Magnetic Anisotropy of Transition Metal Complexes Nicholas F

Magnetic Anisotropy of Transition Metal Complexes Nicholas F. Chilton

A thesis submitted to e University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences

2015 School of Chemistry e University of Manchester

Contents

Contents ...... 3

List of abbreviations ...... 5

Abstract ...... 7

Declaration ...... 9

Acknowledgements ...... 13

1. Preface ...... 15

2. Introduction ...... 17

Rationale for the alternative format and organization of thesis ...... 17

Brief of included works and roles of authors ...... 17

Magnetically anisotropic materials ...... 21

3. The electronic and magnetic properties of molecules ...... 23

Magnetic properties of the electron ...... 23

Quantum mechanics and the wavefunction ...... 24

A single electron: the hydrogen atom ...... 26

Multi-electron atoms ...... 28

Magnetic and relativistic effects: the Dirac equation ...... 29

Energy scales ...... 30

Linear algebra ...... 32

Perturbation theory ...... 34

The spherical harmonics ...... 36

More than one way to skin a cat ...... 37

The parametric approach: angular momentum in quantum mechanics ...... 38

The ab initio approach ...... 42

Connection of the ab initio and parametric approaches ...... 45

Experimental observables ...... 47

Conclusion ...... 48

References ...... 48

4. Paper one: “On the Possibility of Magneto-Structural Correlations: Detailed Studies of

Dinickel Carboxylate Complexes” ...... 51

5. Paper two: “Large Zero-Field Splittings of the Ground Spin State Arising from

Antisymmetric Exchange Effects in Heterometallic Triangles” ...... 65

6. Paper three: “An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes” ...... 79

7. Paper four: “The first near-linear bis(amide) f-block complex: a blueprint for a high temperature single molecule magnet” ...... 111

8. Paper five: “Design Criteria for High-Temperature Single-Molecule Magnets” ...... 131

9. Paper six: “Direct measurement of dysprosium(III)···dysprosium(III) interactions in a single-molecule magnet” ...... 149

10. Conclusion ...... 185

11. Outlook ...... 187

12. Appendix: Further Ph.D. publications ...... 189

Total word count: 55,000

List of abbreviations

AO Atomic Orbital

BO Born-Oppenheimer

CASPT2 Complete Active Space Second Order Perturbation Theory

CASSCF Complete Active Space Self-Consistent Field

CF Crystal Field

CI Configuration Interaction

EPR Electron Paramagnetic Resonance

HF Hartree-Fock

LCAO Linear Combination of Atomic Orbitals

MCSCF Multi-Configurational Self-Consistent Field

MF Mean Field

MO Molecular Orbital

NEVPT2 N-Electron Valence State Perturbation Theory

PT Perturbation Theory

QM Quantum Mechanics

RASSCF Restricted Active Space Self-Consistent Field

RASSI Restricted Active Space State Interaction

RS Russell-Saunders

SCF Self-Consistent Field

SMM Single Molecule Magnet

SOC Spin-Orbit Coupling

ZFS Zero Field Splitting

Abstract

Magnetic Anisotropy of Transition Metal Complexes: a thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences.

The study of magnetic anisotropy in molecular systems permeates the physical sciences and finds application in areas as diverse as biomedical imaging and quantum information processing. The ability to understand and subsequently to design improved agents requires a detailed knowledge of their fundamental operation. This work outlines the background theory of the electronic structure of magnetic molecules and provides examples, for elements across the Periodic Table, of how it may be employed to aid in the understanding of magnetically anisotropic molecules.

II III II The magnetic anisotropies of a series of dimetallic Ni 2 complexes and a Ru 2Mn triangle are determined through multi-frequency Electron Paramagnetic Resonance (EPR) spectroscopy and ab initio calculations. The magnetic anisotropy of the former is found to be on the same order of magnitude as the isotropic exchange interactions, while that of the latter is found to be caused by large antisymmetric exchange interactions involving the RuIII ions.

An intuitive electrostatic strategy for the prediction of the magnetic anisotropy of DyIII complexes is presented, allowing facile determination of magnetic anisotropy for low symmetry molecules.

Through the presentation of the first near-linear pseudo-two-coordinate 4f-block complex, a new family of DyIII complexes with unprecedented Single Molecule Magnet (SMM) properties is proposed. Design criteria for such species are elucidated and show that in general any two-coordinate complex of DyIII is an attractive synthetic target.

The exchange interaction between two DyIII ions is directly measured with multi- frequency EPR spectroscopy, explaining the quenching of the slow magnetic relaxation in the pure species compared to the SMM properties of the diluted form. The interpretation of this complex system was achieved with supporting ab initio calculations.

Nicholas F. Chilton

February 2015

Declaration

No portion of the work credited to the author in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning.

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University’s policy on Presentation of Theses.

Acknowledgements

Firstly I would like to thank The University of Manchester for a President’s Doctoral

Scholarship, which has given me the opportunity to perform this work. Furthermore, none of this would have been possible without the support, assistance and advice from my supervisors Prof. Eric McInnes and Prof. Richard Winpenny. I am also extremely grateful for all the sage wisdom provided by Prof. David Collison.

A large debt of thanks is owed all my co-authors and all those in the Molecular

Magnetism group, but specifically to Dr. Andrew Kerridge, Dr Richard Layfield, Dr.

Samantha Magee, Dr. David Mills, Dr. Eufemio Moreno Pineda, Dr. Thomas Pugh, Dr.

Alessandro Soncini, Dr. Floriana Tuna and Dr. James Walsh for all their hard work and advice.

I would also like to acknowledge Prof. Keith Murray, without whom I would never have been here in the first place.

Enormous thanks are due to my friends and family for all their love, support and encouragement over the years. Finally and especially, all my love and thanks to Rachael for always being by my side and for understanding.

1. Preface

The development of parametric models during the 1950’s to the 1970’s for the interpretation of magnetic and spectroscopic data of high symmetry transition metal sites relied on a sound understanding of Quantum Mechanics (QM) and group theory.

These models, largely based around Crystal Field (CF) theory, were very successful in describing the experimental results in a rigorous theoretical framework. The renaissance of 4f-based complexes in molecular magnetism in the new millennium brought with it the use of such parametric CF approaches to model the magnetic data and to understand the electronic structure. However, many of these works were riddled with oversimplifications based on ‘pseudo-symmetric’ environments which were in fact devoid of any symmetry elements. These inappropriate symmetry restrictions resulted in qualitatively incorrect conclusions in many cases, contributing no insight into the origin of the magnetic anisotropy. It was not until the arrival of sophisticated ab initio computational approaches that the treatment of low symmetry lanthanide complexes was remedied. However, unsurprisingly, these complex calculations provided complex answers which were unable to be simply understood.

Despite the plethora of dubious magnetic interpretations, much progress was made in enhancing the barrier to magnetic reversal in lanthanide-based SMMs. However, after the initial flood of new and interesting results, there was no clear direction for how more substantial progress could be made. This stemmed from a lack of understanding of the origin of the magnetic anisotropy in these systems and how it could be controlled.

While it was largely understood that rigorous high symmetry would define and allow prediction of magnetic anisotropy, there was no real grasp on how to engineer the magnetic anisotropy in the much more common cases of low symmetry.

The work presented herein provides some answers to the challenges posed above and begins to suggest how the magnetic anisotropy can be controlled in low symmetry environments.

2. Introduction

Rationale for the alternative format and organization of thesis

The work undertaken during the course of this Ph.D. was of a topical nature which required rapid dissemination in peer-reviewed literature. Therefore all the research contained in this thesis is presented as published peer-reviewed articles.

Following a brief introduction to magnetic anisotropy in molecular magnetism, a detailed, but far from comprehensive, text on the electronic and magnetic properties of molecules is presented. Both the parametric and ab initio approaches to electronic structure are expounded and this text could be considered a primer for students starting a course of study in theoretical molecular magnetism.

The research studies of this Ph.D. have involved numerous research collaborations and investigations in different aspects of molecular magnetism. In order to tell a coherent and compelling story, six key works highlighting the study of magnetic anisotropy across the Periodic Table are presented in a non-chronological manner. Other publications containing work performed during this Ph.D. can be found as citations in the appendix and it is anticipated that further publications resulting from original research during this Ph.D. will be forthcoming.

Brief of included works and roles of authors

Paper one: “On the Possibility of Magneto-Structural Correlations: Detailed Studies of

Dinickel Carboxylate Complexes”.

J. P. S. Walsh, S. Sproules, N. F. Chilton, A.-L. Barra, G. A. Timco, D. Collison, E. J. L.

McInnes and R. E. P. Winpenny, Inorg. Chem., 2014, 53, 8464.

This paper highlights the difficulty of determining the magnetic parameters of even seemingly simple complexes. The study of a closely related family of dimetallic complexes finds that the Zero Field Splitting (ZFS) of the = 1 NiII ions is on the same

푆 17 order of magnitude as the magnetic interactions between them, greatly complicating the analysis. However, magneto-structural correlations between the Ni-O(H2)-Ni bridging angle and the magnetic exchange interaction are revealed.

JPSW synthesized all the complexes (with assistance from GT) and performed the magnetic and EPR measurements (with assistance from SS). High frequency EPR measurements were performed by ALB (with assistance from JPSW). NFC estimated the single-ion magnetic parameters for each dimer, through the use of ab initio calculations, providing a good starting point for the spectral simulations. JPSW and NFC modelled the EPR spectra and the magnetic data. The manuscript was written by JPSW with assistance from the other authors.

Paper two: “Large Zero-Field Splittings of the Ground Spin State Arising from

Antisymmetric Exchange Effects in Heterometallic Triangles”.

S. A. Magee, S. Sproules, A.-L. Barra, G. A. Timco, N. F. Chilton, D. Collison, R. E. P.

Winpenny and E. J. L. McInnes, Angew. Chem. Int. Ed., 2014, 53, 5310.

This paper elucidates the anomalous magnetic anisotropy in the ground state of a

III carboxylate bridged, oxo-centred Ru2Mn triangle. As the Ru ions are strongly antiferromagnetically coupled, the spin ground state of the molecule is owed to the

= 5 2 state of MnII. However, the magnetic anisotropy associated with the spin ground푆 ⁄ state is much too large to originate from manganese alone and was determined to arise through significant antisymmetric exchange interactions with the RuIII ions.

SAM synthesized the complex (with assistance from GT) and performed the magnetic and EPR measurements (with assistance from SS). High frequency EPR measurements were performed by ALB (with assistance from SAM). NFC developed an analytical antisymmetric exchange model and applied it to the coupled spin eigenstates of the isotropic exchange Hamiltonian, deriving an expression for the ground state axial ZFS

18 parameter with second-order perturbation theory. The manuscript was written by

SAM, SS, NFC퐷 and EJLM.

Paper three: “An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes”.

N. F. Chilton, D. Collison, E. J. L. McInnes, R. E. P. Winpenny and A. Soncini, Nat.

Commun., 2013, 4, 2551.

This paper develops an intuitive and simple model to explain and to predict the magnetic anisotropy of low-symmetry DyIII complexes. The aspherical electron density

III distribution of the = ± 15 2 state of Dy , derived from the / spin-orbit 6 multiplet, is employed푚 퐽in conjunction⁄ with a minimal valence bond model퐻15 for2 the partial charges of the ligands and determines the orientation of the main magnetic axis by electrostatic optimization.

NFC developed the idea for this approach in concert with AS. NFC performed all ab initio calculations, wrote a program for the electrostatic analysis and subsequently performed all electrostatic calculations. The manuscript was written by NFC with assistance from the other authors.

Paper four: “The first near-linear bis(amide) f-block complex: a blueprint for a high temperature single molecule magnet”.

N. F. Chilton, C. A. P. Goodwin, D. P. Mills and R. E. P. Winpenny, Chem. Commun.,

2015, 51, 101.

This paper presents the first near-linear pseudo-two-coordinate f-block complex. The near-linear N-Sm-N coordination mode is the ideal environment to exploit the oblate electron density distribution of the large angular momentum states of DyIII and the

i + proposed complex [Dy{N(Si Pr3)2}2] is predicted to have a barrier to magnetization

19 reversal of 1800 . It is predicted that such a complex would display −1 magnetic hysteresis푈푒푓푓 ≈ at temperatures푐푚 above that of liquid nitrogen.

CAPG synthesized the complex (with assistance from DPM) and performed all characterizations. NFC identified the potential magnetic properties of the linear motif and performed all ab initio and CF calculations. The manuscript was written by NFC with assistance from DPM and REPW.

Paper five: “Design Criteria for High-Temperature Single-Molecule Magnets”.

N. F. Chilton, Inorg. Chem., 2015, 54, 2097.

This paper elucidates the requirements for two-coordinate DyIII complexes to show large barriers to magnetization reversal. The work reveals that near-linearity is not a requirement and that any two-coordinate complex of DyIII should show a large barrier to magnetization reversal. Furthermore, the effect of coordinating solvent is shown to be a major detriment to the relaxation barrier.

The entire paper was the work of NFC.

Paper six: “Direct measurement of dysprosium(III)···dysprosium(III) interactions in a single-molecule magnet”.

E. Moreno Pineda, N. F. Chilton, R. Marx, M. Dörfel, D. O. Sells, P. Neugebauer, S.-D.

Jiang, D. Collison, J. van Slageren, E. J. L. McInnes and R. E. P. Winpenny, Nat.

Commun., 2014, 5, 5243.

This paper is the first direct spectroscopic determination of a dysprosium-dysprosium exchange interaction in a coordination complex. It is shown for this asymmetric dimetallic complex that one pocket behaves as a SMM, while the other does not, and that the interaction between the two metals is responsible for quenching the SMM behaviour in the pure complex.

The synthesis, crystallographic characterization and magnetic data collection and processing for all complexes were performed by EMP. Low-frequency EPR data were collected jointly by DOS and EMP. FIR and HF-EPR measurements were performed by

EMP, RM, MD, PN, S-DJ and JvS. NFC performed ab initio calculations for the two dysprosium sites, providing a starting point for the model. EMP and NFC performed fits of the magnetic data as well as simulations of the EPR spectra. The paper was written by EMP, NFC, EJLM and REPW with input from the other authors.

Magnetically anisotropic materials

The study of magnetic anisotropy in transition metal complexes is pertinent to many areas of science, with applications reaching from biomedical imaging to quantum information processing. The magnetic anisotropy of molecular complexes is distinct from the collective anisotropy of ordered ferromagnets and is an intrinsic feature of the molecule. Magnetic anisotropy arises in molecules which possess unquenched orbital angular momentum and therefore large anisotropy is most commonly associated with open shell transition metal complexes. Even if such metal complexes have orbitally non- degenerate ground states, mixing of excited orbitally degenerate configurations into formally spin-only ground states via Spin-Orbit Coupling (SOC) can produce non- negligible magnetic anisotropy in the ground state.

The origin of the magnetic anisotropy in open shell transition metal complexes is the interaction of the coordinating ligands with the orbital magnetic moment, and is therefore directly connected to the spatial arrangement and symmetry of the coordination environment. The magnetic anisotropy results in a physical localization of the magnetic moment and is most apparent in complexes with degenerate orbitals, such as complexes of the d- and f-block. While not directly influenced by the environment, the spin magnetic moment is coupled to the orbital magnetic moment via SOC and therefore the magnetic anisotropy can be enhanced when this relativistic effect is large.

The success of inorganic coordination chemistry in being able to design molecular

21 architectures around metal ions provides a direct handle for tailoring the magnetic anisotropy, thus facilitating the wide range of applications which often require very different magnetic properties.

The difficulty in determining the nature of the magnetic anisotropy varies for different blocks of the Periodic Table. For example, the magnetic anisotropy and exchange interactions can often be of the same order of magnitude for compounds of 3d ions, therefore untangling the two becomes difficult, while for complexes of the 4f ions, the magnetic anisotropy is much more sensitive to the environment. Compounds employing the 4d, 5d and 5f ions often possess magnetic anisotropy and magnetic exchange interactions that are of the same order as other electronic interactions, making the determination of the magnetic properties even more challenging.

The unravelling of the magnetic properties and the understanding of the magnetic anisotropy, while at times a very challenging task, is a requirement for the in-depth understanding of magnetic phenomena. It is only with a complementary suite of experimental techniques and some theoretical insight that the true nature of complex magnetic problems can be revealed. The connection between experiment and theory can be elegantly broached by interpretation of the magnetic properties with a simple parametric model. It will be the subject of this text to outline the basic theory behind the electronic properties of magnetic molecules and present some selected research highlights to show the application of such approaches to complexes of the 3d ions, those of mixed 3d/4d ions, and some of 4f ions. This work shows the great promise held by such approaches to further the fundamental understanding of magnetic interactions in molecules, and provides some insight for how more interesting magnetic molecules may be designed.

3. The electronic and magnetic properties of molecules

Magnetic properties of the electron

e traditional picture of electrons orbiting a positively charged nucleus is an amazingly intuitive picture of the atom, Figure 1. Although the innitely localizable formalism of classical mechanics fails at length scales such as those of the atom or molecule, this classical picture is a useful point of departure for QM. e rst step into the quantum realm is the acknowledgement that this classical picture is inherently awed; radial acceleration due to the circular motion of such an orbit would cause the electron to emit radiation, thus losing energy and eventually spiralling in towards the nucleus, leading to the collapse of matter. ankfully, this catastrophe is prevented by QM due to the quantization of the orbital states, allowing only certain congurations and transitions.

Indeed, the classical concept of simultaneous knowledge of position and momentum is juxtaposed by QM, which is inherently probabilistic in nature. In place of accurate position and momentum, QM dictates dened quantum states which have certain probabilities of possessing given momentum or position.*

l ⇀

+ r ⇀- p ⇀

Figure 1 | Traditional circular electron orbit around the nucleus.

* When measured however, particles can be found to have a given position and momentum simultaneously, which, according to the Copenhagen interpretation of QM, is a deterministic result due to the collapse of the wavefunction under the inuence of the measurement device. 23

Suppose for a moment, however, that accurate knowledge of the position, , and momentum, , of an electron as it orbits the nucleus are available. With this 푟⃑can be associated a classical푝⃑ angular momentum, , as described by (1), where the cross product indicates that the angular momentum 푙⃑vector is perpendicular to both the radial direction and the direction of orbital motion of the electron, Figure 1.

= × (1)

The orbital motion of the electron could푙⃑ also푟⃑ 푝be⃑ considered as a current loop, which generates a magnetic field owing to the Biot-Savart law. Indeed, this magnetic field is also perpendicular to the plane in which the electron orbits and defines a deep connection between angular momentum and magnetism. In the context of a single electron, this is referred to as the orbital magnetic moment which arises from the physical orbital angular momentum of the electron in three-dimensional space. There exists an entirely separate,* additional, form of angular momentum associated with the electron, which is referred to as the ‘spin’ angular momentum. It is a purely quantum property with no classical analogue and resides in an isolated ‘spin space’. Electrons are

= 1 2 particles which may possess either the = ± 1 2 quantization. One should

푠disregard⁄ any association between the spin degree푚푠 of freedom⁄ and a physical spinning motion – the electron cannot be thought to have any axis around which to spin.

The reader should rest assured that the existence and quantization of both the orbital and spin angular momentum are experimentally verified and in fact belong to a group of core observations that led to the inception of QM.

Quantum mechanics and the wavefunction

The wavefunction is an entity that encapsulates all the information about a given particle or group of훹 particles. It is generally a complex valued function of the spatial and spin coordinates of each particle, whose squared magnitude = | | is associated ∗ 2 훹 훹 훹 * The spin is separate in its native state. Under the influence of spin-orbit coupling these two forms of angular momentum cannot be separated. See Magnetic and relativistic effects: the Dirac equation. 24 with the probability distribution of the particle(s).* Given that there is certain probability of finding the particle(s) somewhere in space, the integral of the probability distribution over all coordinates is defined as unity, (2). A key concept is that of an operator, denoted by , which is the mathematical representation of a measurement in the framework of QM.� Wavefunctions that are not altered by a particular operator are said to be eigenfunctions of that operator, and the action of the operator is to reveal a characteristic value of the wavefunction, known as an eigenvalue. These kinds of equations are known as eigenvalue equations and, while not limited to QM, the fundamental eigenvalue equation of QM is Schrödinger’s equation, (3),1 where the

Hamiltonian operator measures the total energy of the wavefunction. To calculate the expectation value 퐻(eigenvalue)� of any operator,퐸 (3) can be pre-multiplied by the complex conjugate and integrated over all coordinates, (4). While the integral ∗ notation of Schrödinger훹 is more traditional, this text will employ the more compact ‘bra-ket’ notation of Dirac, where the ket | is equivalent to the wavefunction and the bra | is its complex conjugate – integration훹⟩ over all variables is implied훹 in the ∗ combination⟨훹 of bra and ket.2 훹

ö : = 1 : | = 1 (2) ∗ 푆푐ℎ푟 푑푖푛푔푒푟 � 훹 훹 푑휏 퐷푖푟푎푐 ⟨훹 훹⟩ 휏 ö : = : | = | (3)

푆푐ℎ푟 푑푖푛푔푒푟 퐻�훹 퐸훹 퐷푖푟푎푐 퐻� 훹⟩ 퐸 훹⟩

ö : = : = (4) ∗ 푆푐ℎ푟 푑푖푛푔푒푟 � 훹 퐻�훹푑휏 퐸 퐷푖푟푎푐 �훹�퐻��훹� 퐸 The absolute positions, velocities휏 or time evolution of the electrons are not of concern for molecular systems. Rather, the interest lies in the bound states of the molecule where the electrons are in a steady-state and the wavefunction can be considered as a standing wave. In such cases the Hamiltonian takes the form (5),3,4 which is constructed to

* Recall that for any complex number = + , the complex conjugate is = and that = ( )( + ) = + = | | . ∗ ∗ 2 2 2푧 푎 푖푏 푧 푎 − 푖푏 푧 푧 푎 − 푖푏 푎 푖푏 푎 푏 푧 25 contain all the interactions involved in the system, comprising the kinetic and potential energy of each particle, and , respectively.*

푃�푖 푉�푖

= 푁 + = 푁 + (5) 2 2 4 2 −ℏ 2 푒 푞푖푞푗 푖 푖 �푖 퐻� ��푃� 푉� � � � 푖 훻 � 0 푖푗 � 푖=1 푖=1 푚 푗≠푖∈푁 휋휀 푟

| = | | (6)

푒 푛 Although the discussion thus far has 훹focussed⟩ 훹 ⟩on훹 electrons,⟩ nuclei are also described by the Hamiltonian (5) and the nuclear wavefunction does indeed exist. However as the rest mass of nucleons are three orders of magnitude larger than that of the electron

( 1840 ) their kinetic energy and magnetic moment are three orders of magnitude푚푝 ≈ 푚푛 smaller.≈ Therefore푚푒 the total wavefunction of an atom or molecule is assumed to be separable into an electronic and nuclear part, (6); this is known as this is the Born-

Oppenheimer (BO) approximation.5 This separation allows the solution for the electronic wavefunction only, where the nuclei are considered as fixed point charges.

A single electron: the hydrogen atom

In view of seeking a general solution to the electronic wavefunction, the special case of the hydrogen atom is considered briefly. In this case, the many-body wavefunction | is replaced with the single-electron wavefunction | , the nucleus defined as the origin훹⟩ and the position vector of the electron is given in휓 spherical푒⟩ coordinates, = ( , , ).

