, O—red, N—blue, C—grey. Hydrogens are omitted for clarity. Reprinted from Cen et al. [103] with permission from John Wiley and Sons

7 Ab Initio Modelling of Lanthanide-Based Molecular Magnets: Where …

351

it from attaining the largest blocking barrier. The unfavourable geometry of 77 (the smallest CShM value from ideal D4d geometry in the complex) and the presence of weak intermolecular π···π interactions might be the reason for stronger transverse anisotropy. In order to increase the blocking barrier, Tang and co-workers have reported a series of mononuclear Dy(III) SIM; [DyLz2 (o-vanilin)2 ].X.solvent (Lz = 6-pyridin2-yl-[1,3,5]triazine-2,4-diamine; X = Br− (78), NO3 − (79), CF3 SO3 − (80)), with a pseudo D4d symmetry (Fig. 7.33) [104]. The Dy(III) ion resides in an {N4 O4 } coordination environment with a square antiprismatic geometry in all the complexes. The four-nitrogen donor centres come from two Lz ligands, and the four oxygen donor centres come from two o-vanilin ligands. The ligands are arranged in such a way that it forms a sandwich-type complex around Dy(III) ion. However, a subtle change in the counter anion leads to a great difference in magnetic properties in these complexes. Complex 79 shows retention of magnetisation up to 7 K with a very high U eff of 484 cm−1 , which is notably larger than 78 and 80. Calculations reveal a dominant mJ = |±15/2⟩ ground state in all the complexes, along with negligible transverse anisotropy in-ground state KD (Fig. 7.33). The gzz axis is found to orient along the highest order C 4 axis in all the complexes (Fig. 7.33). The complex 78, 79 and 80 relaxes via the second, second and first excited KDs, respectively, due to a significant transverse anisotropy which promotes substantial TA-QTM value for magnetisation relaxation (78: gxx = 0.09, gyy = 0.10, gzz = 13.24 in KD3; 79: gxx = 0.18, gyy = 0.24, gzz = 13.92 in KD3; 80: gxx = 0.20, gyy = 0.21, gzz = 16.43 in KD2). Furthermore, the second excited state (dominant mJ = |±11/2⟩) reveals strong mixing of mJ states (first excited state for 80, Fig. 7.33). However, the gzz axis in the first and second excited shows a small deviation (