Then, the potential energy term is simply the spherically symmetric Coulomb푟⃑ potential푟 휃 휙 for the electron’s attraction to the nucleus, and the Hamiltonian reduces to (7).

= (7) 2 2 4 2 −ℏ 2 푒 퐻� 훻� − 푚푒 휋휀0푟

* is the number of particles, is the reduced Planck constant, is the rest mass of particle ,

= + + is the Laplacian operator, is the elementary charge,푖 is the vacuum permittivity, 푁 2 2 2 ℏ 푚 푖 2 휕 휕 휕 2 2 2 훻� is the휕푥 charge휕푦 of particle휕푧 and is the distance푒 between particles and . 휀0 푞푖 푖 푟푖푗 푖 푗 26

When (7) is written in spherical coordinates and | inserted, the radial and angular parts of the equation are separable and may be 휓solve푒⟩ d independently. The angular solutions are the spherical harmonics, (8),* which are the origin of the familiar Atomic

Orbitals (AOs) and are chosen specifically to have well defined orbital angular momentum , (9) and (10), where is the projection of along the z-axis.† An

푙 analytical solution푙 to the radial wavefunction푚 , is not of concern푙 here as there is no general solution and is dependent on the exact�푅푛 problem,푙� ‡ however it is in solving for

, that the principal quantum number, , is introduced and that only one radial

�푅function푛 푙� for each pair of allowed quantum푛 numbers , is required. Finally, a spin function is included in an ad-hoc manner which accounts푛 푙 for the two possible quantizations of the = 1 2 electron spin, (11) and (12). This yields the final single- electron wavefunction푠 (13)⁄, where represents the electron position and its set of unique quantum numbers ( , ,훼, , , ).§

훼 ≡ 푟⃑ 푛 푙 푚푙 푠 푚푠

( , ) | , (8) 푙 푙 �푌푚 휃 휙 � ≡ 푙 푚푙⟩ | , = ( + 1)| , (9) 2 2 푙̂ 푙 푚푙⟩ ℏ 푙 푙 푙 푚푙⟩ | , = | , (10)

푙̂푧 푙 푚푙⟩ ℏ푚푙 푙 푚푙⟩ | , = ( + 1)| , (11) 2 2 푠̂ 푠 푚푠⟩ ℏ 푠 푠 푠 푚푠⟩ | , = | , (12)

푠̂푧 푠 푚푠⟩ ℏ푚푠 푠 푚푠⟩ | ( ) = , | , | , (13)

휓푒 훼 ⟩ �푅푛 푙� 푙 푚푙⟩ 푠 푚푠⟩

* See The spherical harmonics for more information. † Note that the choice of the z-axis for quantization is arbitrary and is simply convention. ‡ Generally, however, the radial function decays exponentially as and is often a highly nodal polynomial function. 푟 → ∞ § Recall the allowed values for the quantum numbers are = 1, 2, 3, …; = 0, 1, 2, … , 1; = , + 1, … , 1, ; = 1 2; and = ± 1 2; and that the Pauli exclusion principle prohibits more 푛 푙 푛 − 푚푙 than one electron possessing the same set of quantum numbers. −푙 −푙 푙 − 푙 푠 ⁄ 푚푠 ⁄ 27

Multi-electron atoms

With more than one electron the Coulomb repulsion between the electrons is an important part of the Hamiltonian. However, as this term is a function of the coordinates of two particles, there are no analytical solutions for the many-body wavefunction. If the electron-electron Coulomb interaction is approximated by assuming each electron moves in a spherically symmetric field that is the average effect of the remaining electrons, the two-electron operator can be omitted and the potential term replaced with a modified single-electron potential , resulting in the Mean Field

(MF) Hamiltonian, (14).* This approximation results in 푈the푖 electrons not sensing each other correctly, and is known as the lack of dynamic correlation.†

= 푁푒 + ( ) (14) 2 2 −ℏ 2 푀퐹 �푖 푖 푖 퐻� � � 푒 훻 푈 푟 � As the MF Hamiltonian only contains푖= single1 푚-electron operators, the approximate many- body wavefunction can be decomposed into single-electron wavefunctions and each solved independently.‡ These single-electron wavefunctions have the same angular form as the solution for the hydrogen atom, however the radial wavefunctions differ owing to the modified potential. However, a simple product of these single-electron wavefunctions does not directly satisfy the Pauli exclusion principle; the wavefunction must be anti-symmetric with respect to a simultaneous position and spin permutation and therefore a special linear combination of the single-electron functions is required.

This can be elegantly achieved with a Slater determinant, (15), which represents a unique orbital population of electrons. Note that the Slater determinant does not necessarily represent an entire electron configuration – cases with orbital degeneracy require multiple determinants to describe the many-body wavefunction, an effect known as static correlation.

* is the number of electrons. † See The ab initio approach for further discussion. 푁푒 ‡ Subject to the constraint of orthogonality. 28

( ) ( ) ( )

1 �휓푒1(훼1)� �휓푒2(훼1)� ⋯ �휓푒푁푒(훼1)� | = (15) ! � � 푒1 2 푒2 2 푒푁푒 2 푒⟩ �휓 훼 � �휓 훼 � ⋯ �휓 훼 � 훹 푒 �푁 � ⋮ ⋮ ⋮ ⋮ � �휓푒1�훼푁푒�� 휓푒2�훼푁푒 �� ⋯ �휓푒푁푒 �훼푁푒�� Magnetic and relativistic effects: the Dirac equation

That astute reader will note that the Schrödinger treatment neglects interactions of magnetic origin and that Einstein’s theory of Special Relativity is also curiously absent.

For = 1 2 particles such as electrons, it is the Dirac equation which is built to include spin,푠 relativity⁄ and electro-magnetic fields, and provides a much more rigorous description of the electron.6 Although it will not be discussed in detail, it is worth mentioning that this remarkable equation does indeed reduce to the time-independent

Schrödinger equation in the limit of non-relativistic electrons. For the purposes of this text, the most important result from the Dirac treatment is that electrons moving in potentials like that of an atomic nucleus experience a coupling of their spin and orbital angular momentum. The so-called Spin-Orbit Coupling (SOC) may have been intuitively expected – the magnetic moment generated by the orbital angular momentum can interact with the intrinsic spin magnetic moment of the electron. SOC combines the orbital and spin angular momenta into a total angular momentum of the electron, = + . Importantly, this renders the individual projections and poor quantum 횥⃑numbers,푙⃑ 푠⃑ replaced by the projection of the total angular momentum푚푙 푚푠 .

Although the Dirac treatment of the electronic wavefunction has many advantages,푚 푗it proves to be a much more challenging problem which has no general solutions. Luckily, it turns out that relativistic effects only become important for elements with large atomic number where the core electrons move at a significant fraction of the speed of light.7,* For this reason, the Schrödinger treatment is usually a good starting point and

* The definition of ‘significant’ depends on the calculation, but here implies compounds containing elements of the fourth period, 19, and beyond.

푍 ≥ 29 the SOC can be included as a perturbation with the SOC Hamiltonian, (16), which is based on the result of the Dirac equation.3,4,*

( ) = 푁푒 (16) 2 2 ℏ 푑푈푖 푟푖 푆푂퐶 2 2 ⃑̂푖 ⃑푖 퐻� � � 푒 푖 푖 푙 ∙ 푠̂ � 푖=1 푚 푐 푟 푑푟 Energy scales

A number of approximations have been introduced when determining the electronic wavefunction, without much justification. The Hamiltonian is a recipe for calculating the energy of a quantum system and therefore any approximations should be judged on their contribution to the total energy. Take for example the BO approximation; it was justified that as the masses of the proton and neutron are three orders of magnitude larger than that of the electron, the kinetic energy of the nuclei are at least three orders of magnitude smaller than that of the electrons. It is these vastly differing energy scales that make ignoring the nuclear wavefunction justifiable. In the case of multi-electron systems the explicit electron-electron Coulomb repulsion was ignored and replaced with an average single-electron potential; the electron-electron repulsion energy is usually at least an order of magnitude smaller than the difference between subsequent electron configurations, see for example Figure 2, and therefore this is often a reasonable approximation. The exclusion of relativistic effects is again justifiable on the basis of energy scales; even the largest relativistic effect, the SOC, is usually an order of magnitude weaker than the electron-electron repulsion. The result of these vastly different energy scales is well visualized in the electronic structure of a multi-electron atom or ion, Figure 2, where the successive inclusion of each correction is a minor departure compared to the preceding stage.

* is the speed of light.

푐 30

1 G0

3 P2 3 P0

1 D2

3 F4 3 F3 3 F2

Figure 2 | Lowest electronic energy levels of a free VIII ion. e le hand levels correspond to the electron congurations, the central levels are the terms split by electronic repulsion and the levels on the right are the resultant SOC states. Magnied region shows SOC states for the lowest energy levels. SOC state labels are omitted for all but the lowest levels.

According to the energy scales outlined above, the electron-electro n Coulomb repulsion acts to remove the degeneracy of the electron congurations into terms that are characterized by the total spin and orbital momenta, commonly given in spectroscopic notation (17).* ese total angular momenta are sums of the individual electron moments, (18) and (19), and this is referred to as the Russell-Saunders (RS) or

LS coupling scheme. e ground term of the ground electron conguration is given by the empirical Hund’s rules8 which state:

1) the ground term will have the maximal spin multiplicity; and

2) for that multiplicity it will have the maxima l orbital momentum.

* Total ( ) or single particle ( ) orbital angular momentum are assigned codes where 0, 1, 2, 3, 4, and are subsequently alphabetic. 31

Within a manifold of degenerate orbitals, the ground term can easily be determined by assigning and quantum numbers to each electron, commencing with =

+1/2; 푚=푠 + , followed푚푙 by = +1/2; = + 1 etc. 푚푠1

푚푙1 푙 푚푠2 푚푙2 푙 − For light elements where the SOC is relatively weak, the LS coupling scheme is not a bad approximation and the SOC can be considered to couple total spin and orbital moments of each term leading to a further splitting into multiplets characterized by a total angular momentum = + .* In cases where the SOC is stronger than the Coulomb repulsion, the so-called퐽⃑ jj coupling퐿�⃑ 푆⃑ scheme is more appropriate, where the SOC is first applied to each electron individually to give a total one-electron angular momentum = + , followed by the inter-electronic Coulomb repulsion. 횥⃑ 푙⃑ 푠⃑

(17) 2푆+1 퐿퐽

= 푁푒 (18)

푆 � 푚푠푖 푖=1

= 푁푒 (19)

퐿 � 푚푙푖 푖=1 Linear algebra

The method most commonly employed for the solution to the Schrödinger wave equation is the equivalent matrix formalism of Heisenberg.3 With a defined basis,† the required integrals, given in Dirac notation as = , form a matrix and are therefore also known as matrix elements. The basis퐻푖푗 in�훹 which푖�퐻��훹 the푗� Hamiltonian matrix is constructed is a complex vector space known as a Hilbert space, which contains all the possible states for the wavefunction.‡ For example, the basis for a particle with total

* The coupling = + is a vector sum and therefore the allowed values are given by = | |, | | + 1, … , + 1, + . 퐽⃑ 퐿�⃑ 푆⃑ 퐽 퐽 퐿 − 푆 퐿 − † Explicitly and generally, | | , | . 푆 퐿 푆 − 퐿 푆 ‡ While Hilbert spaces can in fact be of infinite dimension, only spaces of finite dimension are considered Ψ푖⟩ ∈ 푎⟩ �Ψ푗� ∈ 푎⟩ here. 32 angular momentum (ignoring spatial coordinates for the time being) is given by all possible projections of푗 the angular momentum, (20), which has the dimensionality (21). All elements of the basis are orthonormal* and are said to span the Hilbert space.†

, {| , , | , + 1 , … , | , 1 , | , } (20)

�푗 푚푗� ∈ 푗 −푗⟩ 푗 −푗 ⟩ 푗 푗 − ⟩ 푗 푗⟩ = 2 + 1 (21)

푑푖푚푗 푗 = (22) −1 The language of QM describes the action퐻 푃퐷of the푃 Hamiltonian ‘connecting’ elements of the basis (i.e. the matrix element between two basis states is non-zero) as a ‘mixing’ of the two states. Once the Hamiltonian matrix is constructed, there always exists a unitary‡ transformation which can bring the퐻 matrix to a diagonal form (22).§ This diagonalization is in fact 푃the solution to a special case of the eigenvalue problem where the Hamiltonian matrix must be Hermitian** such that the energy eigenvalues are real- valued observables. The diagonal entries of are the eigenvalues whose corresponding eigenvectors are the columns of . The eigenvectors퐷 must necessarily be orthogonal, leading to the connection of the푃 diagonalization, and therefore solution, of the Hamiltonian matrix as a rotation of the basis.†† Accordingly, such a transformation is also known as a change of basis, where the rotated basis described by is the eigenbasis of the Hamiltonian. 푃

* That is, each element does not share any commonalities with another and that they are normalized to unit magnitude. † A set of basis kets | span the Hilbert space by definition of the closure relation | | = 1 ‡ Unitary matrices are those with the property ( ) = ( ) = , where is the identity matrix. 푎⟩ ∑푎 푎⟩⟨푎 § Or equivalently, = . ∗ 푇 ∗ 푇 퐴 퐴 퐴 퐴 퐼 퐼 ** Hermitian matrices−1 are equal to their conjugate transpose, = = ( ) . 푃 퐻푃 퐷 †† Assuming the initial basis vectors are not already the eigenbasis of† the Hamiltonian.∗ 푇 As the initial basis 퐴 퐴 퐴 vectors span the Hilbert space, the eigenvectors (linear combinations of these basis vectors) representing the eigenbasis of the Hamiltonian must also span the Hilbert space and are therefore also all orthogonal. As the two sets of vectors span the same Hilbert space, all elements of each set are orthogonal to one another and the two sets are not coincident, the eigenbasis is simply a rotation of the initial basis in the Hilbert space. 33

A commonly employed change of basis is the coupling of two angular momenta to form one total angular momentum, having application to, for example, the SOC of a single electron or the coupling of two spins. The coupling coefficients, given for the coupling of two arbitrary angular momenta , and , to form , , , ,* are

1 푗 2 푗 1 2 퐽 known as the Clebsch-Gordan coefficients,�푗 푚 1� (23), �푗which푚 2form� an orthonormal�푗 푗 퐽 푚 �set of eigenvectors spanning the new coupled Hilbert space. It is important to realize that while the basis has been changed, the dimensionality and closure of the Hilbert space remains; merely the matrix representation of the Hamiltonian is now different.

, , ,

�푗1 푗2 퐽 푚퐽� (23) = 푗1 푗2 , , , , , , , , ,

� � �푗1 푚푗1 푗2 푚푗2�푗1 푗2 퐽 푚퐽� �푗1 푚푗1 푗2 푚푗2� 푚푗1=−푗1 푚푗2=−푗2 Perturbation theory

While of course in reality all interactions should enter the Hamiltonian on an equal footing, sometimes it may be desirable to determine how a small perturbation influences the wavefunction of a more important Hamiltonian. With knowledge of the unperturbed or zeroth order wavefunction, which is an eigenfunction of the zeroth order Hamiltonian, and the assumption that the perturbed wavefunction is not too dissimilar (i.e. the perturbation is small), Perturbation Theory (PT) provides an approach to calculate the corrections to both the energies and wavefunctions to arbitrary order. For well-behaved perturbations the series is convergent and at the limit of infinite order the solution for the perturbed states is exact. The beautiful formalism of

PT is well explained in other texts9,10 and will not be explicitly derived here, save for the final results. The zeroth order (unperturbed) Hamiltonian is denoted ( ), with 0 퐻�

* Two independent Hilbert spaces are formally connected with the Kronecker or Tensor product , , , , , , , with dimension = (2 + 1)(2 + 1).

The coupling1 =2 + is 1a vector2 sum and1 therefore2 the allowed values1 2are given by = | �푗1 푚푗 � ⊗ �푗2 푚푗 � ≡ �푗1 푚푗 푗2 푚푗 � ≡ �푚푗 푚푗 � 푑푖푚푗 ⊗푗 푗1 푗2 |, | | + 1, … , + 1, + . The new coupled Hilbert space, spanned by the , , , basis 퐽⃑ 횥⃑1 횥⃑2 퐽 퐽 푗1 − vectors, has exactly the same dimension and contains exactly the same information as the uncoupled 푗2 푗1 − 푗2 푗1 푗2 − 푗1 푗2 �푗1 푗2 퐽 푚퐽� , , , basis.

�푗1 푚푗1 푗2 푚푗2� 34

( ) ( ) eigenvectors and eigenvalues , and the perturbing Hamiltonian denoted as 0 0 푖 푖 ( ). The first�Ψ order� correction to the퐸 energies is given as (24), which is simply the 1 퐻expectation� value (matrix element) for the perturbation in the unperturbed eigenbasis. The second order corrections to the energies are given by (25), which involve the matrix elements that mix the unperturbed functions. The first order correction to the wavefunction is very similar to the second order correction to the energies, as this must of course involve the mixing of the unperturbed functions, (26).

( ) ( ) ( ) = ( ) (24) 1 0 1 0 퐸푖 �훹푖 �퐻� �훹푖 � ( ) ( ) ( ) ( ) 2 0 0 (25) = ( ) 1 ( ) 2 ��훹푗 �퐻� �훹푖 �� 퐸푖 � 0 0 푗≠푖∈푎 푖 푗 ( )퐸 ( −) 퐸( ) ( ) ( ) 0 0 (26) = ( ) 1 ( ) 1 �훹푗 �퐻� �훹푖 � 0 �훹푖 � � 0 0 �훹푗 � Of course the expressions (25) and푗 ≠(26)푖∈푎 become퐸푖 − undefined퐸푗 in the presence of degenerate eigenstates, in which case the numerator is required to be exactly zero such that a singular expression is avoided. Unfortunately this can only be achieved by taking the appropriate linear combination of the unperturbed eigenstates that span the degenerate subspace, which is equivalent to diagonalizing the block of the Hamiltonian matrix containing the degenerate states. As with the case of degeneracy, in the case of near- degeneracy the corrections will be very large due to small denominators and therefore

PT is not valid, so the near-degenerate block must also be diagonalized. Only when the energy separation between states is large will PT yield good approximations to the perturbed states.

Suddenly the picture becomes clear; diagonalization of the matrix is in fact infinite order

PT within the subspace of our chosen basis. For example, the basis space including all electronic configurations is infinite and therefore by restricting the basis to the ground configuration only, the mixing of excited configurations into the ground one by action

35 of the Hamiltonian are ignored.* is has been previously justied, but now it can be seen that diagonalizing the Hamiltonian matrix in the subspace of a single conguration is simply innite order PT in this restricted basis. Any further truncation of the basis, such as only considering the lowest energy term, follows exactly the same protocol,

Figure 3.

Figure 3 | Schematic of the Hamiltonian matrix for the VIII problem. Largest grid shows basis containing all states from the 3d2, 3d14s1 and 3d14p1 congurations, smaller grid shows all terms arising from the 3d2 conguration and black box represents the 21 spin- orbit states of the ground 3F term.

e spherical harmonics

e spherical harmonics are a versatile set of angular functions, dened as the angular solutions to the Laplace Equation in spherical coordinates and formally given by (27) and (28).3,11,† ey are a complete set of orthonormal functions, (29) and (30), and are therefore capable of representing any scalar function on a sphere via a generalized

Fourier Transform, (31) and (32). Note that the Kronecker delta function appearing in

(29) is dened as (33).

* is is either justied by large energy gaps to excited states creating large denominators, small mixing coecients creating small numerators, or both. † e functions are the associated Legendre Polynomials. 36

(2 + 1)( )! ( , ) = ( 1) ( ) | , (27) 4 ( + )! 푙 푚푙 푙 푙 푖푚푙휙 푚푙 푙 푙 − 푚 푚푙 푙 푌 휃 휙 − � 푙 푃 푐표푠 휃 푒 ≡ 푙 푚 ⟩ 휋 푙 푚 ( , ) = ( 1) ( , ) (28) 푙 ∗ 푚푙 푙 푌푚푙 휃 휙 − 푌−푚푙 휃 휙

, | , = 휋 휋 ( , ) ( , ) = , , (29) ′ ∗ ′ ′ 푙 ′ 푙 ′ ′ ⟨푙 푚푙 푙 푚푙⟩ � � 푌푚푙 휃 휙 푌푚푙 휃 휙 푠푖푛 휃 푑휃 푑휙 훿푙 푙훿푚푙 푚푙 휃=0 휙=−휋

휋 휋 ( , ) = 1 (30) 푙 2 � � �푌푚푙 휃 휙 � 푠푖푛 휃 푑휃 푑휙 휃=0 휙=−휋

= 휋 휋 ( , ) ( , ) (31) 푙 푙 ∗ 푓푚푙 � � 푓 휃 휙 푌푚푙 휃 휙 푠푖푛 휃 푑휃 푑휙 휃=0 휙=−휋

( , ) = ∞ 푙 ( , ) (32) 푙 푙 푓 휃 휙 � � 푓푚푙 푌푚푙 휃 휙 푙 푙=0 1푚 = −푙 = = , 0 (33) 푖푓 훼 훽 훿훼 훽 � 푖푓 훼 ≠ 훽 More than one way to skin a cat

Under the MF approximation discussed earlier, the single-electron wavefunctions for a multi-electron atom have the same angular form as the electron in hydrogen, which is capable of exact analytical evaluation. However, the unknown form of radial function defies a general analytical solution and historically this problem has been approached in two distinct ways. Experimentalists have preferred a parametric approach where the radial integrals of the Hamiltonian are treated as parameters to be determined by experiment, while theorists have preferred an ab initio approach and sought a numerical approximation to the radial function. Both approaches have their distinct advantages and disadvantages, and it will be the subject of the remainder of this text to describe the two techniques and how they can be employed in concert to gain useful insight into the electronic and magnetic properties of molecular systems.

The parametric approach: angular momentum in quantum mechanics

The great usefulness of treating the unknown radial integrals as parameters while exactly solving the angular part is that calculations are much more facile and can be easily compared with experiment. To illustrate the power of the parametric approach, the SOC in the free ion configuration [Ar]3d1 will be examined. Owing to the Pauli exclusion principle, all angular momentum within closed shells will cancel* and only electrons in open shells contribute to the magnetic properties. Furthermore, it is only the energy differences to the excited states which are of importance for the magnetic and spectroscopic properties and therefore these paired electrons, which only contribute to the absolute energy of the ground state, can be neglected.† Because the free ion has spherical symmetry, all five 3d orbitals are degenerate and the electron is free to occupy any of these orbitals, leading to 10 possible states for the electron once the spin degree of freedom is accounted for. These states, where = 2 and = 1 2 and labelled | , for brevity, form a basis for the angular part of 푙the wavefunction푠 ⁄.‡ Due to the separation푚푙 푚푠⟩ of variables, the matrix elements (integrals) that must be computed when evaluating the

SOC Hamiltonian are of the type (34). With a reminder of the implication of Dirac notation, (35),§ the radial integral can be replaced with a single parameter , the single- electron SOC parameter for the 3d electron, leading to the simplified SOC 휁Hamiltonian (36) which can be expressed in an angular momentum only basis, (37). It is important to note that in the evaluation of matrix elements such as those in the final expansion of

(37), the orbital operators only act on the orbital part of the wavefunction and the spin operators only on the spin part. This is perhaps best expressed in the full Kronecker product expansion of the operator acting on the basis kets, (38).

* i.e. the sums (18) and (19) are zero when each state of angular momentum is doubly occupied. † The zero of the energy scale can be arbitrarily set and thus only the orbital manifold with unpaired 푚푙 푙 electrons needs to be considered. ‡ Recall that this combined Hilbert space is actually the Kronecker product of the two independent spaces for the orbital and spin angular momentum. The dimension of the Hilbert space is the product of the dimensions of the constituent spaces, which in this case is 5 × 2 = 10. § Note that the infinitesimal volume element for integration, commonly referred to as the Jacobian, in spherical coordinates is = sin . 2 푑푉 푟 푑푟 휃 푑휃 푑휙 38

( ) ( ) ′ 1푒 ( ) 푆푂퐶 푒 (34) = �휓 훼 �퐻� �휓 훼, � , 2 2 , , ℏ 푑푈 푟 ′ ′ 2 2 �푅푛 푙� �푅푛 푙� �푚푙 푚푠 �푙⃑̂ ∙ 푠⃑̂�푚푙 푚푠� 푚푒 푐 푟 푑푟 1 ( ) 1 ( ) , , = ∞ , = (35) 2 2 2 2 ℏ 푑푈 푟 ℏ 2 푑푈 푟 2 2 2 푛 푙 푛 푙 2 2 푛 푙 푒 �푅 � �푅 � 푒 � 푅 푟 푑푟 휁 푚 푐 푟 푑푟 푚 푐 푟=0 푟 푑푟 = (36)

퐻�푆푂퐶 휁푙⃑̂ ∙ 푠⃑̂ ( ) ( ) = , , ′ ′ ′ (37) �휓푒 =훼 �퐻�푆푂퐶, �휓푒 훼 � + 휁 �푚푙+ 푚푠 �푙⃑̂ ∙ 푠⃑,̂�푚푙 푚푠� ′ ′ 휁�푚푙 푚푠 �푙̂푥푠̂푥 푙̂푦푠̂푦 푙̂푧푠̂푧�푚푙 푚푠� | , = [| | ] = | [ | ] (38)

To determine 푙̂훼the푠̂훼 푚matrix푙 푚푠⟩ elements푙̂훼⨂푠̂훼 of푚 푙⟩the⨂ 푚Hamiltonian푠⟩ �푙̂훼 푚 푙⟩matrix,�⨂ 푠̂훼 푚the푠⟩ action of the operators on the basis kets must be determined; these rules are given for a general angular momentum , but are equally applicable for any orbital angular momentum or spin angular momentum푗 . The first two relations (39) and (40) are no more than 푙the original quantization rules푠 for the definition of the angular momentum and therefore the basis kets are unsurprisingly eigenstates of and . Due to the Heisenberg 2 uncertainty principle, definite values of all components횥̂ of 횥̂the푧 angular momentum are not available simultaneously and therefore the basis kets are not eigenstates of the and

operators. These two operators are most simply expressed as functions of the r횥̂aising푥

횥̂and푦 lowering operators, (41) – (44). These operators either add or subtract a quantum of angular momentum from the state on which they act. Accordingly then, no more angular momentum can be added to or subtracted from states already possessing either the maximal = or minimal = , respectively, (45).

푚푗 푗 푚푗 −푗

, = ( + 1) , (39) 2 2 횥̂ �푗 푚푗� ℏ 푗 푗 �푗 푚푗� , = , (40)

푧 푗 퐽 푗 횥̂ �푗 푚 �1 ℏ푚 �푗 푚 � = ( + ) (41) 2 횥푥̂ 횥+̂ 횥−̂ 39

1 = ( ) (42) 2 푦 + − 횥̂ 횥̂ − 횥̂ , = ( + 1)푖 + 1 , + 1 (43)

+ 푗 푗 푗 푗 횥̂ �푗 푚 � ℏ�푗 푗 − 푚 �푚 ��푗 푚 � , = ( + 1) 1 , 1 (44)

횥−̂ �푗 푚푗� ℏ�푗 푗 − 푚푗�푚푗 − ��푗 푚푗 − � | , = | , = 0 (45)

Therefore, the matrix elements of횥+̂ (37)푗 푗⟩ can횥 −̂be푗 further−푗⟩ expanded and evaluated as (46) –

(49). The only remaining components then are the integrals of the type

, | , which, due to the orthonormality of the spherical harmonics* and the ′ ′ spin⟨푚푙 functions,푚푠 푚푙 푚 simplify푠⟩ to Kronecker deltas, (50).

1 , , = , + + , (46) 2 ′ ′ ′ ′ �푚푙 푚푠 �푙⃑̂ ∙ 푠⃑̂�푚푙 푚푠� �푚푙 푚푠 � �푙̂+푠̂− 푙̂−푠̂+� 푙̂푧푠̂푧�푚푙 푚푠� , ,

= ( + 1) ′( ′+ 1) ( + 1) ( 1) (47) �푚푙 푚푠 �푙̂+푠̂−�푚푙 푚푠� 2 × , | + 1, 1 푙 푙 푠 푠 ℏ �푙 푙 − 푚′ 푚 ′ �푠 푠 − 푚 푚 − ⟨푚푙 푚푠 푚푙 푚푠 − ⟩ , ,

= ( + 1) ′( ′ 1) ( + 1) ( + 1) (48) �푚푙 푚푠 �푙̂−푠̂+�푚푙 푚푠� 2 × , | 1, + 1 푙 푙 푠 푠 ℏ �푙 푙 − 푚′ 푚 ′− �푠 푠 − 푚 푚 푙 푠 푙 푠 ⟨푚 푚 푚 − 푚 ⟩ , , = , | , (49) ′ ′ 2 ′ ′ �푚푙 푚푠 �푙̂푧푠̂푧�푚푙 푚푠� ℏ 푚푙푚푠⟨푚푙 푚푠 푚푙 푚푠⟩ , | , = , , (50) ′ ′ ′ ′ 푙 푙 푠 푠 Therefore the SOC Hamiltonian⟨푚푙 푚 푠 푚푙 푚푠 expressed⟩ 훿푚 푚 in훿 푚our푚 chosen basis takes the form 2 (51), where the bras and kets are labelled퐻�푆푂퐶⁄ℏ as , | and | , , and the labels are simplified as + + 1 2 and 1 2. ⟨Once푚푙 푚 푠diagonalized,푚푙 푚푠 ⟩the SOC 푚Hamiltonian푠 reveals two characteristic≡ ⁄ eigenvalues− ≡ − of ⁄ 3 /2 and which are four- and six-fold 2 2 degenerate, respectively, corresponding −to ℏthe휁 two spinℏ -휁orbit manifolds = 3/2 and

= 5/2 of the term of the [Ar]3d1 configuration. In fact, the coupled푗 , , , 2 푗 퐷 �푠 푙 푗 푚푗�

* See The spherical harmonics. 40 basis expressed by the Clebsch-Gordan coefficients (23) is exactly the basis which diagonalizes the single particle SOC Hamiltonian matrix and indeed the Hamiltonian could have been constructed in this basis to begin with, leading to immediate identification of the eigenvalues.

| 2, | 1, |0, |+1, |+2, | 2, + | 1, + |0, + |+1, + |+2, + 2, | 0 0 0 0 0 0 0 0 0 − −⟩ − −⟩ −⟩ −⟩ −⟩ − ⟩ − ⟩ ⟩ ⟩ ⟩ 1, | 0 0 0 0 0 0 0 0 ⟨− − 휁 2 휁 ⟨− − 휁 3 0, | 0 0 0 0 0 0 0 0 0 2 ⟨ − � 휁 3 +1, | 0 0 0 0 0 0 0 0 2 2 휁 ⟨+2, −| 0 0 0− 0 0 0� 0 휁 0 (51) 2, +| 0 0 0 0 0 0 0 0 ⟨ − −휁 휁 3 ⟨−1, +| 0 0휁 0 0−휁 0 0 0 0 2 2 휁 ⟨− � 휁 − 3 0, +| 0 0 0 0 0 0 0 0 0 2 ⟨ � 휁 +1, +| 0 0 0 0 0 0 0 0 2 +2, +| 0 0 0 0 0 0 0 0 0휁 ⟨ 휁 More generally,⟨ the flexibility of the parametric approach allows the treatment휁 of one or more metal ions in molecular environments, by considering the loss of spherical symmetry for the individual ions as well as any magnetic interactions between them. A generalized angular momentum Hamiltonian accounting for some common features of molecular systems can be written in the basis of the ground Russell-Saunders terms,

(52), accounting for the SOC, CF, spin exchange and Zeeman interactions, respectively.*

= 푁푆 + 푁푆 푘 2 , , 푞 푞 푖 푖 푖 푖 푖푗 푗 퐻� � 휆 퐿�⃑ ∙ 푆⃑̂ � � � 퐵푘 푖푂�푘 푖 − � 푆⃑̂ ∙ 퐽 ∙ 푆⃑̂ (52) 푖=1 푖=1 푘=2 4 6 푞=−푘 푖<푗∈푁푆 + 푁퐼 +

휇퐵 � �퐿�⃑푖 ∙ 퐼 푆⃑̂푖 ∙ 푔푖� ∙ 퐵�⃑ Although the parameters appearing푖= 1in such a general Hamiltonian have well defined origins, they should be considered purely phenomenological. For example, the CF

* = ± is the multi-electron SOC constant, where is the total spin of the ground term and the 휁푖 negative휆푖 2푆sign푖 is for shells that are more than half full,푆 푖 are the CF operators in Stevens operator equivalent notation, is the electronic Bohr magneton, is푞 the identity matrix and is the g-matrix. 푂�푘 푖 휇퐵 퐼 푔푖 41 parameters have a clear physical interpretation – they represent the loss of 푞 degeneracy of퐵 푘the metal orbitals upon complexation – however they cannot be derived. Likewise, the spin exchange terms result from a host of physical interactions such as the dipole and superexchange mecha퐽푖푗nisms, yet are not simply separable.

The ab initio approach

The aim of the ab initio approach is a direct numerical approximation for the elusive form of the radial function. The simplest and perhaps most widely used ab initio approach is the Hartree-Fock (HF) method. With the many-body wavefunction decomposed into simpler single-electron functions, the HF method constructs the wavefunctions for each Molecular Orbital (MO) from a combination of atom-centred

AOs; this is the Linear Combination of Atomic Orbitals (LCAO) approach.* In a further computational approximation, the radial parts of the atomic basis functions are composed of multiple Gaussian functions.† These basis sets can contain only a few or a very large number of AOs, which provide increased flexibility in the MOs they seek to describe and therefore with a larger basis set the MOs approach the true HF spatial wavefunctions. The HF method seeks a numerical solution by iteratively minimizing the total energy with respect to the MO expansion coefficients; the variational principle provides that the HF wavefunction is given at the stationary point. This procedure, known as the Self-Consistent Field (SCF), usually converges well for molecules that are aptly described by HF theory, providing the initial trial MOs are of ample quality.

Despite these numerical approximations, and not forgetting the previous approximations, HF theory can provide useful results for closed shell molecules at equilibrium geometry in the electronic ground state. The approximations contained in the HF solution to the non-relativistic electronic many-body wavefunction, excluding the necessity of a finite basis set, are the static and dynamic correlation effects. More

* This is only one approach to construct the wavefunction; any complete set of functions, such as plane waves for example, can also be used. † Sometimes other functions, such as Slater functions, are also used. 42 advanced ab initio calculations seek to remedy these deficiencies in a number of different manners, however only one of these that is of particular use for magnetically anisotropic systems is examined here.

Static correlation arises as a limitation of using a single Slater determinant to describe the electronic wavefunction. Cases with complete or near orbital degeneracy simply cannot be accounted for in the HF method and multiple determinants are required, an approach generally known as Multi-Configurational SCF (MCSCF). In reality, the electrons in a molecule can occupy any orbital (whilst obeying the Pauli exclusion principle); a situation referred to as Configuration Interaction (CI). The CI wavefunction is a set of linear combinations of all the possible determinants and directly accounts for the static correlation by allowing the electrons to freely occupy the degenerate or low-lying MOs. In fact, in the limit of full CI the dynamic correlation is also completely accounted for, and if coupled with an infinite basis set, the numerical solution approaches the true solution to the non-relativistic BO electronic many-body wavefunction. While performing this calculation would be the perfect scenario, the number of determinants rises extraordinarily quickly with the number of electrons and

MOs, as described by the Weyl formula (53),12,* becoming computationally prohibitive and therefore this limit is never achieved in practice. The calculation then becomes a balancing act; achieving the desired level of accuracy while encompassing the necessary physics is not always a simple task.

2 + 1 + 1 + 1 = (53) + 1 표 +표 + 1 푆 푁2 2푁 푑푒푡 푒 푒 푁 표 �푁 � �푁 � 푁 − 푆 푆

* The terms in brackets are binomial coefficients and is the number of orbitals.

푁표 43

Luckily, there are a number of strategies which can be employed to assist with such a task – the focus here is on the use of Complete Active Space SCF (CASSCF) calculations.

In the CASSCF method, a subset of the total MO space is chosen to be ‘active’ in which full CI is performed, while the remaining doubly occupied MOs are ‘inactive’ and those unoccupied are ‘external’, Figure 4. e benet of this approach is that only determ inants with signicant contributions to the wavefunction are included, greatly adding to the describable physics over HF theory while drastically reducing the number of determinants from full CI. In some cases it is a simple task to determine which orbitals should be active and in others it can be much more dicult, however the choice is highly dependent on the information required from the calculation.

Figure 4 | Schematic of the MO space used in the CASSCF approach.

Although the CASSCF method is very successful at accommodating static correlation in the many-body wavefunction, the dynamic correlation is not fully accounted for due to the use of a one-electron MF potential. With a good approximation to the zeroth order wavefunction, the eects of dynamic correlation can be accounted for with PT, where the eect of full CI can be approximated. Approaches such as Complete Active Space

Perturbation eory (CASPT2) 13 or N-Electron Valence State Perturbation eory

(NEVPT2)14 generally consider a large number of single and double electronic 44 excitations on top of the CASSCF wavefunction in order to approximate the full CI energies, thus recovering the dynamic correlation.

Relativistic effects become important for the electronic structure of molecules containing heavy elements or even just molecules with unquenched orbital angular momentum. Fully relativistic calculations with the Dirac equation are only currently tractable for molecules of a few atoms and therefore most approaches employ pseudo- relativistic approximations. On top of the non-relativistic approach, this usually consists of two steps resulting from the Douglas-Kroll-Hess transformation:15

1) use of a contracted basis set to account for the scalar relativistic effects; and

2) inclusion of the SOC.

While the use of an appropriate basis set is straightforward, the inclusion of SOC is somewhat more troublesome. Ideally the SOC would be employed during the orbital and CI optimization of the CASSCF wavefunction, however this is not currently a widespread approach in electronic structure packages. More commonly the SOC is applied after the CASSCF wavefunction in either a perturbative or non-perturbative manner.

Connection of the ab initio and parametric approaches

In the limit of a perfect ab initio calculation with no approximations, all interactions and effects would be implicitly accounted for and the magnetic properties could be directly obtained. However, such a calculation is a distant dream and the approximation techniques outlined previously must be used. Due to these approximations, results from even the most high-level ab initio calculations cannot match reality. By contrast, the parametric approach is simply designed to reflect experiment and therefore the errors are confined to the simplifications within the chosen model. In complicated situations however, the number of model parameters is much greater than can be reliably determined from the available experimental data and the problem is said to be over- parameterized. Simplifications are usually made so that some parameters can be

45 removed, but even so, it is not always possible to find a unique solution. In such cases, ab initio calculations can yield approximate values for the parameters such that it may be possible to find a solution for the parametric model. From another perspective, it could be viewed that due to the approximations made in the inherently inflexible ab initio approach, the parametric model provides an opportunity to correct the calculation by direct comparison to experiment. In reality, it is often a combination of the two perspectives that highlight the synergistic manner in which the parametric and ab initio approaches work together.

The restriction of the parametric basis to that of the angular momentum of the metal ion(s) implies an ionic form of the metal ligand interactions, i.e. the open shell orbitals remain purely metallic in character, which is in stark contrast to the ab initio MO approach. However, even in MO approaches, the magnetic orbitals usually maintain mostly metal character. Therefore in the simplest approximation, only the interactions with the d- or f-orbital populations are required, leading a relatively straightforward

CASSCF approach in which the nascent metal orbitals define the active space. This type of active space is described as minimal because it is likely that the electrons involved in the coordination bonds to the metal ion (i.e. the ligand lone pairs) have some metal character and therefore should really also enter the multi-configurational wavefunction.

Despite this, the minimal active space is usually quite successful in describing qualitative, and sometimes quantitative, features of the system. Indeed, this technique is very successful for 3d and 4f complexes, which form the vast majority of studied systems, because the orbitals do not mix too much with those of the ligand. Complexes of the 4d, 5d and 5f elements are not so straightforward, as these metals possess more radially extensive magnetic orbitals which may have significant ligand interactions. In these cases more care must be taken in the choice of active space to correctly describe the magnetic properties.

Experimental observables

The electronic structure, once determined by the methods outlined in the preceding sections, can be utilized to calculate many experimental observables and thus can be judged against available data. The magnetization of any electronic state is defined as the gradient of its energy as a function of magnetic field and thus the equilibrium molar magnetization is the thermally averaged sum over each state, (54).* The molar magnetic susceptibility is the derivative of the molar magnetization vector and therefore defines a second-rank tensor, (55).16,†

1 = 푑푖푚 (54) −퐸푖 푖 푘퐵푇 훼 휕퐸 푀 퐵 � − 훼 푒 푍휇 푖=1 휕퐵

푑푖푚 푑푖푚 −퐸푖 2 −퐸푖 ⎡ 휕퐸푖 휕퐸푖 푘퐵푇 휕 퐸푖 푘퐵푇 ⎤ = 퐵 , 푍 �� 훼 훽 푒 − 푘 푇 � 훼 훽 푒 � (55) 10 ⎢ 푖=1 휕퐵 휕퐵 푖=1 휕퐵 휕퐵 ⎥ 푁퐴 ⎢ ⎥ 휒훼 훽 2 푑푖푚 푑푖푚 푘퐵푇푍 ⎢ −퐸푖 −퐸푖 ⎥ ⎢ 휕퐸푖 푘퐵푇 휕퐸푖 푘퐵푇 ⎥ − �� 훼 푒 � �� 훽 푒 � ⎢ 푖=1 휕퐵 푖=1 휕퐵 ⎥ EPR spectra can also be calculated,⎣ considering the net absorption of ⎦microwave intensity by the sample, (56) and (57), where , , , is a lineshape function of the energy difference between and , the microwave푓�퐸푖 퐸푗 휂푖푗 frequency푣� and the linewidth

.‡ 퐸푖 퐸푗 푣

휂푖푗

, −퐸푖 −퐸푗 퐵 퐵 = 푖 푗∈푑푖푚 + 푘 푇 푘 푇 (56) 2 2 �푒 − 푒 � ′ ′ 퐼 � ��푗�푇��푥��⃑��푖�� 푗�푇��푦��⃑��푖�� 푖<푗 × , , , 푍 푓�퐸푖 퐸푗 휂푖푗 푣�

* Note that the magnetization is a vector and thus (54) expresses a single Cartesian component, . = −퐸푖 is the partition function, is the Boltzmann constant and is the temperature. 훼 † is the푑푖푚 Avogadro푘퐵푇 constant. 푍 ∑푖=1 푒 푘퐵 푇 ‡ The and unit vectors are mutually orthogonal to the static magnetic field in unpolarized 푁퐴 perpendicular′ mode′ EPR. 푥��⃑ 푦��⃑ 47

= 푁퐼 + (57) 휇퐵 푇��퐵�⃑� � �퐿�⃑푖 ∙ 퐼 푆⃑̂푖 ∙ 푔푖� ∙ 퐵�⃑ �퐵�⃑� 푖=1 Conclusion

This text has outlined some ways in which electronic, and hence magnetic, properties may be calculated. While not an exhaustive survey of all electronic structure methods, a flavour of some of the most common approaches that are particularly relevant to magnetism theory has been given. The subsequent chapters present some examples of how such techniques may be applied in the study of magnetic molecules, highlighting the pivotal role of electronic structure in the understanding of complex magnetic phenomena.

References

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2 P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, 1958.

3 E. U. Condon and G. Shortley, The Theory of Atomic Spectra, Cambridge University

Press, 1951.

4 B. G. Wybourne, Spectroscopic properties of rare earths, Interscience Publishers, 1965.

5 T. Veszpremi,́ Quantum chemistry: fundamentals to applications, Springer, 1999.

6 P. A. M. Dirac, Proc. R. Soc. Lond. Ser. A, 1928, 117, 610.

7 P. Pyykko, Chem. Rev., 1988, 88, 563.

8 W. Kutzelnigg and J. D. Morgan III, Z. Phys., 1996, 36, 197.

9 A. Z. Capri, Nonrelativistic Quantum Mechanics, World Scientific, 2002.

10 J. J. Sakurai and S. F. Tuan, Modern quantum mechanics, Addison-Wesley Longman,

2010.

11 A. R. Edmonds, Angular momentum in quantum mechanics, Princeton University

Press, 1974.

12 J. Paldus, Phys. Rev. A, 1976, 14, 1620.

13 K. Andersson, P. A. Malmqvist, B. O. Roos, A. J. Sadlej and K. Wolinski, J. Phys.

Chem., 1990, 94, 5483.

14 C. Angeli, S. Borini, M. Cestari and R. Cimiraglia, J. Chem. Phys., 2004, 121, 4043.

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16 R. Boča, Theoretical Foundations of Molecular Magnetism, Elsevier, 1999.

4. Paper one: “On the Possibility of Magneto-Structural Correlations:

Detailed Studies of Dinickel Carboxylate Complexes”

J. P. S. Walsh, S. Sproules, N. F. Chilton, A.-L. Barra, G. A. Timco, D. Collison, E. J. L. McInnes and R. E. P. Winpenny, Inorg. Chem., 2014, 53, 8464.

Article

pubs.acs.org/IC

On the Possibility of Magneto-Structural Correlations: Detailed Studies of Dinickel Carboxylate Complexes † † ‡ † § † James P. S. Walsh, Stephen Sproules, , Nicholas F. Chilton, Anne-Laure Barra, Grigore A. Timco, † † † David Collison,*, Eric J. L. McInnes, and Richard E. P. Winpenny † School of Chemistry and Photon Science Institute, The University of Manchester, Manchester M13 9PL, United Kingdom ‡ WestCHEM, School of Chemistry, University of Glasgow, Glasgow G12 8QQ, United Kingdom § Laboratoire National des Champs Magnetiqueś Intenses, 25, rue des Martyrs, B.P. 166, 38042 Grenoble Cedex, France

*S Supporting Information

μ ABSTRACT: A series of water-bridged dinickel complexes of the general formula [Ni2( 2- μ t t ′ t ′ t OH2)( 2-O2C Bu)2(O2C Bu)2(L)(L )] (L = HO2C Bu, L =HO2C Bu (1), pyridine (2), 3- methylpyridine (4);L=L′ = pyridine (3), 3-methylpyridine (5)) has been synthesized and structurally characterized by X-ray crystallography. The magnetic properties have been probed by magnetometry and EPR spectroscopy, and detailed measurements show that the axial zero-ﬁeld splitting, D, of the nickel(II) ions is on the same order as the isotropic exchange interaction, J, between the nickel sites. The isotropic exchange interaction can be related to the angle between the nickel centers and the bridging water molecule, while the magnitude of D can be related to the coordination sphere at the nickel sites.

■ INTRODUCTION significantly, groups now routinely report field-dependent magnetization alongside variable-temperature susceptibility, With growing interest in the use of molecular nanomagnets in 1 and there is also now significant use of inelastic neutron applications such as quantum computing and magnetocaloric scattering (INS), largely driven by Güdel in the first instance.11 refrigeration,2 it is becoming increasingly important to be able The number of parameters used in the spin Hamiltonian has to relate the magnetic properties of molecular nanomagnets to also increased, primarily due to the realization that the their chemical structure. For spin-only systems, where the anisotropy of the single ions is frequently of significance in orbital angular momentum is quenched, there has been some 3−6 determining physical behavior. Recent work has also suggested success in relating the form of the exchange interaction that the form of the exchange interactions used is also vital: e.g., (either ferro- or antiferromagnetic) to structural parameters, 12 anisotropic exchange in {Cr7M} rings and antisymmetric and such studies often use the term magneto-structural 13 exchange within {Ru2M} triangles. correlations to describe these relationships. A major goal in the field of single-molecule magnets (SMMs) An important question has been which structural parameter is to increase the size of the barrier to magnetic relaxation. The to use in these correlations. Two of the most commonly height of this barrier depends upon both the total ground state encountered parameters are the bridging bond angle, as in spin of the molecule, S, and its axial anisotropy, D, according to fi ’ Hat eld s famous correlation for hydroxide-bridged copper(II) the equation dimers,7 and the bond distance between the metal and the 5 ̈ 2 bridge, as used by Gorun and Lippard and developed by Gudel ΔUDSeff = (1) and Weihe8 and later Christou and co-workers.9,10 These correlations have been of huge importance in developing the Given the apparent dominance of the spin term in this area of molecular magnetism, but they were all derived during a relationship, much of the early effort in the field was spent time when chemists generally only measured variable-temper- trying to maximize S. However, it was soon realized that the ature susceptibility as a magnetic observable, and when the only anisotropy could not be neglected and that even a huge ground- terms in the spin Hamiltonian were the Zeeman term and the state spin could result in a poor SMM if the anisotropy was 14 exchange interaction. This modeling approach makes the negligible. This has encouraged a shift toward the use of ions assumption that the exchange interaction, J, is dominant and with a large intrinsic anisotropy, since these are more likely to is often called the “strong-exchange” limit, or the “giant spin translate to a large D in clusters (the cluster D approximates to approximation” (GSA). Now, encouraged by friendly physicists, our physical Received: May 6, 2014 measurements are more comprehensive. Perhaps most Published: July 25, 2014

Inorganic Chemistry Article

Table 1. Crystallographic Data for Compounds 3−6

34 5 6 · formula C40H58N4Ni2O9 C42H72N2Ni2O13 C44H66N4Ni2O9 C2H3NC40H78Mg2−xNixO17 fw 856.3 930.4 953.5 879.64 cryst syst monoclinic monoclinic orthorhombic monoclinic

space group P21/cC2/c Pna21 P21/n a,Å 10.7520(4) 24.2188(8) 20.204(1) 12.0516(6) b,Å 20.1962(5) 19.6774(9) 10.7563(6) 19.9828(8) c,Å 42.546(1) 10.4136(5) 23.1760(1) 22.9732(12) β, deg 92.358(2) 98.088(4) 90 103.790(5) V,Å3 9231.1(5) 4913.4(4) 5036.6(5) 5373.1(4) T,K 100(2) 100(2) 150(2) 150(2) Z 84 4 4 ρ −3 calcd,gcm 1.232 1.258 1.257 1.087 λ,Å/μ,mm−1 0.71073/0.867 0.71073/0.825 0.71073/0.802 0.71073/0.104 fl θ no. of r ns collected/2 max, deg 36116/52.74 9491/52.74 12471/52.74 17587/52.74 no. of rflns unique/I >2σ(I) 18841/13977 5031/4074 6920/5113 10874/6792 no. of params/restraints 1064/23 282/0 584/2 568/0 R1/goodness of fit 0.0598/1.050 0.0489/1.050 0.0569/1.039 0.0643/1.029 wR2 (I >2σ(I)) 0.1496 0.1332 0.1263 0.1415 residual density, e Å−3 0.91/−0.69 1.28/−0.52 0.58/−0.84 0.43/−0.36 a tensor sum of the single-ion anisotropy terms). Lanthanides a literature procedure,19 whereas 2 and 3 were prepared by 20 have proven to be very promising in this regard, with many of modification of published methods. μ μ t t t 2 the recent energy barrier record holders utilizing their high [Ni2( 2-OH2)( 2-O2C Bu)2(O2C Bu)2(HO2C Bu)2(C5H5N)2]( ). A light 15 green solution of 1 (1.00 g, 1.10 mmol) in Et2O (10 mL) was treated intrinsic spin and anisotropy terms. with pyridine (0.18 mL, 2.32 mmol). After 1 h of stirring at ambient Similar success might be possible using transition-metal temperature, MeCN (4 mL) was added and the solution left to stand elements, with the added advantage of a more diverse chemistry overnight, after which time diffraction-quality crystals had formed. and a significantly greater natural abundance. Unfortunately, These were collected by filtration, washed with cold MeCN, and dried fl the most anisotropic of the d-block ions are, by definition, not under a ow of N2. Yield: 0.33 g (33%). Anal. Found: C, 53.11; H, 16 well described using spin-only models, and certainly not with 7.79; N, 3.36. Calcd for C40H68N2Ni2O13: C, 53.24; H, 7.60; N, 3.10. μ μ t t 3 models based on the GSA. There is therefore a real need for [Ni2( 2-OH2)( 2-O2C Bu)2(O2C Bu)2(C5H5N)4](). The same procedure was used as described for 2 using excess pyridine (0.50 mL, 6.45 chemists and physicists to devote more research toward mmol). Yield: 0.41 g (44%). Anal. Found: C, 55.98; H, 6.93; N, 6.43. understanding exactly which factors dictate the exchange Calcd for C40H58N4Ni2O9: C, 56.10; H, 6.83; N, 6.54. μ μ t t t 4 interactions between anisotropic ions. The question arises: [Ni2( 2-OH2)( 2-O2C Bu)2(O2C Bu)2(HO2C Bu)2(CH3C5H4N)2](). which, if any, magnetic parameter should be used within these The same procedure was used as described for 2 using 3- correlations? This is especially important, as it dictates the methylpyridine (0.20 mL, 2.26 mmol). Yield: 0.30 g (29%). Anal. Hamiltonian used to fit the data. Found: C, 54.20; H, 7.86; N, 2.91. Calcd for C42H72N2Ni2O13:C, 54.22; H, 7.80; N, 3.01. A well-cited magneto-structural study by Halcrow et al. μ μ t t 5 [Ni2( 2-OH2)( 2-O2C Bu)2(O2C Bu)2(CH3C5H4N)4](). The same reveals a linear relationship between isotropic exchange, J, and procedure was used as described for 2 using excess 3-methylpyridine − − 9 the Ni O Ni angle in oxo-bridged nickel(II) cubanes but (0.50 mL, 5.65 mmol). Yield: 0.30 g (29%). Anal. Found: C, 57.94; H, does so using only the temperature-dependent susceptibility, 7.51; N, 6.06. Calcd for C44H66N4Ni2O9: C, 57.92; H, 7.44; N, 6.14. μ μ t t t 6 under the assumption that the effect of the zero-field splitting [Mg2−xNix( 2-OH2)( 2-O2C Bu)2(O2C Bu)2(HO2C Bu)4](). · · · · (ZFS) is negligible at higher temperatures: i.e., it implicitly uses 4MgCO3 Mg(OH)2 4H2O (8.00 g, 17.1 mmol), 2NiCO3 3Ni(OH)2 4H2O (0.5285 g, 0.8994 mmol), and pivalic acid (40.0 g, 39.2 mmol) the GSA. More recent work by Hill and co-workers has fl ° revealed that the magnitude of the ZFS in nickel(II) is actually were stirred under re ux (160 C) for 24 h. The mixture was then cooled to room temperature and dissolved fully in an excess of diethyl non-negligible in these compounds.17 A similar correlation of J − − ether (200 mL). MeCN (30 mL) was added with thorough stirring, with Ni O Ni angle has also been proposed by Thompson and the solution was left to stand partially open to allow slow 18 and co-workers. evaporation. Large single crystals formed after 2 days. Yield: 6.32 g Here we report a magnetostructural study on a family of five (39.9%). Anal. Found: C, 54.47; H, 8.61; Mg, 5.65; Ni, 0.58. Calcd for structurally related nickel(II) dimetallics where the zero-field C40H78Mg1.95Ni0.05O17: C, 54.40; H, 8.90; Mg, 5.23; Ni, 0.66. splitting is on the same order of magnitude as the exchange X-ray Crystallography. The single-crystal structures of 1 and 2 have been reported previously.19,20 Single crystals of 3−6 were interaction between the ions. The aim is to examine whether ff fi mounted in the nitrogen cold stream of an Oxford Di raction we can still nd similar magneto-structural correlations when a XCalibur 2 diffractometer. Graphite-monochromated Mo Kα radiation more complex Hamiltonian is required, and where more data (λ = 0.71073 Å) was used throughout. Final cell constants were are available. obtained from least-squares fits of all measured reflections. The structures were solved by direct methods using SHELXS-97.21 Each Δ ■ EXPERIMENTAL SECTION structure was completed by iterative cycles of F syntheses and full- matrix least-squares refinement. All non-H atoms were refined Synthesis. All reagents, metal salts, and ligands were used as anisotropically. Difference Fourier syntheses were employed in μ μ obtained from Sigma-Aldrich. [Ni2 ( 2 -OH2 )( 2 - positioning idealized methyl hydrogen atoms, which were assigned t t t O2C Bu)2(O2C Bu)2(HO2C Bu)4](1) was synthesized by following isotropic thermal parameters (U(H) = 1.5Ueq(C)) and allowed to ride

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Figure 1. Structures of the neutral complexes 1−6 in the crystal state. Data for 1 and 2 are from refs 26 and 20, respectively. All hydrogens are omitted, with the exception of those found crystallographically on the bridging water. Hydrogen bonds are indicated by dashed lines. on their parent C atoms (C−H 0.93 Å). Some pivalate groups bridging pivalates, each metal also bears a monodentate pivalate exhibited rotational disorder. This was modeled by allowing for two ligand that provides a stabilizing hydrogen bond at ∼2.5 Å to conformations of the tert-butyl group and refining their occupancy fi 2 21 the bridging water molecule. The coordination sphere of each factors. All re nements were against F and used SHELXL-97. metal ion is completed by two pivalic acid groups, and the Crystallographic data are collected in Table 1. CCDC reference numbers: 926884−926886 (3−5) and 999472 (6). complex has an overall neutral charge. Physical Measurements. Electronic absorption spectra were Substitution of the terminal pivalic acid groups in 1 occurs in collected on a PerkinElmer Lambda 1050 spectrophotometer. IR two stages. The pivalic acid cis to the bridging water, which is spectra of neat powders were recorded using a Thermo Scientific more labile than its trans counterpart, departs first. Compounds Nicolet iS5 FTIR spectrometer equipped with an iD5 ATR. Variable- 2 and 4 are generated by treatment with 2 equiv of pyridine and temperature (2−300 K) magnetic susceptibility measurements were 3-methylpyridine, respectively, in 30% yield (Figure 1). Further fi recorded in a 0.1 T magnetic eld on a SQUID magnetometer substitution utilizes an excess of pyridine and 3-methylpyridine (Quantum Design MPMS-XL). The experimental magnetic suscept- to afford 3 and 5, respectively, in similar yields (Figure 1). ibility data were corrected for underlying diamagnetism using ’ Infrared (IR) spectra of this series do not display any tabulated Pascal s constants, and the simulations of both magnetization ν and susceptibility were performed using PHI.22 Q-band EPR data were terminal (OH) stretches from either the bridging water or collected on a Bruker EMX spectrometer, and high-frequency, high- pivalic acid ligands. This is due to the formation of field EPR spectra were recorded at the LNCMI-CNRS at Grenoble on intramolecular hydrogen bonds, with the aforementioned a home-built spectrometer.23 EPR spectra were simulated using interaction between the bridging water and the available 24 EasySpin. Analytical data were obtained by the microanalytical oxygen atom of the monodentate pivalate ligands, and also service of The University of Manchester. between the pivalic acid protons and their neighboring Computational Details. All CASSCF calculations were performed 25 monodentate and bridging pivalate groups. This results in a with MOLCAS 7.8 using the RASSCF, RASSI, and SINGLE_- − ν ANISO modules. In all cases the ANO-RCC basis sets were used, weakening of the O H bond, shifting it to the (CH) region − −1 where the metal ion of interest was treated with TZVP quality, the first (2800 3000 cm ; Figure S1, Supporting Information). coordination sphere (and bridging water hydrogen atoms) was treated The change in electronic structure upon substituting pivalic with VDZP quality, and all other atoms were treated with VDZ quality. acid for stronger field pyridine ligands is evident in the The two electron integrals were Cholesky decomposed with the electronic spectra of this series (Figure 2). The low-energy default settings. region (<25000 cm−1) shows three ligand field (LF) transitions at ∼9000, ∼15000, and ∼25000 cm−1 (Table 2). This profile ■ RESULTS AND DISCUSSION 2+ 28 bears a striking resemblance to that of [Ni(OH2)6] , and the Synthesis and Characterization. The entry point for this spectra have been interpreted assuming approximate octahedral μ μ t t t 8 3 → 3 series is [Ni2( 2-OH2)( 2-O2C Bu)2(O2C Bu)2(HO2C Bu)4] symmetry at each d ion. The lowest energy A1g T2g (1), whose preparation involves heating nickel carbonate or excitation is a measure of the LF, following the trend 1 < 2, 4 < nickel hydroxide in pivalic acid.19 The compound contains two 3, 5. The additional methyl substituent in 4 and 5 has no effect nickel(II) ions bridged by one water and two pivalate ligands on the transition energies. − (Figure 1).26 Structures of this type are well-known for nickel The LF splitting of 1 at 8440 cm 1 matches that of the 27 28 and a host of other divalent metal ions. In addition to the hexaaquanickel(II) ion, which is not unexpected for a NiO6

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comparison to NiO5N and NiO4N2), thus decreasing the ZFS effects. Crystal Structures. The structures of 3−6 have been determined by single-crystal X-ray diffractometry and con- trasted with the structures of 1 and 2. Salient metric parameters are collated in Table 3. Compounds 1−5 all contain the μ t μ {Ni2( 2-O2C Bu)2( 2-OH2)} core and vary only in the remaining two coordination sites at each nickel center, which are incrementally changed from two pivalic acid groups to two pyridine ligands. The oxygen atom of the water molecule (Ow) adopts a pseudotetrahedral geometry. The water protons are μ t aligned essentially parallel with each {Ni( 2-O2C Bu)Ni} plane, which in turn lie at an angle of ∼80° to each other. The − Figure 2. Electronic absorption spectra of 1 5 recorded in Et2O hydrogen bond between the water molecule and the terminal solutions at ambient temperature. pivalates is invariantly ∼2.5 Å across the series. The nickel − a centers display only slight distortion from regular octahedral Table 2. Assignment of LF Transitions in 1 5 geometry, with angles between adjacent donors less than 5° 12345 away from normal. ff 3A → 3T (F) 8440 8830 9380 8830 9360 The e ect of introducing pyridine ligands is assessed by 2g 2g μ 3 → 1 monitoring the structural parameters of the {Ni2( 2-OH2)} A2g Eg(D) 13350 13380 13370 13430 13310 fi 3 → 3 unit (Table 3). The rst substitution forming 2 and 4 is A2g T1g(F) 14580 15100 15800 15120 15810 − − ··· 3 → 3 accompanied by an elongation of the Ni Ow,Ni Ot, and Ni A2g T1g(P) 24830 25190 26160 25280 26230 ∼ ∼ ∼ a −1 Ni distances by 0.07, 0.06, and 0.1 Å, respectively. In Energy in cm . contrast, the Ni−O bonds with the bridging and terminal pivalate ligands are essentially unchanged, with the terminal − ff coordination sphere. A uniform increase of ∼400−500 cm 1 is pivalates cis to the substitution site slightly more a ected than observed when the π-donating pivalic acid is replaced by a σ- those in the trans position. The addition of pyridine decreases donating pyridine to generate NiNO and NiN O centers in 2 the overlap between the metal and bridging water, lengthening 5 2 4 − − and 4 and in 3 and 5, respectively. The two higher energy the bond and slightly reducing the Ni Ow Ni angle. It also transitions are similarly shifted. Each complex also exhibits a weakens the bond with the remaining pivalic acid ligand, fl 3 → 1 peak that we assign as the spin- ip A2g Eg excitation, whose promoting a second substitution. The pyridines lie parallel to 3 → intensity is enhanced by proximity to the spin-allowed A2g each other at distances typical for this ligand, and the additional 3 T1g transition. This is most clearly seen in the spectrum of 1, methyl substituents in 4 have no bearing on the overall − ··· − − where the peaks at 13350 and 14580 cm 1 appear to have the topology. The Ni Ni distance exceeds 3.5 Å, and the Ni Ow ∼ ° same intensity and are difficult to differentiate. However, in 3 Ni angle expands to 115 when all four pivalic acids are − − and 5, the stronger ligand field blue-shifts the spin-allowed replaced by pyridine. If the Ni Ow Ni angle were the excitation, leaving a weak shoulder to lower energy. Because dominant structural parameter, then we would expect this peak is essentially independent of the crystal field, it is compounds 1, 2, and 4 to be similar to each other and 3 → 1 fi ff assigned as the A2g Eg transition in all ve compounds. di erent from compounds 3 and 5. fi | | 3 Given that the zero- eld splitting, D , of the A2g term is, to a Compound 6 is the nickel-doped magnesium analogue of first approximation, inversely proportional to the magnitude of compound 1 and was prepared so that we could directly the ligand field, we would expect this zero-field splitting to be measure the single-ion parameters of nickel in a near-identical larger in 2 and 4 than in 3 and 5, and this is indeed what is environment, but in the absence of exchange coupling. The observed (vide infra). The smaller magnitude of |D| in 1, metric parameters are not expected to be identical between 1 despite having a ligand field strength smaller than that of 2 and and 6, given the significant difference in the metal radii between 4, might be explained by 1 having a coordination environment magnesium and nickel and the fact that the space group is ff (NiO6) that is more appropriately treated as octahedral (in slightly di erent (Pbca in 1, P21/n in 6), but they are actually

Table 3. Salient Average Bond Distances (Å) and Angles (deg) for 1−6a

123456 Ni···Ni 3.361(1) 3.465(2) 3.5092(6) 3.4760(7) 3.511(1) 3.462(1) Ni−O−Ni 111.2(1) 110.7(3) 115.8(1) 110.6(1) 114.7(2) 112.5(1) − Ni Ow1 2.037(2) 2.106(4) 2.075(3) 2.087(2) 2.085(5) 2.080(2) − Ni Ob2 2.018(3) 1.992(5) 2.045(3) 2.034(2) 2.061(5) 2.032(2) − Ni Ob3 2.032(3) 1.983(5) 2.027(3) 1.998(2) 2.023(5) 2.063(2) − Ni Ot3 2.053(3) 2.069(5) 2.069(3) 2.070(2) 2.058(5) 2.092(2) − Ni L1 2.070(3) (O) 2.142(5) (O) 2.096(4) (N) 2.118(2) (O) 2.086(7) (N) 2.073(2) − Ni L2 2.080(3) (O) 2.095(7) (N) 2.128(4) (N) 2.089(3) (N) 2.135(7) (N) 2.097(2) a ∑ Values were calculated using ( xi/n), where x is the bond metric and n is the number of values averaged. The subscript w denotes the oxygen of the bridging water group, b the bridging pivalate, and t the terminal pivalate. L represents the atoms at the labile terminal positions. 1−3 denote pairs trans to each other.

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Figure 3. (A) Overlay of the temperature dependence of the magnetic susceptibility recorded on powders of 1−5 under a static field of 0.1 T. (B) Field-dependent magnetization of compounds 1−5 measured at 2 and 4 K. Solid lines in all graphs represent simulations using the parameters in Table 6. quite close and fall within the ranges set by 1−5. We can equivalent, even though this is only strictly true for the therefore be confident that the parameters we obtain will be of symmetry-related ions in 2 and 4, and the exchange is treated as relevance to the studies on the pure compounds. isotropic. Magnetometry. The temperature dependence of the χ χ ̂ ̂ ̂ ⃗ ̂ ̂ ̂ ̂ ̂ ̂ product MT (where M = molar magnetic susceptibility and H = gSβ()12+ S· H+ S 1122··DD S+ S·· S− 2 JSS 12· (2) T = temperature) for 1−5 are shown in Figure 3A. For all χ compounds, the room-temperature MT values of around 2.5 Good fits to the data can be obtained using this model but − − cm3 K 1 mol 1 correspond well to the expected spin-only value are ambiguous with regard to the sign of the axial ZFS − for two uncoupled S = 1 ions with giso = 2.2 2.3. The value is parameter, D, and are unable to offer a conclusive measure of constant until around 80 K for all compounds. At low the exchange interaction, which appears to be smaller than the temperature, the plot follows markedly different profiles for 1 ZFS in 2−4. In fact, the isotropic exchange parameter, J, can be and 4, which rise, suggesting ferromagnetic coupling, in fi varied over a range of values (keeping the tted D and giso comparison to those for 2, 3, and 5, which exhibit a sharp values fixed) before adversely affecting the“goodness” of the drop at low temperatures. It is tempting to attribute this drop simulations. These ranges for the value of J (in units of cm−1) to an antiferromagnetic exchange (indeed, this was done by are as follows: 1, 2.30−2.90; 2, 0.10−0.35; 3, −0.05 to −0.20; others29 for compound 2), but this neglects the effect of the − − − fi 4, 0.55 0.85; 5, 0.25 to 0.45. zero- eld splitting (ZFS) or at least assumes that it is much As a result, it is impossible to deduce the exchange (and, smaller than the exchange. To illustrate this point, we modeled importantly, to compare values across the series) from fitting the susceptibility alone for 2 and obtained a J value of −0.5 −1 − the magnetic data alone; further data are clearly required to cm (using the 2J convention). In reality, the ZFS can easily understand the magnetic behavior of these simple compounds be on the order of the exchange interaction in compounds fi unambigiously. Here, the additional data are EPR spectroscopy containing octahedral nickel(II) and may even be signi cantly at 331 GHz on pure and doped samples and CASSCF larger.30 calculations to yield estimates of the g values and anisotropy In compound 1, the upturn in the susceptibility below 50 K − − parameters for single-ion sites. culminating in a χ T value of 3.9 cm3 K 1 mol 1 at 2.5 K, M INS data for compound 1 were actually collected over a before a sharp downturn due to ZFS, is consistent with an S =2 decade ago;19 INS and magnetic susceptibility were used to ground state of two ferromagnetically coupled nickel(II) ions. model the exchange and ZFS parameters, but the poor quality We rule out the possibility of weak intermolecular interactions fi as the cause of this downturn by noting that in compounds 1−5 of the INS data meant that the rhombic term of the zero- eld the intermolecular Ni···Ni distances are never below ∼9Å. splitting tensor, E, was neglected and only isotropic exchange χ was employed, a scheme that we will show is inappropriate for Compound 4 also exhibits an upturn in MT upon cooling below ∼40 K that is likely due to a ferromagnetic interaction, these systems (vide infra). The INS data are reproduced in the albeit weaker than that in 1, with ZFS again resulting in the Supporting Information and are consistent with the model we onset of a sharp downturn (∼10 K) that prevents the arrive at from the present EPR studies. susceptibility reaching the value for a pure S = 2 state. Single-Ion Anisotropy: EPR Spectroscopy. To deter- The field-dependent magnetization curves are given in Figure mine the single-ion parameters (g, D, and E)in1 without the 3B. Compound 1 exhibits a sharp rise at low fields, with the 2 K added complexity of exchange coupling, we synthesized a data reaching saturation above 5 T. In contrast, compound 2 diamagnetic magnesium analogue doped with 5% nickel (6). At exhibits a very shallow rise that fails to reach saturation even at this doping level, the amount of pure Ni−Ni molecule is 7 T. Compounds 3−5 exhibit behavior that falls between these expected to be virtually undetectable in comparison to the two extremes. amount of Mg−Mg (diamagnetic) and Mg−Ni molecules, and For all five compounds, we begin by fitting the temperature- thus the spectrum should be that of the individual nickel ions dependent susceptibility alongside the field-dependent magnet- (i.e., the exchange coupling interaction is effectively turned off). ization using the spin Hamiltonian given in eq 2. In this model, The powder HFEPR spectrum (Figure 4 and Figure S3 the single-ion anisotropies of the two nickel ions are assumed (Supporting Information)) reveals sharp transitions localized at − − − to be axial (Dxx = Dyy = D/3 and Dzz = 2D/3) and also 5.0 5.5 T, with broader features over the range 8.5 13.5 T,

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called “double shell eﬀect”) proved only to aﬀect the results in a minor fashion (Table 5), and therefore this was not investigated

Table 5. Calculated g, D, and E Parameters for the S =1 Ground State with 8 Electrons in 10 Orbitals, for 1

−1 −1 compd, site gx gy gz D (cm ) E (cm ) 1, Ni1 2.32 2.32 2.29 +4.0 −0.3 1, Ni2 2.31 2.31 2.29 +3.0 −0.2

for the remaining complexes. It is also possible to extract the orientations of the D tensor and the g matrix (see the Supporting Information). The calculated parameters for 1 and the parameters obtained Figure 4. Powder electron paramagnetic resonance spectra of 6 from simulations of 6 differ, chiefly in the magnitude of D (ab measured at 331.2 GHz at 5 and 15 K, with the simulation shown in initio, +4.4/+3.2 cm−1; experimental, +1.6/+1.1 cm−1). The red. gx1 = gy1 = 2.32, gz1 = 2.25, gx2 = gy2 = 2.25, gz2 = 2.19, D1 = +1.6 −1 −1 −1 −1 observed axiality of g is reproduced and is of the same sense (gz cm , E1 = +0.35 cm , D2 = +1.1 cm , E2 = +0.20 cm . Asterisks 32,33 indicate the appearance of a pair of features upon warming to 15 K < gx = gy), which is consistent with a positive D. What is that is recreated in the simulated spectrum. most noticeable about the calculated parameters is the extreme sensitivity of D to the nickel coordination environment. This parameter varies from +8.7 to 3.0 cm−1 with only minor and resembles the spectrum expected of an S = 1 ion with a changes in coordination sphere. This covers as wide an energy rhombic ZFS. range as the observed exchange interactions in nickel These spectra can be modeled as two nonequivalent nickel compounds.9,18 The experimental D value for the isolated sites with the following parameters: gx1 = gy1 = 2.32, gz1 = 2.25, nickel(II) ions in 6 suggests that the calculated values are −1 −1 gx2 = gy2 = 2.25, gz2 = 2.19, D1 = +1.6 cm , E1 = +0.35 cm , D2 somewhat overestimated. The calculated g values are also much −1 −1 = +1.1 cm , E2 = +0.20 cm . Reassuringly, the simulation less sensitive to coordination environment than is found recreates the appearance of a pair of features at around 11.5 T experimentally. (marked with an asterisk) upon warming from 5 to 15 K, which Dinickel Compounds: EPR Spectroscopy. Compounds can be attributed to transitions into the mS = 1 state from the 1−5 were first measured in the powder state at lower increasingly thermally populated mS = 0 state. frequencies (see the Supporting Information). These spectra Single-Ion Anisotropy: Computational Modeling. are extremely difficult to interpret, due to a very limited number Taking inspiration from previous success with cobalt analogues of transitions falling within the available magnetic field range. 31 of these complexes, we modeled compounds 1−5 using Such complexity often arises in compounds where a significant complete active space calculations to generate estimates for the ZFS (which is typical for nickel) is acting on weakly coupled, anisotropic g values and individual site ZFS parameters. non-Kramers ions. As an example we show the Q-band Compounds 2 and 4 possess nickel(II) sites that are related spectrum of compound 1 (Figure 5). Examples for the other by symmetry (2-fold rotation), while compounds 1, 3, and 5 compounds are given in the Supporting Information. have independent sites, with compound 3 having not two but four independent nickel sites (two molecules per asymmetric unit). In all cases, the nickel site that is not the focus of the calculation is replaced by a diamagnetic zinc(II) ion. The active space was chosen as the five 3d orbitals of the nickel(II) ion, where all 10 S = 1 and 15 S = 0 configuration state functions were calculated and mixed by spin−orbit coupling. From these calculations, the gx, gy, gz, D, and E values for the S = 1 ground multiplet can be extracted (Table 4). Expanding the active space to include the 4d orbitals of the nickel(II) ion (the so-

Table 4. Calculated g, D, and E Parameters for the S =1 Ground State with 8 Electrons in 5 Orbitals, for 1−5

−1 −1 compd, site gx gy gz D (cm ) E (cm ) Figure 5. Q-band EPR of 1 measured in the powder state. The red 1, Ni1 2.34 2.34 2.31 +4.4 −0.3 trace is a simulation using the parameters in Table 6. 1, Ni2 2.34 2.33 2.31 +3.2 −0.2 2, Ni1 2.35 2.33 2.28 +8.7 −1.3 3, Ni1 2.29 2.30 2.31 −2.4 +0.5 To allow for the large ZFS, we need to move to frequencies 3, Ni2 2.28 2.29 2.31 −3.0 +0.7 where hν is larger than the ZFS. We therefore collected powder 3, Ni3 2.31 2.30 2.29 −1.7 +0.5 spectra (Figure 6) at a much higher frequency (331.2 GHz) 3, Ni4 2.29 2.29 2.28 +2.0 −0.4 and over a much larger ﬁeld range (0−16 T). Although the 4, Ni1 2.34 2.33 2.28 +6.4 −1.3 deﬁnition of the features is greatly improved, the spectra are 5, Ni1 2.31 2.30 2.29 +3.0 −0.9 still remarkably complicated for such simple compounds, and 5, Ni2 2.30 2.30 2.28 +2.5 −0.6 the variation between spectra is considerable.

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to keep gi, Di, and J coincident in all compounds to reduce the number of parameters and neglected the fact that the metal centers in 3 and 5 were crystallographically nonequivalent (i.e., the parameters for all sites were taken as identical, assuming a pseudosymmetry). fi χ We started by simultaneously tting the MT(T)and M(H,T) data to eq 2 to obtain initial values of J and Di, before introducing these parameters into eq 3 to calculate fi HFEPR spectra, now introducing Ei and re ning J and Di.Asa next step, we introduced Euler angles relating the Di tensors (assuming they are related by a 2-fold axis as in 2 and 4): these angles were fixed from CASSCF calculations (see the Supporting Information). This only made significant improve- ments inand hence were only retained forcompounds 2 and 4, which we assume is because of their much larger Di values (and possibly because they have true crystallographic equivalence of the centers). The final parameters (Table 6) χ were then used to recalculate MT(T) and M(H,T) (Figure 3). For compound 1, where we had the doped materials available, the single-ion parameters were fixed from the doping study. Because we have more information for this complex (hence fewer free variables), we investigated the effect of a small anisotropic component to J (on the order of the dipolar interaction) and found this gave an improvement to the fi − Figure 6. Solid-state high- eld EPR (331.2 GHz) spectra of 1 5. calculated EPR spectra. For 2−5, where we do not have the Experimental traces are in black, with simulations using the parameters doped analogues, J was held as isotropic. in Table 6 plotted in red. The final parameter sets used to generate all of the simulations shown in this paper (including the magnetic Compounds 1, 3, and 5 exhibit their most intense features data) are given in Table 6. The fits to the observed HFEPR between 9−14 T, with effectively no transitions at lower fields. spectra are remarkably good for 1, and the main features of 2− Although compounds 2 and 4 display transitions in this same 5 are also simulated. However, in each case additional region, they also exhibit features at lower fields, between 1 and experimental features are observed that do not arise from this 4 T, that are of a much higher intensity. These low-field simple model. The predicted trend in the magnitude of D from transitions are a signature of significantly larger ZFS terms in ab initio calculations (2, 4 > 1, 3, 5) is supported, but the these examples. calculated values themselves do not give good simulations. A The combination of the doped study and computational key result of this study is that the low-field features observed for − work gives us a guide to the single-ion parameters. To interpret 2 and 4 can only be reproduced with |D| >5cm 1. the complex spectra of 1−5, we have used the spin Hamiltonian In 2−5, there is no obvious benefit to using anisotropic g − − − given in eq 3, where now Dxx = D/3 + E, Dyy = D/3 E, and values, and so isotropic g values are retained. The small Dzz =2D/3. anisotropy in the g values is predicted by ab initio methods, and the magnitude of the g values from both experimental and ab Ĥ = β()SSHSSSSJSŜ ·+gĝ · ⃗ + ̂ ·· D̂ + ̂ ·· D̂ − 2̂ · ̂ 1 1 2 2 112212 12 initio methods are within the expected range for nickel(II) ions, (3) albeit generally overestimated in the latter. This is almost identical with the Hamiltonian used to fit the Experimentally, the sign of D is positive for all compounds, magnetic data, except that we have introduced rhombic ZFS which is in general agreement with ab initio methods, which (E) terms and also the possibility for anisotropic exchange. In predict a positive D in all compounds except for 3, where in fact all cases, the reference frames for the gi matrix and Di tensors three of the nonequivalent centers are predicted to be negative are coincident for a given metal site; however, the reference and the remaining one positive. frames may differ between different nickel sites. Additionally, J Assessing the simulation parameters given in Table 6 is always fixed in the global reference frame. We initially chose alongside the structural parameters in Table 3 allows us to

Table 6. Electronic Parameters Used in the Global Simulations of 1−5

1 2 345

gxx, gyy 2.32/2.25 2.24 2.20 2.26 2.26

gzz 2.25/2.19 2.24 2.20 2.26 2.26 D (cm−1) +1.60/+1.10 +7.40 +2.10 +5.40 +1.80 E (cm−1) +0.35/+0.20 +2.45 +0.10 +1.40 +0.10 −1 − − Jxx, Jzz (cm ) +2.40 +0.35 0.1 +0.70 0.3 −1 − − Jyy (cm ) +2.55 +0.35 0.1 +0.70 0.3 R (deg)a 0, 0, 0 +109, −88.5, −71.0 0, 0, 0 +70.6, +102, −109 0, 0, 0 aEuler rotations of one nickel site in relation to the other in the ZY′Z″ convention.

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Inorganic Chemistry Article comment on a number of possible correlations. First, the ■ ASSOCIATED CONTENT magnitude of the axial parameter, D, is significantly larger in 2 * fi S Supporting Information and 4. This can be explained by the presence of a well-de ned Figures and tables giving infrared spectra for 1−5, full-range axis along the nitrogen atom in the NiNO5 coordination sphere HFEPR spectra for 6, low-temperature Q-band EPR spectra for for these compounds. The addition of a further nitrogen atom 2−5, Euler rotations of g/D from CASSCF calculations, and in 3 and 5, yielding NiN2O4 with oxygen atoms cis to each the INS spectrum of 1 using data from ref 19. This material is other, is accompanied by a reduction in D, as there is no longer available free of charge via the Internet at http://pubs.acs.org. a unique metal−donor atom direction. This magneto-structural ff correlation is very clear. The dramatic di erence in the ■ AUTHOR INFORMATION measured EPR spectra of 1 in comparison with those of 2 and 4 is then due to the much larger change in D between the Corresponding Author * complexes. Second, the exchange is ferromagnetic in nature for E-mail for D.C.: [email protected]. − − 1, 2, and 4, which have the smallest Ni Ow Ni angles of the Notes series. It is worth noting that, despite the ferromagnetic The authors declare no competing financial interest. χ exchange observed in 2, the product MT falls at low temperature, which is due to D being 20 times as large as J. ■ ACKNOWLEDGMENTS This research was funded by the EPSRC (UK); R.E.P.W. holds ■ CONCLUSION a Royal Society Wolfson Research Merit Award. J.P.S.W. thanks Detailed magnetic and EPR studies carried out on five closely the North West Nanoscience DTC for a Ph.D. studentship. ’ related dinickel compounds and on a doped diamagnetic N.F.C. thanks the University of Manchester for a President s analogue of the parent compound have shown that incredibly Doctoral Scholarship. We thank the EPSRC UK National rich and diverse data can be obtained from seemingly simple Electron Paramagnetic Resonance Service at the University of compounds. At first glance, there is no simple correlation Manchester. between structure and magnetic or spectroscopic behavior, e.g. − − ■ REFERENCES compounds 1 and 2 have very similar Ni Ow Ni angles, but in the former case variable-temperature susceptibility measure- (1) Leuenberger, M. N.; Loss, D. Nature 2001, 410, 789−793. χ − ments show an upturn in MT at low temperature and in the (2) Sharples, J. W.; Collison, D. Polyhedron 2013, 54,91 103. latter a downturn. (3) (a) Ohba, S.; Kato, M.; Tokii, T.; Muto, Y.; Steward, O. W. Mol. − However, detailed analysis shows that in these five simple Cryst. Liq. Cryst. 1993, 233, 335 344. (b) Ribas, J.; Escuer, A.; Monfort, M.; Vincente, R.; Cortes, R.; Lezama, L.; Rojo, T. Coord. compounds there are two correlations. The major correlation is − fi Chem. Rev. 1999, 193, 1027 1068. (c) Ruiz, E.; Cano, J.; Alvarez, S.; between the axial zero- eld splitting parameter, D, and the Alemany, P. J. Am. Chem. Soc. 1998, 120, 11122−11129. coordination geometry; where there is a single unique axis, due (d) Rodriguez-Fortea, A.; Alemany, P.; Alvarez, S.; Ruiz, E. Chem. to the presence of a N donor in a NiO5N donor set, the D value Eur. J. 2001, 7, 627−637. (e) Mialane, P.; Duboc, C.; Marrot, J.; is around 3 times larger than in a NiO6 donor set or in a cis Riviere, E.; Dolbecq, A.; Secheresse, F. Chem. Eur. J. 2006, 12, 1950− NiO4N2 donor set. The measured thermodynamic and 1959. (f) Thompson, L. K.; Mandal, S. K.; Tandon, S. S.; Brisdon, J. spectroscopic properties vary most due to this correlation. N.; Park, M. K. Inorg. Chem. 1996, 35, 3117−3125. When this is allowed for, we find that the correlation9,18 (4) (a) Escuer, A.; Aromi, G. Eur. J. Inorg. Chem. 2006, 4721−4736. (b) Gomez,́ V.; Corbella, M.; Roubeau, O.; Teat, S. J. Dalton Trans. between bridging angle and the sign of the magnetic exchange − interaction still appears to be present. This supports the 2011, 40, 11968 11975. (5) Gorun, S. M.; Lippard, S. J. Inorg. Chem. 1991, 30, 1626−1630. hypothesis that the dominant superexchange pathway in these 31 (6) (a) Gregoli, L.; Danieli, C.; Bara, A.-L.; Neugebauer, P.; compounds is via the bridging water molecule. Pellegrino, G.; Poneti, G.; Sessoli, R.; Cornia, A. Chem.Eur. J. 2009, Magneto-structural correlations continue to appear, but these 15, 6456−6467. (b) Maganas, D.; Krzystek, J.; Ferentinos, E.; Whyte, should be treated with considerable skepticism unless one A. M.; Robertson, N.; Psycharis, V.; Terzis, A.; Neese, F.; Kyritsis, P. Hamiltonian is used to simulate all the data. For example, we Inorg. Chem. 2012, 51, 7218−7231. (c) Maurice, R.; de Graaf, C.; could easily have simulated the magnetic susceptibity data of Guihery, N. J. Chem. Phys. 2010, 133, 084307. (d) Barra, A.-L.; 1−5 to a simple isotropic Zeeman plus exchange Hamiltonian Caneschi, A.; Cornia, A.; Gatteschi, D.; Gorini, L.; Heiniger, L. P.; − and described the trends in J with respect to structure. This Sessoli, R.; Sorace, L. J. Am. Chem. Soc. 2007, 129, 10754 10762. would have had no physical meaning, because modeling the (7) Crawford, V. H.; Richardson, H. W.; Wasson, J. R.; Hodgson, D. J.; Hatfield, W. E. Inorg. Chem. 1976, 15, 2107−2110. magnetization data (requiring local anisotropy terms that are (8) Weihe, H.; Güdel, H. U. J. Am. Chem. Soc. 1997, 119, 6539− larger than J) shows not only that such a Hamiltonian is ffi 6543. insu cient but also that the J values so determined are wrong (9) Halcrow, M. A.; Sun, J.-S.; Huffman, J. C.; Christou, G. Inorg. (even giving the wrong sign). This would not be helped by Chem. 1995, 34, 4167−4177. fitting the magnetization data to a second, different (10) Cañada-Vilalta, C.; O’Brien, T. A.; Brechin, E. K.; Pink, M.; Hamiltonian (often based on a giant spin approximation), to Davidson, E. R.; Christou, G. Inorg. Chem. 2004, 43, 5505−5521. determine global zero-field splitting parameters, as this is only (11) Basler, R.; Boskovic, C.; Chaboussant, G.; Güdel, H. U.; Murrie, ≫ M.; Ochsenbein, S. T.; Sieber, A. ChemPhysChem 2003, 4, 910−926 appropriate when J Di. Hence, care needs to be taken in attempting to correlate spin Hamiltonian parameters to and references therein. (12) (a) Piligkos, S.; Bill, E.; Collison, D.; McInnes, E. J. L.; Timco, structure when the anisotropy terms are larger than or 34 G. A.; Weihe, H.; Winpenny, R. E. P.; Neese, F. J. Am. Chem. Soc. comparable to the exchange, and this requires that the data − 35 2007, 129, 760 761. (b) Piligkos, S.; Weihe, H.; Bill, E.; Neese, F.; El are treated with a single Hamiltonian. Any meaningful model Mkami, H.; Smith, G. M.; Collison, D.; Rajamaran, G.; Timco, G. A.; should fit both variable-temperature susceptibility, variable-field Winpenny, R. E. P.; McInnes, E. J. L. Chem. Eur. J. 2009, 15, 3152− magnetization, and other spectroscopic data where available. 3167.

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(13) Magee, S. A.; Sproules, S.; Barra, A.-L.; Timco, G. A.; Chilton, N. F.; Collison, D.; Winpenny, R. E. P.; McInnes, E. J. L. Angew. Chem., Int. Ed. 2014, 53, 5310−5313. (14) Ako, A. M.; Hewitt, I. J.; Mereacre, V.; Clerac,́ R.; Wernsdorfer, W.; Anson, C. E.; Powell, A. K. Angew. Chem., Int. Ed. 2006, 45, 4926− 4929. (15) See for example: Blagg, R. J.; Ungur, L.; Tuna, F.; Speak, J.; Comar, P.; Collison, D.; Wernsodrfer, W.; McInnes, E. J. L.; Chibotaru, L. F.; Winpenny, R. E. P. Nat. Chem. 2013, 5, 673−678. (16) Zadrozny, J. M.; Xiao, D. J.; Atanasov, M.; Long, G. J.; Grandjean, F.; Neese, F.; Long, J. R. Nat. Chem. 2013, 5, 577−581. (17) Wilson, A.; Lawrence, J.; Yang, E.-C.; Nakano, M.; Hendrickson, D. N.; Hill, S. Phys. Rev. B 2006, 74, 140403. (18) (a) Nanda, K. K.; Thompson, L. K.; Bridson, J. N.; Nag, K. J. Chem. Soc., Chem. Commun. 1992, 1337−1338. (b) Thompson, L. K.; Brisdon, J. N.; Nag, K. J. Chem. Soc., Chem. Commun. 1994, 1337− 1338. (19) Chaboussant, G.; Basler, R.; Güdel, H.-U.; Ochsenbein, S.; Parkin, A.; Parsons, S.; Rajaraman, G.; Sieber, A.; Smith, A. A.; Timco, G. A.; Winpenny, R. E. P. Dalton Trans. 2004, 17, 2758−2766. (20) Eremenko, I. L.; Nefedov, S. E.; Sidorov, A. A.; Moiseev, I. I. Russ. Chem. Bull. 1999, 48, 405−416. (21) Sheldrick, G. M. Acta Crystallogr., Sect. A 2008, A64, 112−122. (22) Chilton, N. F.; Anderson, R. P.; Turner, L. D.; Soncini, A.; Murray, K. S. J. Comput. Chem. 2013, 34, 1164−1175. (23) Barra, A. L.; Brunel, L. C.; Robert, J. B. Chem. Phys. Lett. 1990, 165, 107−109. (24) Stoll, S.; Schweiger, A. J. Magn. Reson. 2006, 178 (1), 42−55. (25) (a) Karlström, G.; Lindh, R.; Malmqvist, P.-Å.; Roos, B. O.; Ryde, U.; Veryazov, V.; Widmark, P.-O.; Cossi, M.; Schimmelpfennig, B.; Neogrady, P.; Seijo, L. Comput. Mater. Sci. 2003, 28, 222−239. (b) Veryazov, V.; Widmark, P.; Serrano-Andres,́ L.; Lindh, R.; Roos, B. O. Int. J. Quantum Chem. 2004, 100, 626−635. (c) Aquilante, F.; De Vico, L.; Ferre,́ N.; Ghigo, G.; Malmqvist, P.; Neogrady,́ P.; Pedersen, T. B.; Piton̆ak,́ M.; Reiher, M.; Roos, B. O.; Serrano-Andres,́ L.; Urban, M.; Veryazov, V.; Lindh, R. J. Comput. Chem. 2010, 31, 224−247. (26) Rajaraman, G.; Christensen, K. E.; Larsen, F. K.; Timco, G. A.; Winpenny, R. E. P. Chem. Commun. 2005, 3053−3055. (27) Aromí, G.; Batsanov, A. S.; Christian, P.; Helliwell, M.; Parkin, A.; Parsons, S.; Smith, A. A.; Timco, G. A.; Winpenny, R. E. P. Chem. Eur. J. 2003, 9, 5142−5161. (28) Cotton, F. A.; Wilkinson, G. Chemistry of the Transition Elements. In Advanced Inorganic Chemistry, 4th ed.; Wiley: New York, 1980; p 786ﬀ. (29) Eremenko, I. L.; Golubnichaya, M. A.; Nefedov, S. E.; Sidorov, A. A.; Golovaneva, I. F.; Burkov, V. I.; Ellert, O. G.; Novotortsev, V. M.; Eremenko, L. T.; Sousa, A.; Bermejo, M. R. Russ. Chem. Bull. 1998, 47, 704−718. (30) Titis,̌ J.; Boca,̆ R. Inorg. Chem. 2010, 49, 3971−3973. (31) Boeer, A. B.; Barra, A.-L.; Chibotaru, L. F.; Collison, D.; McInnes, E. J. L.; Mole, R. A.; Simeoni, G. G.; Timco, G. A.; Ungur, L.; Unruh, T.; Winpenny, R. E. P. Angew. Chem., Int. Ed. 2011, 50, 4007−4011. (32) Mabbs, F. E.; Collison, D. Electron Paramagnetic Resonance of d Transition Metal Compounds; Elsevier: Amsterdam, 1992; p 482. (33) Atanasov, M.; Comba, P.; Helmle, S.; Müller, D.; Neese, F. Inorg. Chem. 2012, 51, 12324−12335. (34) Inglis, R.; Jones, L. F.; Milios, C. J.; Datta, S.; Collins, A.; Parsons, S.; Wernsdorfer, W.; Hill, S.; Perlepes, S. P.; Piligkos, S.; Brechin, E. K. Dalton Trans. 2009, 18, 3403−3412. (35) Herchel, R.; Boca,̆ R.; Krzystek, J.; Ozarowski, A.; Duran,́ M.; van Slageren, J. J. Am. Chem. Soc. 2007, 129 (34), 10306−10307.

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SUPPORTING INFORMATION

On the possibility of magneto-structural correlations: Detailed studies of di-nickel carboxylate complexes

James P. S. Walsh,† Stephen Sproules,†,‡ Nicholas F. Chilton,† Anne-Laure Barra,§ Grigore A. Timco,† David Collison,†* Eric J. L. McInnes† and Richard E. P. Winpenny†

†School of Chemistry and Photon Science Institute, University of Manchester, Manchester, M13 9PL, United Kingdom. ‡WestCHEM, School of Chemistry, University of Glasgow, Glasgow, G12 8QQ, United Kingdom. §Laboratoire National des Champs Magnétiques Intenses, 25, rue des Martyrs, B.P. 166, 38042 Grenoble Cedex, France. *[email protected]

−1 1 Table S1. Summary of the vibrational spectra (cm ) in the methyl ν(CH) region for 1–5. 2 1 2 3 4 5 3 i 2966.0 2980.9 2970.8 2972.7 2970.3 2961.6 2949.6 2959.2 2951.0 4 ii 2930.3 2926.0 2921.1 2924.0 2920.2 5 iii 2905.7 2904.3 2898.0 2902.8 2899.0 iv 2870.5 2870.5 2963.8 2868.6 2864.3

4000 3600 3200 2800 2400 2000 1600 1200 800 Table S2. Eigenstates (in cm−1) of the lowest energy manifold -1 Wavenumber (cm ) calculated for 1 using the parameters in Table 6. Figure S1. Overlay of the IR spectra of 1–5 S Energy (in cm−1) measured using crushed polycrystalline 0 15.99 samples. 1 11.79 10.73 10.47 2 2.280 331.2 GHz, 15 K 2.188 1.104 '' / dB 331.2 GHz, 5 K

d 0.2624 0

Table S3. Euler rotations of one nickel site in relation to the other in the ZY´Z ´´ convention. Extracted from CASSCF calculations as 0 2 4 6 8 10 12 14 16 detailed in the text. Field (T) Figure S2. Powder electron paramagnetic Compound R (°) resonance spectra of 6 measured at 331.2 1 +48.1,–139,–144 GHz at 5 and 15 K, with the simulation 2 +109,–88.5,–71.0 shown in red. gx1 = gy1 = 2.32, gz1 = 2.25, gx2 −1 3 +75.0,–34.0,–6.71 = gy2 = 2.25, gz2 = 2.19, D1 = +1.6 cm , E1 = −1 −1 −1 +123,–102,+2.81 +0.35 cm , D2 = +1.1 cm , E2 = +0.20 cm . 4 +70.6,+102,–109 5 +24.9,–18.8,–140

34.13 GHz, 10 K 2 3 34.09 GHz, 10 K '' / dB '' / dB d d

400 800 1200 1600 0 400 800 1200 1600 Field (mT) Field (mT)

34.23 GHz, 10 K 4 5 34.13 GHz, 5 K '' / dB '' / dB d d

0 400 800 1200 1600 0 400 800 1200 1600 Field (mT) Field (mT)

Figure S3. Q-band EPR spectra of 2 (top left), 3 (top right), 4 (bottom left), and 5 (bottom right).

−1 Energy transfer (cm ) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0

0.06 5 K 0.05

0.04

0.03

0.02 Intensity (arb. units) 0.01

0.06 1.5 K 0.05

0.04

0.03

0.02 Intensity (arb. units) 0.01

0.00

0.0 0.5 1.0 1.5 2.0 Energy transfer (meV) Figure S4. INS spectrum of 1 (data reproduced from Ref. 19). Absorptions occur over the energy range of 7.5–14.0 cm−1, which is consistent with the lowest energy manifold calculated using the parameters for 1 in Table 6, and shown in Table S2. 2

5. Paper two: “Large Zero-Field Splittings of the Ground Spin State

Arising from Antisymmetric Exchange Effects in Heterometallic

Triangles”

S. A. Magee, S. Sproules, A.-L. Barra, G. A. Timco, N. F. Chilton, D. Collison, R. E. P. Winpenny and E. J. L. McInnes, Angew. Chem. Int. Ed., 2014, 53, 5310.

Angewandte. Communications

DOI: 10.1002/anie.201400655 Magnetic Anisotropy Large Zero-Field Splittings of the Ground Spin State Arising from Antisymmetric Exchange Effects in Heterometallic Triangles** Samantha A. Magee, Stephen Sproules, Anne-Laure Barra, Grigore A. Timco, Nicholas F. Chilton, David Collison, Richard E. P. Winpenny, and Eric J. L. McInnes*

5 [3, 4] Abstract: [Ru2Mn(O)(O2CtBu)6(py)3] has an S = /2 ground values. The unusual EPR signatures of some trimetallic state with a very large zero-field splitting (ZFS) of D = Cu[7] and FeS[8] enzymes can be explained by these effects. 1 1 2.9 cm , as characterized by EPR spectroscopy at 4– ASE effects on S > =2 states are much less studied, although it 330 GHz. This is far too large to be due to the MnII ion (D has been proposed as a mechanism for otherwise-forbidden 1 < 0.2 cm ), as shown from the {Fe2Mn} analogue, but can be magnetization quantum tunneling steps in Mn12 and other modeled by antisymmetric exchange effects. single molecule magnets[9] and also for the origin of high- order ZFS effects.[10] The magnetic anisotropy in transition-ion clusters is of Belinsky[11] and Tsukerblat et al.[12] have calculated the 3 fundamental importance in areas such as molecular magnet- effects of the ASE on the maximum S = /2 state (as ground or [1] II IV ism, for example giving rise to memory effects, through to excited state) in Cu 3 and V 3 triangles. They showed that [2] the characterization of metalloenzyme active sites. When components of the ASE within the M3 plane could break the the ground state can be described by a total electron spin S > degeneracy, that is, introduce a ZFS. However, in-plane 1 =2, arising from dominant isotropic exchange, the magnetic components are symmetry forbidden when the M3 triangle anisotropy tends to be dominated by the zero-field splitting lies on a mirror plane.[5] They further showed that symmetry- (ZFS) of the (2S + 1)-fold multiplet. This is generally assumed allowed components normal to the triangle do not split the to be dominated by the projection of the local ZFSs of the quartet unless there is a large isosceles distortion.[11] Here we [3] 1 1 metal ions. When the local spins are s = =2 (and have no show that this can explain the huge ZFSs in the S > =2 ground 6 ZFS) or when they are intrinsically isotropic, such as the S state of the heterometallic triangle [Ru2Mn(O)- II Mn ion, it is well understood that anisotropic components of (tBuCO2)6(py)3] ({Ru2Mn}, py = pyridine; Figure 1), which the exchange are the main contributions to the ZFS. In we have characterized by multifrequency EPR spectroscopy contrast, the general significance of the antisymmetric from 4 to 330 GHz. 1 component of the exchange interaction on S > =2 states is {Ru2Mn} and its {Fe2Mn} analogue both crystallize in the

not so clear: here we introduce an example where it provides P21 space group with one molecule per asymmetric unit (ESI): the source of very large ZFS effects. the Mn site is not crystallographically resolved. They have the Antisymmetric exchange (ASE; also known as Dzyalosh- classic structure of basic metal carboxylate triangles,[13] with inski–Moriya exchange) is the origin of spin canting (weak bridging pivalate and terminal pyridine groups (Figure 1). ferromagnetism) in extended lattices. In terms of molecular The acetate analogues of these complexes have been systems, ASE was first observed and discussed in trigonal reported,[14,15] and we find similar magnetic susceptibility (c) [4,5] 3 1 clusters of half-integer spins. This is because they provide behavior. {Ru2Mn} has cT= 4.42 cm Kmol (Figure 2), the the simplest discrete systems in which spin frustration can be studied,[6] since antiferromagnetic coupling in an equilateral 1 triangle leads to two degenerate S = =2 lowest energy states (2E term). ASE provides a mechanism for breaking the 2E degeneracy, and these effects can be quantified by EPR spectroscopy as they are manifested as unusual effective g-

[*] S. A. Magee, Dr. S. Sproules, Dr. G. A. Timco, N. F. Chilton, Prof. D. Collison, Prof. R. E. P. Winpenny, Prof. E. J. L. McInnes School of Chemistry and Photon Science Institute The University of Manchester Oxford Road, Manchester, M13 9PL (UK) E-mail: [email protected] Dr. A.-L. Barra Laboratoire National des Champs Magntiques Intenses UPR 3228 CNRS, UJF-INSA-UPS, BP 166 38042 Grenoble Cedex 9 (France) [**] This work was supported by the EPSRC (UK), including the National [26] Figure 1. Molecular structure of {Ru2Mn}. Scheme: Ru/Mn (large EPR Facility and Service, and The University of Manchester. spheres), O (gray), N (white), C (black), H omitted for clarity. Average

Supporting information for this article is available on the WWW M···M and M-O distances: 3.36(4) and 1.94(4) in {Ru2Mn} and

under http://dx.doi.org/10.1002/anie.201400655. 3.32(4) and 1.91(5) in {Fe2Mn}, respectively.

5310 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2014, 53, 5310 –5313

Angewandte Chemie

Figure 2. cT(T) values for {Ru2Mn} (circles) and {Fe2Mn} (squares) measured in an applied magnetic field (H) of 0.1 T, with simulations (lines) as described in the text. Inset: M(H) at 2 and 4 K.

5 value expected for an isolated S = /2 state, hence the magnetic properties appear at first sight to be those of the isolated MnII ^ ion. Using the isotropic Hamiltonian Hiso [Eq. (1)]: X Figure 3. EPR spectra of {Ru2Mn} at 10 K, measured at: a) 3.87 GHz H^ ¼ g bsˆ H 2J sˆ sˆ 2J sˆ sˆ þ sˆ sˆ ð1Þ iso i i i 1 1 2 2 1 3 2 3 on a CH2Cl2/toluene solution; b) 220.8; and c) 331.2 GHz on a poly- ^ ^ crystalline sample. Simulations (red) based on Hamiltonian Hiso þ Hanti with the parameters in the text; the S-band simulation included an 1 III 5 5 II where s = s = = (Ru , low-spin d ) and s = / (Mn , high- isotropic 55Mn hyperfine coupling (Asˆ Iˆ ; I = 5/ )of 1 2 2 3 2 3 3 2 5 4 1 3 [20] spin d ), this corresponds to the j 0,s3i ground state (j S12,Si). A j A j =85 10 cm (typical for this coordination environment ). Gaussian linewidth parameters: a)15 (x,y) and 3 (z)G, with an A-strain simple Kamb treatment gives this ground state for J1/J2 > 3.5, of 2%; b,c) 1500G Gaussian linewdith with 5% strain in dz (see text with the first excited state (j 1,3/2i) at a relative energy of 2 for definition of the z direction). 7J22J1. Test calculations show that j J1 j must be greater than 3 1 ca. 10 cm and that J1/J2 must be greater than ca. 10 for there to be no evidence of excited-state population in cT(T) at high ions with O,N donor sets are j D j< 0.2 cm1.[20] To test the temperature. Large j J1 j values are justified by the large radial model we have prepared and studied the equivalent 3d II extent of the 4d wavefunctions, and indeed direct overlap as complex {Fe2Mn} in which the Mn ion has the same ^ opposed to super-exchange is possible. Couplings of this coordination environment. Fitting cT(T) for {Fe2Mn} to Hiso III [16] 5 1 magnitude have been observed in homometallic Ru cages. with s1 = s2 = /2 gives J1 = 63.5 and J2 = 21.9 cm with g = Fitting the low-temperature drop in cT(T), and magnet- 2.0 (Figure 2), similar to its acetate analogue.[21] This gives the 5 ^ ization (M) data (Figure 2), for an isolated S = /2 with HZFS j 1,3/2i ground state, as confirmed by low-temperature [Eq. (2)], gives the axial ZFS parameter j D j= 3.0 cm1 (E = magnetization and EPR measurements. Modeling the latter [17] ^ 3 1 0, with fixed g = 1.98). with HZFS for S = /2 gives D =+0.25, j E j= 0.04 cm (fixed g = 2.0; Figures 2 and 4). While the j 1,3/2i ZFS has contri- ^ ^2 SðS þ 1Þ ^2 ^2 butions from all three metal ions, assuming it arises entirely HZFS ¼ gibSˆ H þ D S þ E S S ð2Þ z 3 x y II 1 from Mn (an over-estimation) gives DMn =+0.13 cm . This is in the range known for O,N-donor six-coordinate MnII,[20]

This is much larger than estimated previously for the but negligible compared to the ground-state ZFS of {Ru2Mn}. [15] 1 acetate analogue by low-frequency EPR spectroscopy; Hence {Ru2Mn} has a contribution of about 3 cm to its ZFS therefore, we have measured the value of D directly by from another source. high-frequency EPR spectroscopy.[18] Spectra at low frequen- Given the predictions for ASE effects in the high-spin 1 1 [11] cies are those of an axial effective spin /2 with geff,? 6 and state of s = =2 triangles, we have investigated this model by 1 ^ ^ [4] geff,k 2, consistent with a ground-state M = /2 Kramers the Hamiltonian Hiso þ Hanti, where di are the ASE vectors, 55 ^ doublet, with resolved Mn hyperfine for frozen solution and Hanti is defined in Equation (3). samples (Figure 3a and see Figure S1 in the Supporting Information). Spectra recorded at 220 and 330 GHz H^ d s s d s s s s 3 5 anti ¼ 1 ˆ1 ˆ2 þ 2 ˆ2 ˆ3 þ ˆ3 ˆ1 ð Þ (Figure 3) unambiguously define the S = /2 multiplet, and [19] ^ 1 simulation with HZFS gives D =+2.9 cm (E = 0) with g = 1.98 (see Figure S2 in the Supporting Information). If we neglect the torsion angles of the terminal pyridine

In this simple model the ZFS of the j 0,5/2i state should ligands, then {Ru2Mn} is an isosceles triangle with C2v correspond to that of the isolated MnII site. However, this is symmetry. We define the normal to the trimetallic plane as absurd: the largest values reported for six-coordinate MnII the z axis to maintain consistency with the literature reports

Angew. Chem. Int. Ed. 2014, 53, 5310 –5313 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 5311

Angewandte. Communications

There are no first-order corrections to the eigenvalues, but each ground-state M component mixes with all excited states with DM = 0. The three Kramers doublets are separated and, 1 3 3 5 to second order, the j M j= /2 to /2 gap is half the /2 to /2 gap. 5 This is the form of an isolated S = /2 state under Hamil- tonian (2), with separations of 2D and 4D, respectively. To second order, D is then given by Equation (4), 2dzJ þ dzJ 2 D ¼ 2 1 1 2 ð4Þ ðÞ2J1 7J2 ðÞ2J1 2J2 ðÞ2J1 þ 5J2

where the denominator is the product of the gaps to the three excited states. Excellent simulations of the EPR spectra (Figure 3 and see Figure S3 in the Supporting Information), including variable-temperature and hyperfine effects, are obtained by ^ ^ full diagonalization of Hiso þ Hanti (with gMn = 1.98 and gRu = 2.0; the simulations are insensitive to the latter), and fits to the low-temperature cT(T) and M(H) are indistinguishable from those in Figure 2. For these calculations we have taken 1 J1 = 1000 cm with J1/J2 = 10 (see above). Clearly, more than one set of dz,dz components can generate a given Figure 4. EPR spectra of polycrystalline {Fe2Mn} measured at 10 K and 1 2 z z 1 a) 9.76 and b) 33.97 GHz (5 K), with simulations (red) based on D value. If we take d1 ¼ 0, then we find d2 ¼66 cm ^ z Hamiltonian HZFS with the parameters in the text. Gaussian linewidth (Figure 3; calculations are insensitive to the sign of di ); of 300 G. z much larger values of d1 are required to generate equivalent D values [see Eq. (4)]. Adjustment of J1 and J2 will give z on M3 triangles. In this case, all three metal–metal vectors lie different di values, but the conclusion does not change: very 1 on a mirror plane and the only non-zero components of di are large ZFS effects (several cm ) have been introduced to the z 1 di . Such components do not lead to ZFS in the high-spin S > =2 ground state by ASE effects in second-order. (There [11,12] 5 states of equilateral triangles. In our case we have are also higher-order contributions: the calculated j M j= /2 to 3 3 1 a system that is isosceles not only in terms of the strength of /2 gap is not exactly double the /2 to /2 gap from ^ ^ the J coupling (often seen in homometallic triangles), but in diagonalization using Hiso þ Hanti.) Moreover, the simple

the identity of the spins themselves. description of the magnetic properties of {Ru2M} as being ^ Hamiltonian Hiso gives the 24 24 energy matrix compris- “those of the isolated M ion” is wrong.

ing the ground j 0,5/2i (with eigenvalue + 3J1/2) and excited The ASE parameters we have found here are entirely z 1 j 1,3/2i (J1/2 +7J2), j 1,5/2i (J1/2 +2J2), and j 1,7/2i (J1/ reasonable: di values of greater than 100 cm have been ^ z II [5,7,22] 25J2) states. Applying Hanti, with only di 6¼ 0, as a perturba- found for antiferromagnetically coupled Cu triangles. It tion to this coupled basis gives the non-zero matrix elements has been argued that the latter are due to the favorable [7] (labeling states as j S12,S,Mi): alignment of ground- and excited-state d orbitals, because ASE arises from exchange between the electronic ground state of one ion and the excited state of another through spin–

orbit coupling (SOC). In {Ru2M}, the ASE will be favored both by the large SOC of RuIII, and the strong exchange interactions arising from the large radial extent of the 4d orbitals. The perturbative expressions above show that this can also be viewed as mixing of the ground-state MnII

functions (S12 = 0) with the “ferromagnetic” excited state

(S12 = 1) of the {Ru2} unit. This is equivalent to an alternative description of the electronic structure proposed for some oxo- bridged RuIII oligomers, where direct exchange between the 4d functions results in delocalized singlet and “low lying” (in electronic spectroscopy terms) triplet states.[23,24] In summary, we have shown the ASE interaction can lead to very large spin ground-state ZFSs in polymetallic complexes, even through a second-order perturbation on the isotropic exchange. The model reproduces spectroscopic observations across two orders of magnitude in field/frequency regime of EPR spectroscopy. The results show that these effects cannot be ignored, particularly when 2nd and 3rd

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Angewandte Chemie

row metal ions are involved, as is becoming popular as a route Jones, E. K. Brechin, V. Mosser, W. Wernsdorfer, Phys. Rev. B to introduce magnetic anisotropy into molecular magnets. 2008, 78, 132401. Furthermore, it is possible that ASE is much more widespread [10] N. Kirchner, J. van Slageren, B. Tsukerblat, O. Waldmann, M. Dressel, Phys. Rev. B 2008, 78, 094426. than generally imagined in coupled systems, and assigned to [11] M. I. Belinksy, Inorg. Chem. 2008, 47, 3521; M. I. Belinksy, Inorg. anisotropic exchange or local ZFS effects; for example, Chem. 2008, 47, 3532. 1 a ground state D = 0.25 cm for {Fe2Mn} can equally well be [12] B. Tsukerblat, A. Tarantul, A. Mller, Phys. Lett. A 2006, 353, 48. z 1 derived from d2 ¼0.67 cm , similar to values observed in [13] R. D. Cannon, R. P. White, Prog. Inorg. Chem. 1988, 36, 195. [5] {Fe3} triangles. This is important because different aniso- [14] A. Ohto, Y. Sasaki, T. Ito, Inorg. Chem. 1994, 33, 1245; H. tropy terms in the spin Hamiltonian result in different mixing Kobayashi, N. Ury, R. Miyamoto, Y. Ohba, M. Iwaizumi, Y. between ground and excited states, which have important Sasaki, A. Ohto, T. Ito, Bull. Chem. Soc. Jpn. 1995, 68, 2551. [15] H. Kobayashi, N. Ury, I. Mogi, R. Miyamoto, Y. Ohba, M. consequences for phenomena such as magnetic relaxation, Iwaizumi, Y. Sasaki, A. Ohto, M. Suwabe, T. Ito, Bull. Chem. including, as discussed by others, a possible origin of Soc. Jpn. 1996, 69, 3163. “forbidden” quantum tunneling steps and avoided crossings [16] A. Upadhyay, J. Rajprohit, M. K. Singh, R. Dubey, A. K. observed in several molecular nanomagnets.[9,25] Srivastava, A. Kumar, G. Rajaraman, M. Shanmugam, Chem. Eur. J. 2014, DOI: 10.1002/chem.201304826. Received: January 21, 2014 [17] Modeling of magnetic data used PHI software, see N. F. Chilton, Published online: April 15, 2014 R. P. Anderson, L. D. Turner, A. Soncini, K. S. Murray, J. Comput. Chem. 2013, 34, 1164. [18] On a home-built instrument: A. L. Barra, A. K. Hassan, A. Keywords: antisymmetric exchange · electronic structure · . Janoschka, C. L. Schmidt, V. Schnemann, Appl. Magn. Reson. EPR spectroscopy · exchange coupling · zero-field splitting 2006, 30, 385. [19] EPR analysis and simulations used Weihes SimEPR, with some calculations using routines as described by Piligkos et al., see [1] R. Sessoli, D. Gatteschi, J. Villain, Molecular Nanomagnets, C. J. H. Jacobsen, E. Pederson, J. Villadsen, H. Weihe, Inorg. Oxford University Press, Oxford, 2006. Chem. 1993, 32, 1216; S. Piligkos, E. Bill, D. Collison, E. J. L. [2] W. R. Hagen, Biomolecular EPR Spectroscopy, CRC Press, McInnes, G. A. Timco, H. Weihe, R. E. P. Winpenny, F. Neese, J. Taylor and Francis Group, 2009. Am. Chem. Soc. 2007, 129, 760; S. Piligkos, H. Weihe, E. Bill, F. [3] A. Bencini, D. Gatteschi, EPR of Exchange Coupled Systems, Neese, H. El Mkami, G. M. Smith, D. Collison, G. Rajaraman, Springer, Berlin, 1990. G. A. Timco, R. E. P. Winpenny, E. J. L. McInnes, Chem. Eur. J. [4] B. S. Tsukerblatt, M. I. Belinskii, A. V. Ablov, Dokl. Akad. Nauk 2009, 15, 3152. SSSR 1971, 201, 1410; M. I. Belinskii, B. S. Tsukerblatt, A. V. [20] C. Duboc, M.-N. Collomb, F. Neese, Appl. Magn. Reson. 2010, Ablov, Mol. Phys. 1974, 28, 283; B. S. Tsukerblat, B. Y. Kuyav- 37, 229. skaya, M. I. Belinskii, A. V. Ablov, V. M. Novotortsev, V. T. [21] A. B. Blake, A. Yavari, W. E. Hatfield, C. N. Sethulekshmi, J. Kalinnkov, Theor. Chim. Acta 1975, 38, 131; Y. V. Rakitin, Y. V. Chem. Soc. Dalton Trans. 1985, 2509. Yablokov, V. V. Zelentsov, J. Magn. Reson. 1981, 43, 288. [22] For recent examples, see A. Escuer, G. Vlahopoulou, F. Lloret, [5] For a recent review, see R. Bocˇa, R. Herchel, Coord. Chem. Rev. F. A. Mautner, Eur. J. Inorg. Chem. 2014, 83. 2010, 254, 2973. [23] T. R. Weaver, T. J. Meyer, S. A. Adeyemi, G. M. Brown, R. P. [6] O. Kahn, Chem. Phys. Lett. 1997, 265, 109. Eckberg, W. E. Hatfield, E. C. Johnson, R. W. Murray, D. [7] J. Yoon, E. I. Solomon, Inorg. Chem. 2005, 44, 8076; J. Yoon, Untereker, J. Am. Chem. Soc. 1975, 97, 3039. E. I. Solomon, Coord. Chem. Rev. 2007, 251, 379. [24] J. A. Baumann, D. J. Salmon, S. T. Wilson, T. J. Meyer, W. E. [8] Y. Sanakis, A. L. Macedo, I. Moura, J. J. G. Moura, V. Papaef- Hatfield, Inorg. Chem. 1978, 17, 3342; Y. Sasaki, A. Tokiwa, T. thymiou, E. Mnck, J. Am. Chem. Soc. 2000, 122, 11855; F. Ito, J. Am. Chem. Soc. 1987, 109, 6341. Tiago de Oliveira, E. L. Bominaar, J. Hirst, J. A. Fee, E. Mnck, [25] F. Cinti, M. Affronte, A. G. M. Jansen, Eur. Phys. J. B 2002, 30, J. Am. Chem. Soc. 2004, 126, 5338. 461. [9] M. I. Katsnelson, V. V. Dobrovitski, B. N. Harmon, Phys. Rev. B [26] CCDC 926362 and 926363 contain the supplementary crystallo- 1999, 59, 6919; I. Chiorescu, R. Giraud, A. G. M. Jansen, A. graphic data for this paper. These data can be obtained free of Caneschi, B. Barbarba, Phys. Rev. Lett. 2000, 85, 4807; C. M. charge from The Cambridge Crystallographic Data Centre via Ramsey, E. del Barco, S. Hill, S. J. Shah, C. C. Beedle, D. N. www.ccdc.cam.ac.uk/data_request/cif. Hendrickson, Nat. Phys. 2008, 4, 277; S. Bahr, C. J. Milios, L. F.

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Supporting Information Wiley-VCH 2014

69451 Weinheim, Germany

Large Zero-Field Splittings of the Ground Spin State Arising from Antisymmetric Exchange Effects in Heterometallic Triangles** Samantha A. Magee, Stephen Sproules, Anne-Laure Barra, Grigore A. Timco, Nicholas F. Chilton, David Collison, Richard E. P. Winpenny, and Eric J. L. McInnes* anie_201400655_sm_miscellaneous_information.pdf

Supplementary Information

Synthesis. Solvents and reagents were from commercial sources and used without further purification, with the exception of manganese pivalate which was made by literature methods.[1]

{Ru2Mn}: Pivalic acid (22.1 g, 219 mmol) and RuCl3.xH2O (0.15 g, 0.57 mmol) in 4:1 EtOH/H2O (25 mL) were heated at 70 °C for 10 min. After cooling, manganese pivalate (0.52 g, 2.01 mmol) was added with stirring. After standing overnight, the purple solution was filtered, the solvent removed under vacuum, and the resultant oil filtered to remove excess manganese pivalate. The filtrate was diluted with MeCN (10 mL) and excess pivalic acid removed by successive cooling (-40 °C) and filtration. The solvent was removed, the residue stirred in distilled water (200 mL) for 30 min then filtered to give a dark purple solid which was washed with 4:1 H2O/MeCN (3 × 10 mL) and air dried (0.1 g). The solid was dissolved in MeCN (30 mL) and pyridine (10 mL), stirred overnight and filtered. The solvent was removed and the purple residue recrystallised from a minimum amount of pyridine (25 mg, 26%). Microanalysis: (C45H69MnN3O13Ru2) calcd: C, 48.38; H, 6.23; N, 3.76; Mn, 4.92%. Found: C, 48.38; H, 6.32; N, 3.79; Mn 4.47%. IR (KBr): 1405, 1369, 1356 sh cm–1. ESI-MS + 1 +ve (m/z): 1018 [M – (C5H9O2)] . H NMR (CDCl3): δ 2.04 s (36H, CH3), δ 3.13 s (18H, CH3) ppm.

{Fe2Mn}: Pivalic acid (50.0 g, 490 mmol) and Fe(NO3)3.9H2O (10.0 g, 24.8 mmol) were heated for 4 h with stirring at 160 °C, in a 2-neck round-bottom flask with condenser, to remove NO2. Manganese acetate (3.00 g, 12.2 mmol) was added under N2, and acetic acid removed by addition of toluene (2 x 100 mL) and distillation of the azeotrope. After cooling to room temperature, MeCN (50 mL) was added and the solution stirred for 15 min. The solid was filtered off and washed with MeCN (3 × 15 mL). The solid was dissolved in CH2Cl2 (0.2 g in 20 mL) and excess pyridine (200 mg) with stirring for 20 min. After removing the solvent under vacuum, the residue was recrystallised from diethyl ether/MeCN to give large brown crystals of {Fe2Mn} (0.102 g, 53%). Microanalysis:

(C45H69Fe2MnN3O13) calcd: C, 52.64; H, 6.77; N, 4.09; Fe, 10.88; Mn, 5.35%. Found: C, 52.60; H, 6.89; N, 4.14; Fe, 10.76; Mn, 5.20%. IR (KBr): 1408, 1371, 1357 sh cm–1. ESI-MS +ve (m/z): 846 [M + 1 – (C5H9O2) – (C5H5N)] . H NMR (CDCl3): δ 3.09 s (18H, CH3), δ 4.33 s (36H, CH3) ppm.

X-ray crystallography. X-ray diffraction data were collected on an OXFORD Diffraction XCaliber2 CCD diffractometer using Mo Kα radiation. The structures were solved with SHELXS, and SHELXL was used for the refinement.[2] Crystallographic information files were produced using OLEX2.[3]

Table S1. Crystallographic Data for 1 and 2

1 2

Empirical formula C45H69MnN3O13Ru2 C45H69Fe2MnN3O13

Formula weight 1117.11 1026.67

T, K 100(2) 100(2)

λ, Å 0.71073 0.71073

Crystal system monoclinic monoclinic

Space group P21 P21

a, Å 11.6515(5) 11.6211(5)

b, Å 19.772(1) 19.7837(6)

c, Å 11.8487(6) 11.9437(4)

β, ° 106.661(5) 106.955(4)

3 V, Å 2607.1(2) 2626.3(2)

Z 2 2

–3 Density (calculated), Mg m 1.423 1.298

3 Crystal size, mm 0.5 × 0.5 × 0.3 0.5 × 0.3 × 0.3

Theta range for data collection, ° 3.08 – 28.51 2.98 – 28.02

Reflections collected 10925 19757

Independent reflections 7574 [R(int) = 0.0363] 9845 [R(int) = 0.0476]

Completeness, % 99.7 99.8

Data / restraints / parameters 7574 / 2 / 598 9845 / 2 / 598

2 Goodness-of-fit on F 0.993 1.025

Final R indices [I > 2σ(I)] R1 = 0.0363, wR2 = 0.0843 R1 = 0.0524, wR2 = 0.1043

R indices (all data) R1 = 0.0446, wR2 = 0.0871 R1 = 0.0859, wR2 = 0.1353

–3 Largest diff. peak and hole, e Å 0.930 and -0.560 0.551 and -0.659

Physical measurements. Magnetic measurements were performed in the temperature range 1.8 – 300 K using a Quantum Design MPMS-XL SQUID magnetometer equipped with a 7 T magnet. Corrections for diamagnetism were made using Pascal’s constants and magnetic data were corrected for diamagnetic contributions from the sample holder. Fits were performed using the program PHI.[4] S-, X-, Q- and W-band EPR spectra were measured on Bruker instrumentation at the EPSRC National UK EPR Facility and Service at The University of Manchester. High-frequency, high-field EPR spectra were recorded at the LNCMI-CNRS at Grenoble on a home-built spectrometer.[5] Simulations and data analysis were performed using software written by Weihe,[6] Piligkos[7] and some test calculations performed with Stoll’s EasySpin.[8]

[1] M. A. Kiskin, I. G. Fomina, G. G. Aleksandrov, A. A. Sidorov, V. M. Novotortsev, Y. V. Rakitin, Z. V. Dobrokhotova, V. N. Ikorskii, Y. G. Shvedenkov, I. L. Eremenko, I. I. Moiseev, Inorg. Chem. Commun. 2005, 8, 89. [2] G. M. Sheldrick, Acta Cryst. 2009, A64, 112. [3] O. V. Dolomanov, L. J. Bourhis, R. J. Gildea, J. A. K. Howard, H. Puschmann, J. Appl. Cryst. 2009, 42, 339. [4] N. F. Chilton, R. P. Anderson, L. D. Turner, A. Soncini and K. S. Murray, J. Comput. Chem. 2013, 34, 1164. [5] A.L. Barra, A.K. Hassan, A. Janoschka, C.L. Schmidt, V. Schünemann, Appl. Magn. Reson. 2006, 30, 385. [6] C. J. H. Jacobsen, E. Pederson, J. Villadsen and H. Weihe, Inorg. Chem. 1993, 32, 1216. [7] S. Piligkos, E. Bill, D. Collison, E.J.L. McInnes, G.A. Timco, H. Weihe, R.E.P. Winpenny and F. Neese, J. Amer. Chem. Soc. 2007, 129, 760; S. Piligkos, H. Weihe, E. Bill, F. Neese, H. El Mkami, G. M. Smith, D. Collison, G. Rajaraman, G. A. Timco, R. E. P. Winpenny and E. J. L. McInnes, Chem. Eur. J. 2009, 15, 3152; [8] S. Stoll and A. Schweiger, J. Magn. Reson. 2006, 178, 42.

Fig. S1 Multi-frequency EPR spectra of {Ru2Mn} at 10 K recorded in CH2Cl2/toluene solution at S- (3.874 GHz), X- (9.388 GHz) and Q-band (33.95 GHz), and as a polycrystalline solid at W-band (93.99 GHz).

Fig. S2 331.2 GHz EPR spectrum (black) of polycrystalline {Ru2Mn} at 10 K, with simulation (red) -1 according to Hamiltonian �!"# (2) with S = 5/2, giso = 1.98 and D = +2.9 cm , E = 0 and a Gaussian linewidth of 1500 G with 5% D-strain.

Fig. S3 Variable temperature 331.2 GHz EPR spectra (black) of polycrystalline {Ru2Mn} at (top to bottom) 5, 10 and 15 K. Simulations (red) with Hamiltonian �!"# + �!"#$ and the parameters in the ! main text (Gaussian linewidth of 1500 G with 5% strain in �!).

Fig. S4 Variable temperature 331.2 GHz EPR spectra of polycrystalline {Fe2Mn}.

6. Paper three: “An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes”

N. F. Chilton, D. Collison, E. J. L. McInnes, R. E. P. Winpenny and A. Soncini, Nat. Commun., 2013, 4, 2551.

ARTICLE

Received 15 Jul 2013 | Accepted 4 Sep 2013 | Published 7 Oct 2013 DOI: 10.1038/ncomms3551 An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes

Nicholas F. Chilton1, David Collison1, Eric J.L. McInnes1, Richard E.P. Winpenny1 & Alessandro Soncini2

Understanding the anisotropic electronic structure of lanthanide complexes is important in areas as diverse as magnetic resonance imaging, luminescent cell labelling and quantum computing. Here we present an intuitive strategy based on a simple electrostatic method, capable of predicting the magnetic anisotropy of dysprosium(III) complexes, even in low symmetry. The strategy relies only on knowing the X-ray structure of the complex and the well-established observation that, in the absence of high symmetry, the ground state of dysprosium(III) is a doublet quantized along the anisotropy axis with an angular momentum 15 quantum number mJ ¼ ± /2. The magnetic anisotropy axis of 14 low-symmetry monometallic dysprosium(III) complexes computed via high-level ab initio calculations are very well reproduced by our electrostatic model. Furthermore, we show that the magnetic anisotropy is equally well predicted in a selection of low-symmetry polymetallic complexes.

1 School of Chemistry and Photon Science Institute, The University of Manchester, Oxford Road, Manchester M19 3PL, UK. 2 School of Chemistry, University of Melbourne, Parkville, Victoria 3010, Australia. Correspondence and requests for materials should be addressed to N.F.C. (email: [email protected]) or to A.S. (email: [email protected]).

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3551

he fascinating magnetic properties of the lanthanides have complexes, the ground Kramers doublet of DyIII is strongly continued to be a highly topical and strongly multi- axial with the principal values of the g-tensor approaching those ±15 6 Tdisciplinary research area for over 60 years. Such is the of the mJ ¼ /2 levels of the atomic multiplet H15/2 diversity of this field that their application reaches from magnetic (gx ¼ gy ¼ 0, gz ¼ 20). This empirical observation suggests that a resonance imaging and cell labelling1,2, to potential building simple, but appropriate, variational ansatz for the many-electron blocks of quantum computers3. The pursuit of such applications ground state wavefunction of these low-symmetry complexes relies on detailed knowledge of the magnetic anisotropy, which, consists of the atomic functions w±(a,b) corresponding to the 15 6 while being completely defined in cases of high symmetry, mJ ¼ ± /2 states of the multiplet H15/2. The variational is difficult to elucidate in low-symmetry complexes. Much parameters to be optimized in w±(a,b) consist of the two polar recent work, where 4f complexes have shown slow magnetic angles a and b, which specify the orientation of the quantization 4–7 relaxation and unprecedented non-collinear magnetic textures axis with respect to the low-symmetry crystal field VCF defined by at the single-molecule level8–10, also depends on understanding the ligands. To determine these angles, and hence the full g-tensor the orientation of the magnetic anisotropy. Of all the lanthanide of the ground Kramers doublet of the DyIII complex, we can use III ions, it is Dy that continues to prove the most interesting, the variational principle and minimize the energy E 15=2ða; bÞ¼ providing unexpected examples of new magnetic phenomena, by hiw ða; bÞjVCF j w ða; bÞ with respect to all possible virtue of its unique magnetic anisotropy11. However, because of orientations (a,b) of the quantization axis. the intricate electronic structure of lanthanide complexes, simple This proposed strategy is readily mapped onto a classical models that can predict magnetic anisotropy in molecular solids electrostatic energy minimization problem. Following the work of of low symmetry are still missing. Sievers21, the many-electron wavefunctions w±(a,b) can be ðÞa;b The single-ion properties of 4f metal ions, whether in mono- or described by an electron density distribution r 15=2ðÞy; f , where polymetallic complexes, are difficult to elucidate owing to the y and f are polar angles defined in the reference frame of VCF, shielded nature of the 4f orbitals giving rise to weak interactions expressing the angular dependence of the axially symmetric with the surrounding environment. Recent advances in post aspherical electron density. This aspherical f-electron density can Hartree-Fock multi-configurational ab initio methodology have be written as a linear combination of three spherical harmonics made accurate quantum chemical calculations on paramagnetic 4f Y2,0(y,f), Y4,0(y,f) and Y6,0(y,f), where the coefficients of each compounds possible12. The Complete Active Space Self Consistent are fully determined by angular momentum coupling and average Field (CASSCF) method can accurately predict the magnetic atomic radial multipole moments21. In the particular case of 13,14 III ðÞa;b properties of lanthanide complexes , and calculations of this Dy , r 15=2ðÞy; f can be approximated by an oblate spheroid type have become an indispensable tool for the explanation of distribution owing to the dominant contribution of the 6,15–17 22,21 increasingly interesting magnetic phenomena .These quadrupolar term Y2,0(y,f) to the expansion . calculations are especially useful in cases of low symmetry, where As the crystal field is a one-electron potential, the many- previous methods have provided intractable, over parameterized electron variational integral E±15/2(a,b) can be exactly recast into problems18,19. Although CASSCF ab initio calculations are a simple electrostatic energy integral, describing the interaction extremely versatile and implicitly include all effects required to between the electric potential generated by the crystal field ðÞa;b elucidate the magnetic properties, the results offer little in the way VCF(y,j) and the Sievers charge density r 15=2ðÞy; f associated of chemically intuitive explanations and to obtain reliable results with the f-electrons in the central DyIII ion (equation (1)). requires considerable intervention by expert theorists equipped with Zp Z2p access to powerful computational resources. ðÞa;b E 15ðÞa; b ¼ VCFðÞy; j r ðÞy; f sinðÞy dydj ð1Þ Recently, some of us have applied a simple electrostatic model 2 15=2 to rationalize the unexpected direction of the calculated magnetic y¼0 j¼0 anisotropy in two related sets of monometallic 4f complexes20. III This was based on the aspherical electron density distributions of Thus, we arrive at the hypothesis that in low-symmetry Dy the lanthanide ions, pioneered by Sievers21, and the design complexes, the many-electron ground state wavefunction and, principles for the exploitation of f-element anisotropy outlined by hence, the orientation of the magnetic anisotropy axis, can be Rinehart and Long22. Other groups have also been coming to determined simply by solving a classical electrostatic energy similar conclusions17. Although the use of crystal field methods to minimization problem. model anisotropic magnetic data is widespread23–25, models for the prediction of magnetic anisotropy in low symmetry are Constructing the crystal field potential of charged ligands.To few26,27. These methods are based on the diagonalization of a use this hypothesis, we must determine the explicit form of crystal field Hamiltonian, which, especially in cases of low VCF(y,j) appearing in equation (1), by using an appropriate symmetry, requires a large number of parameters that often can model for the charge distribution on the ligands. This may appear only be reliably determined by fitting experimental data. Such an a difficult problem; here, we use a simple model for charged approach can obscure the rationalization of magnetic anisotropy ligands that are common in many low-symmetry DyIII com- and its predictive power is uncertain. plexes. The charge on the ligands is expected to have a dominant Here we report a quantitative method based on a straightfor- role in the determination of the electrostatic potential experienced ward electrostatic energy minimization for the prediction of the by DyIII, and thus, we can calculate the electrostatic field pro- orientation of the ground state magnetic anisotropy axis of duced by charge on the ligands within a minimal valence bond dysprosium(III) ions, which does not rely on the fitting of (VB) model. Within this model, the charge is delocalized as a experimental data, requiring only the determination of an X-ray resonance hybrid that can be seen as a weighted sum of all crystal structure. ‘chemically stable’ Lewis structures (Fig. 1); this is a representation of the leading contribution to the full VB wavefunction. By taking the sum of the partial charges accumulated by each atom Results in the VB resonance hybrid, qn, we arrive at a very simple frac- Many electron wavefunction and electrostatic minimization. tional charge distribution for the ligand, where, typically, very few An increasing number of ab initio CASSCF calcula- atoms will accommodate a charge and most will remain neutral. tions6,14,15,20,28 have shown that in most low-symmetry Our strategy is to construct the crystal field potential solely from

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3551 ARTICLE

1 1 H 1 H 1 H 1 2 – / + / + /5 + /5 – / – /3 2 5 C N C 3 CH3 O HC C C C O +1 – paaH* +1 NO3 Na HC NH O O N C 1 1 O O 2 1 N O 1 1 – /3 – /3 2 – / – /2 1 + /5 H + /5 – /3 3 – /2 1 O – /2 N O +1 +1 Na – Na3DOTA Na O N 1 – /2 1 O H 1 – 1 2 – /2 1 1 – /3 2 acac : R ,R = CH3 – /2 R C R O N C C tfpb–: R1 = CF3, R2 = Ph – 1 2 O O tta : R = CF3, R = 2-thienyl 1 1 – 1 2 O – / – /3 hfac : R ,R = CF3 1 3 – /2 Br

1 H 1 Ph – /5 C – /5 –1 HCC Ph Si S Ph SiS– 1 – 3 – /5 MeCp HC CH Ph NH O 1 –1/ –1 – /5 5 –1 3– O teabmpH3 –1 N O H N HN Br teaH2– –1 –1 i – i PrO N O Pr O HO O –1 Br

Figure 1 | Partial charges assigned to the formally charged ligands in complexes 1–17. The zwitterionic N-(2-pyridyl)-acetylacetamide (paaH*) has two formal charges owing to the deprotonation of the a-carbon and protonation of the pyridyl nitrogen. Each pendant carboxylate arm of the macrocyclic 0 00 000 trisodium 1,4,7,10-tetraazacyclododecane N,N ,N ,N -tetraacetate ligand (Na3DOTA ) has a single negative charge that is delocalized evenly over the two oxygen atoms; three of the four acetate arms bind sodium cations, which each have a single positive charge. The aromatic anion of methylcyclopentadienyl (MeCp ) has a single negative charge that is delocalized evenly over the five cyclic carbon atoms. Not shown: Compounds 12 and 13 contain Zn2 þ ions, which have formal charges of positive two in our model. Compound 17 contains a central oxide (O2 ) ligand, which has a formal charge of negative two in our model. the fractional charges determined by the VB resonance hybrid, determination of magnetic anisotropy in cases of low excluding neutral atoms entirely (see Fig. 1 illustrating the frac- symmetry. In high symmetry, the orientation of the ground tional charges of the resonance hybrids for the ligands of interest Kramers doublet is pre-determined; note that in this case if ðÞa;b here). The partitioning of the charge without any need for VCF(y,j) does not stabilize the r = ðÞy; f electron density 15 2 15 computation illustrates the elegance of our model. along the symmetry axis, then mJ ¼ ± /2 will not be the Once the partitioning of the charge over the ligand is ground state. There is no such restriction in low-symmetry determined, the resulting partial charges are arranged around environments. the central DyIII ion using the known X-ray crystal structure of the complex, allowing the electrostatic potential to be easily Monometallic complexes. The energy spectra and g-tensors of calculated using crystal field theory29 (equation (2), where the ground 6H multiplets for compounds 1–14, as calculated (R ,y ,j ) are the spherical coordinates of the nth charged atom, 15/2 n n n ab initio, are given in Supplementary Tables S1–S18. Table 1 see Methods section). presents, for each complex, the principal value of the diagonal g- X Xk m tensor of the ground Kramers doublet (gz) and the electrostatic 4pð1Þ k VCFðÞy; j ¼ hr iYk;mðÞy; j deviation angle, defined as the angle between the electrostatic ; ; 2k þ 1 anisotropy axis and the ab initio anisotropy axis along which g is k¼2 4 6 m¼k ð2Þ z X defined. qnYk;mðyn; j Þ n We note that the anisotropy axis is accurately predicted by Rk þ 1 n n employing this minimal VB model, without taking into account different electron withdrawing or donating groups in the charged Minimization of the electrostatic energy in equation (1) in ligands, for example, hfac will have less electron density at the conjunction with this minimal VB model yields an orientation oxygen donor atoms compared with acac because of the of the anisotropy axis, which compares remarkably well electron withdrawing nature of the CF groups in the former with that obtained via rigorous ab initio calculations 3 ligand. (Table 1). To exemplify this correlation, we have calculated the magnetic properties of 14 low-symmetry monometallic dysprosium(III) complexes using the CASSCF ab initio Polymetallic complexes. Calculating the ab initio properties of methodology (see Methods) and compare them directly with the following polymetallic complexes is extremely computation- our electrostatic model. In this work, we focus on the ally expensive. Hence, to demonstrate the power of our simple

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3551

Table 1 | Comparison of ab initio and electrostatic calculations for DyIII complexes.

w Compound gz Electrostatic deviation (°) Reference 34 1 [Dy(acac)3(H2O)2] 19.62 10.9 43 2 [Dy(acac)3(phen)] 19.55 10.0 44 3 [Dy(acac)3(dpq)] 19.42 2.9 44 4 [Dy(acac)3(dppz)] 19.57 6.1 45 5 [Dy(tfpb)3(dppz)] 19.48 9.0 35 6 [Dy(tta)3(bipy)] 19.76 12.4 35 7 [Dy(tta)3(phen)] 19.66 8.0 46 8 [Dy(tta)3(pinene-bipy)] 19.81 6.9 47 9 [Dy(hfac)3(dme)] 19.65 11.0 3 þ 20 10 [Dy(paaH*)2(H2O)4] 19.78 2.4 1 þ 20 11 [Dy(paaH*)2(NO3)2(MeOH)] 19.68 7.4 1 þ 7 12 [DyZn2(teabmpH3)2(MeOH)] 19.98 6.6 1 þ 7 13 [DyZn2(teabmpH3)2] 19.90 8.0 2 þ z y 14,37 14 [Dy(DOTA)(H2O)Na3] 19.46 14.8

bipy, 2,20-bipyridine; dme, dimethoxyethane; dpq, dipyridoquinoxaline; dppz, dipyridophenazine; pinene-bipy, 4,5-pinene bipyridine; phen, 1,10-phenanthroline. wAngle between anisotropy axis calculated by ab initio and electrostatic methods. z Average gz value over four calculations, see Supplementary Tables S14–S18. yAngles between the experimentally determined and the ab initio and electrostatic anisotropy axes are 3.9° and 12.1°, respectively.

approach, we have performed a semi-quantitative comparison interact strongly with the two basal oxygen atoms. If, however, between the electrostatic calculations and published ab initio the quantization axis was parallel with the base and top of the results for three compounds: [Dy(MeCp)2(Ph3SiS)]2 15 (ref. 28) trapezium, there would be less interaction with the negative i [Dy6(teaH)6(NO3)6] 16 (refs 30,31) and [Dy5O( PrO)13] 17 charges thus stabilizing the orientation. This is the orientation of (refs 32,33). The methodology for the calculation of the the anisotropy axis (Fig. 2c and Supplementary Fig. S9) calculated electrostatic anisotropy axes in polymetallic complexes is by our electrostatic model and it provides a simple explanation identical to that of monometallic complexes and is performed for the ab initio results. Analogous arguments can be made for for each DyIII ion independently. The DyIII ions that are not the compound 11 where the b-diketonate oxygen donors are in a focus of the calculation are treated as part of the ligand and are similar trapezium-shaped arrangement, however, the coordina- given a þ 3 charge. The charged ligands in complexes 15–17 are tion environment now contains two chelating nitrate anions. The given in Fig. 1, which describes the charge partitioning based on oxygen atoms in NO3 have a larger negative partial charge than the minimal VB model. those in the b-diketonates, but this is offset by the positive charge on the nitrogen atom, which has an attractive effect on the electron density. Therefore, more-or-less the same anisotropy axis Discussion as in compound 10 is observed for compound 11, along the qn The form of the potential (equation (2)) contains terms of Rk þ 1 for diketonate-diketonate vector (Supplementary Fig. S10). each charged atom in the ligand, which implies that the closern to Compounds 1–10 have distorted square anti-prismatic geo- the DyIII ion and larger the magnitude of the charge, the greater metries and the calculated anisotropy axis of the ground state is its effect on the orientation of the anisotropy axis. Complexes 1–9 not found to be coincident with the pseudo-fourfold axis. This contain three b-diketonate ligands in a ‘paddle-wheel’-like observation, shown here to be due to simple electrostatic arrangement, with two b-diketonate ligands trans- to each other arguments, is contrary to many reports in the literature that and the third trans- to a neutral ligand (Fig. 2a). If the employ a fourfold axial interpretation to model the magnetic data ðÞa;b quantization axis of the r 15=2ðÞy; f electron density was along (refs 25,34–36). Clearly, in these cases, the electrostatics are more the ‘paddle-wheel’ axis, then the radial plane of the approximately important than pseudo-symmetry. ‘oblate’ density would be coincident with all three charged Compounds 12 and 13, chosen as a departure from b- ligands, representing a high-energy orientation. Therefore, the diketonate-based complexes in addition to their very interesting anisotropy axis is perpendicular to the ‘paddle-wheel’ axis, and we magnetic properties, are intimately related and can be inter- find that it passes through the two trans-b-diketonate ligands. converted reversibly by drying or soaking in methanol, via a single The radial plane of the oblate electron density is thus coincident crystal to single crystal transformation7. The difference between with only one charged ligand as opposed to two (Fig. 2b and the dysprosium(III) coordination environments is the removal of Supplementary Figs S1–S8). a terminal methanol molecule, changing the local symmetry from For compound 10, the negatively charged oxygen atoms of the a distorted pentagonal bipyramid to a distorted octahedron. two b-diketonate ligands are the closest to DyIII and therefore Considering the local DyIII coordination environment, a pair of have the greatest effect on the orientation of the anisotropy axis. trans- phenoxo oxygen atoms in both 12 and 13 have much These four atoms are roughly coplanar with the dysprosium(III) shorter Dy–O bond lengths than all others, at 2.21(2) Å compared ion and are arranged in a trapezium (Fig. 2c). The oblate electron with 2.39(2) Å for 12 and 2.186(4) Å compared with 2.3(1) Å for ðÞa;b density r 15=2ðÞy; f will be in a high-energy configuration when 13, thus defining both geometries as axially compressed. In both the quantization axis is normal to this plane of four negative cases, the three metal atoms are roughly coplanar with the charges, and therefore the minimum electrostatic energy and equatorial planes of the coordination polyhedra. The close oxygen anisotropy axis will lie in the plane of the b-diketonate oxygen atoms define an axially repulsive potential for dysprosium(III), atoms. The DyIII ion is much closer to the base of the trapezium which, coupled with the attractive nature of the Zn2 þ cations in and therefore if the quantization axis was to bisect the two the equatorial plane, explains the observed magnetic anisotropy parallel edges, then the radial plane of the electron density would axes (Supplementary Figs S11 and S12).

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a H c R1 C R2 C C HC

H O O C R1 R1 C H

C O O C C Dy NH HC CH 3 H N C O O C CH R2 R2

EE C 2.765 OO

H C H C 2.752 2.766 Dy b C C OO 4.308 CH H N

H C

C H

Figure 2 | Magnetic anisotropy and electrostatic potential of b-diketonate species. (a) Idealized ‘paddle-wheel’ geometry of complexes 1–9 viewed down the ‘paddle-wheel axis’. (b) Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy axes for the ground Kramers doublet in 1, ðÞa;b 21 with the form of the oblate r 15=2ðÞy; f f-electron density using the representation of Sievers , magniﬁed approximately 5 for clarity; Dy ¼ green, O ¼ red, C ¼ grey and H ¼ white. (c) Idealized electrostatic potential generated by a pair of paaH* ligands in the trapezium-shaped coordination environment present in the bis-b-diketonate-dysprosium(III) plane of compounds 10 and 11 and the resultant anisotropy axis (green line); red ¼ positive potential, blue ¼ negative potential.

600 a

400 a a

) b –1 200 360 c 270 180 0 b 135 Energy (cm 180 (°) b 90 –200 90 (°) c 45 0 0 –400 c

ðÞa;b Figure 3 | Orthogonal configurations for the magnetic anisotropy axis in compound 14. The form of the oblate r 15=2ðÞy; f f-electron density, visualized in the central structural diagrams, follows the representation of Sievers21, and has been magnified approximately 5 for clarity; Dy ¼ green, Na ¼ yellow, O ¼ red, N ¼ blue, C ¼ grey and H ¼ white. The rods in the central structural diagrams represent the magnetic anisotropy axes, identified by various methods: green ¼ electrostatic calculation, dark blue ¼ ab initio calculation (this work), light blue ¼ ab initio calculation14 and pink ¼ experimentally determined. The energies of each configuration have been calculated using our electrostatic approach. The electrostatic energy surface is calculated by considering all possible orientations (a,b) of the anisotropy axis in the potential generated by the charged ligands of compound 14. The electrostatic energy ðÞa;b surface shows double degeneracy of the minimum, maximum and saddle points due to the axial symmetry of the r 15=2ðÞy; f electron density.

2 þ In [Dy(DOTA)(H2O)Na3] 14, for which the magnetic conﬁguration (Fig. 3a). Therefore, the anisotropy axis is anisotropy axis has been determined experimentally, the perpendicular to the pseudo-tetragonal axis of the molecule. dysprosium(III) ion is encapsulated by the macrocyclic DOTA4 The radial plane of the ‘oblate’ density is attracted by two Na þ ligand (ref. 37). The H atoms of the apical water molecule were ions more strongly than just a single Na þ ion (Fig. 3b), thus not found experimentally, so were placed in calculated positions determining the observed orientation of the anisotropy axis based on crystallographically characterized water molecules (Fig. 3c), which agrees well with the experimentally determined bound to LnIII ions (ref. 20). If the anisotropy axis was and ab initio axes (Table 1). We have rotated the apical water coincident with the pseudo-tetragonal axis, the radial plane of molecule in the ab initio calculations and found no dependence the oblate electron density would have a large interaction with the on the orientation of the ground state anisotropy axis to this four negatively charged acetate groups, yielding a high-energy perturbation, contrary to ref. 14.

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Compound 15 contains two dysprosium(III) ions bridged by presented, we propose that this model can be used to aid in the two anionic Ph3SiS ligands and each capped by two classical rational design of molecular architectures displaying novel organometallic ligands, MeCp (ref. 28). The two ions are magnetic properties, exploiting and stabilizing the strong axiality related by inversion symmetry and therefore possess the same of the ground state of low-symmetry dysprosium(III), through single-ion electronic structure. The electrostatic potential at each the use of formally charged ligands. The simplicity of the proposal paramagnetic ion is dominated by the two MeCp ligands, is so profound that the model resonates strongly with the which are closer to the DyIII ion than the sulfur atoms at conclusions drawn in the 1950s and 1960s that the bonding of the 2.65(3) Å (average Dy–C distance) compared with 2.76(2) Å. This lanthanides is almost purely ionic (refs 38,39). leads to the anisotropy axis of the ground state lying By entirely neglecting the influence of neutral ligands in our perpendicular to the Dy-S-S-Dy plane (Supplementary model, we have shown the dominant nature of charged ligands in Fig. S13), in good agreement with ab initio calculations (ref. 28). the determination of the magnetic anisotropy of dysprosium(III) The hexametallic dysprosium(III) wheel (16) contains highly complexes. Compounds lacking any charged ligands are rare, but anisotropic paramagnetic centres, yet due to crystallographic S6 would likely show magnetic anisotropies that are much more symmetry, possesses a diamagnetic ground state with a toroidal sensitive to the type of ligands present, with contributions due to moment. Each DyIII ion is encapsulated by the teaH2 ligand dipoles and higher order multipoles as well as the spatial extent of and also has one chelating nitrate anion30. Applying our ligand electron density becoming important. The minimal VB electrostatic model to this compound yields the anisotropy axis model for the partitioning of charges on ligands works well for of each dysprosium(III) site in excellent agreement with the ab the formally charged ligands presented here. Other more general initio results (Supplementary Fig. S14)31. The anisotropy axis for schemes for the partitioning of atomic charges over the ligands each DyIII ion is canted around the ring in an alternating up/ are also being investigated, which may offer an improvement over down manner, which, due to the S6 symmetry of the molecule, the minimal VB model. causes the net cancellation of the out-of-plane magnetization in We are also extending the method to other lanthanide ions, the ground state31, leading to a toroidal moment8, similar to that examining whether this approach can work for other oblate ions 9,10 III observed in a Dy3 triangle . It is remarkable that such a simple (for example, Tb ) and whether the reverse electrostatic principles electrostatic approach can rationalize such complex physics. will apply to prolate ions (for example, ErIII). Although the Compound 17, with one of the highest energy barriers to the treatment presented here cannot be rigorously applied to non- reversal of the ground state magnetization, contains five Kramers ions in low-symmetry environments, TbIII complexes that III dysprosium(III) ions arranged in a pyramid, with each Dy possess a pseudo-doublet ground state with mJ ¼ ±6(gx ¼ gy ¼ 0, ion at the centre of an axially compressed octahedron. The gz ¼ 18) should follow similar electrostatic arguments to those equatorial plane of each ion is formed by four bridging discussed here for the determination of the magnetic anisotropy. isopropoxide (iPrO ) ligands and the axial positions are Conversely, preliminary results suggest that the ground state III occupied by the single m5-oxide bridge at the centre of the wavefunctions of Er ions in low-symmetry environments are not 32 molecule and a terminal isopropoxide ligand . The oxygen atom well defined and consist of strongly mixed mJ states, precluding the of the terminal iPrO ligand is substantially closer to the DyIII application of the treatment presented here. ion than all other donor atoms, at 2.04(1) Å compared with The work presented here is an advance not only for the chemistry 2.35(8) Å, and the central m5-oxide has a double negative charge. and physics communities involved with molecular magnetism, but These two features define a strongly repulsive axial potential for also for all areas concerned with the magnetic and spectroscopic ðÞa;b the r 15=2ðÞy; f electron density, where the energy is minimized properties of the lanthanides. Specifically, this work provides some when the quantization axis is coincident with this direction much-needed insight into the complex and continually intriguing (Fig. 4). The presence of the other charged ligands and the magnetic properties of dysprosium(III). High-level quantum trivalent dysprosium ions is a small perturbation in this case, chemical calculations such as CASSCF are computationally because of the strongly directional nature of the almost linear expensive and require intervention by experts to produce reliable iPrO -Dy-O2 axis. The results obtained using our electrostatic results. The complementary approach we outline here is available to model compare extremely well with those obtained using ab initio any chemist with minimal computational requirements. calculations33 (see Supplementary Table S19). Calculation of the ground state magnetic anisotropy axis of III Methods Dy in low-symmetry environments, employing an electrostatic Electrostatic calculations. The electrostatic calculations were performed on the minimization strategy, shows how simple chemical intuition can complete monometallic structures (excluding lattice solvent or non-coordinated aid in the understanding of a complex problem of electronic counter ions) using the reported X-ray geometry with no optimization. The charges structure. Given the success of the electrostatic model in the cases were assigned to the ligands as described in the text and all other atoms did not contribute to the potential. The angles describing the orientation of the ðÞa;b r 15=2ðÞy; f electron density with respect to the experimental geometry, (a,b), were then optimized to minimize the electrostatic energy. In applying this strategy, we have evaluated the uncertainty associated with using X-ray coordinates by moving the atomic positions by a random factor within the estimated standard deviations and found that the change in the orientation of the anisotropy axis over all monometallic compounds studied is on the order of 1°. We have elected to use the Freeman and Watson values for the average radial integrals (ref. 40), /rkS,and have investigated the effect of these values on the calculated anisotropy direction and also found deviations on the order of 1° when their values are altered non- systematically by up to 20%. The electrostatic calculations were implemented in a FORTRAN program, MAGELLAN, which is available from the authors upon request.

Figure 4 | Ground state magnetic anisotropy of compound 17. The blue Ab initio calculations. CASSCF calculations were performed with MOLCAS 7.6 rods represent the orientations of the anisotropy axes for each of the ﬁve (refs 12,41,42) on the same geometry as used for the electrostatic calculations. The III ANO-RCC-VTZP, VTZ and VDZ basis sets were used for the dysprosium ion, ﬁrst Dy ions in complex 17 as calculated by our electrostatic model; coordination sphere atoms and all other atoms, respectively. The calculations Dy ¼ green, O ¼ red, C ¼ grey and H ¼ white. employed the second order Douglas–Kroll–Hess Hamiltonian, where scalar

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Chem. Soc. Rev. 40, 3092–3104 (2011). figures. R.E.P.W. thanks The Royal Society for a Wolfson research merit award. 19. Sessoli, R. & Luzoń, J. Lanthanides in molecular magnetism: so fascinating so challenging. Dalton Trans. 41, 13556–13567 (2012). Author contributions 20. Chilton, N. F. et al. Single molecule magnetism in a family of mononuclear N.F.C. and A.S. designed the research and performed the calculations. E.J.L.M., D.C. and b-diketonate lanthanide(III) complexes: rationalization of magnetic anisotropy R.E.P.W. provided detailed critique of the approach. N.F.C. and A.S. wrote the manu- in complexes of low symmetry. Chem. Sci. 4, 1719–1730 (2013). script with input from the other authors. 21. Sievers, J. Asphericity of 4f-shells in their Hund’s rule ground states. Z. Phys. B Con. Mat. 45, 289–296 (1982). 22. Rinehart, J. D. & Long, J. R. Exploiting single-ion anisotropy in the design of Additional information f-element single-molecule magnets. Chem. Sci. 2, 2078–2085 (2011). Supplementary Information accompanies this paper at http://www.nature.com/ 23. Jiang, S.-D. et al. Series of lanthanide organometallic single-ion magnets. Inorg. naturecommunications Chem. 51, 3079–3087 (2012). 24. Yamashita, K. et al. A luminescent single-molecule magnet: observation of Competing financial interests: The authors declare no competing financial interests. magnetic anisotropy using emission as a probe. Dalton Trans. 42, 1987–1990 Reprints and permission information is available online at http://npg.nature.com/ (2013). reprintsandpermissions/ 25. Pointillart, F. et al. A series of tetrathiafulvalene-based lanthanide complexes displaying either single molecule magnet or luminescence—direct magnetic and How to cite this article: Chilton, N.F. et al. An electrostatic model for the determination photo-physical correlations in the ytterbium analogue. Inorg. Chem. 52, of magnetic anisotropy in dysprosium complexes. Nat. Commun. 4:2551 doi: 10.1038/ 5978–5990 (2013). ncomms3551 (2013).

Supplementary Figure S1 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 2. Dy = green, O = red, N = blue, C = grey and H = white.

Supplementary Figure S2 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 3. Dy = green, O = red, N = blue, C = grey and H = white.

Supplementary Figure S3 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 4. Dy = green, O = red, N = blue, C = grey and H = white.

Supplementary Figure S4 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 5. Dy = green, F = yellow, O = red, N = blue, C = grey and H = white.

Supplementary Figure S5 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 6. Dy = green, S = purple, F = yellow, O = red, N = blue, C = grey and H = white.

Supplementary Figure 6 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 7. Dy = green, S = purple, F = yellow, O = red, N = blue, C = grey and H = white.

Supplementary Figure S7 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 8. Dy = green, S = purple, F = yellow, O = red, N = blue, C = grey and H = white.

Supplementary Figure S8 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 9. Dy = green, S = purple, F = yellow, O = red, N = blue, C = grey and H = white.

Supplementary Figure S9 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 10. Dy = green, O = red, N = blue, C = grey and H = white.

Supplementary Figure S10 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 11. Dy = green, O = red, N = blue, C = grey and H = white.

Supplementary Figure S11 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 12. Dy = green, Br = yellow, Zn = teal, O = red, N = blue, C = grey and H = white.

Supplementary Figure S12 – Comparison of the ab initio (blue rod) and electrostatic (green rod) anisotropy directions for the ground Kramers doublet in 13. Dy = green, Br = yellow, Zn = teal, O = red, N = blue, C = grey and H = white.

Supplementary Figure S13 – The ground state anisotropy directions (blue rods) for the DyIII ions in 15, calculated with the electrostatic model. Dy = green, S = purple and C = grey; H atoms not shown for clarity.

100

Supplementary Figure S14 – The ground state anisotropy directions (blue rods) for the DyIII ions in 16, calculated with the electrostatic model. Dy = green, O = red, N = blue and C = grey; H atoms not shown for clarity.

101

Supplementary Table S1 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 1. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.00 0.01 19.62 - 2 167.4 0.17 0.27 15.89 7.1 3 252.7 1.81 2.43 11.75 11.7 4 309.7 1.40 2.00 7.56 14.5 5 338.3 2.65 7.73 11.63 89.5 6 431.3 0.01 0.28 15.29 86.3 7 475.9 0.09 0.22 17.69 70.7 8 549.3 0.02 0.04 19.11 60.2

Supplementary Table S2 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 2. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.00 0.01 19.55 - 2 142.2 0.16 0.24 15.84 16.4 3 211.0 1.20 1.59 12.42 17.8 4 262.3 3.82 5.37 8.44 14.0 5 310.5 2.25 4.03 9.68 73.6 6 350.2 1.12 2.37 16.52 72.3 7 457.2 0.01 0.06 17.61 85.0 8 530.1 0.01 0.03 18.90 57.0

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Supplementary Table S3 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 3. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.01 0.02 19.42 - 2 133.3 0.37 0.52 15.90 7.1 3 202.5 2.67 4.75 11.17 25.0 4 240.6 2.07 5.78 7.32 89.4 5 284.0 0.52 3.30 10.05 67.1 6 313.1 1.05 3.05 15.36 76.6 7 444.4 0.00 0.02 19.21 84.2 8 545.1 0.00 0.02 19.71 52.9

Supplementary Table S4 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 4. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.00 0.01 19.57 - 2 159.8 0.17 0.26 15.72 1.8 3 239.0 1.95 2.96 11.54 12.4 4 276.4 1.95 5.75 10.95 72.4 5 321.3 1.11 4.26 9.42 69.4 6 365.1 1.56 3.38 15.85 67.3 7 448.3 0.05 0.29 18.81 89.7 8 519.1 0.04 0.09 19.47 52.0

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Supplementary Table S5 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 5. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.01 0.02 19.48 - 2 134.1 0.61 1.60 14.88 6.9 3 163.9 0.65 2.15 16.66 89.3 4 211.7 1.02 4.42 9.70 9.5 5 243.8 3.58 4.60 11.42 51.8 6 273.7 0.63 1.76 16.46 84.0 7 336.8 0.02 0.04 19.67 89.5 8 476.2 0.00 0.01 19.86 49.6

Supplementary Table S6 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 6. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.00 0.01 19.76 - 2 142.4 0.09 0.65 16.50 63.7 3 161.7 1.67 2.00 13.63 22.9 4 201.1 3.28 4.62 12.42 44.7 5 234.2 2.47 3.73 12.04 67.1 6 305.9 0.74 1.07 16.12 79.9 7 415.0 0.02 0.21 17.53 80.5 8 470.6 0.05 0.20 18.57 51.5

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Supplementary Table S7 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 7. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.01 0.01 19.66 - 2 134.2 0.72 0.91 15.79 11.2 3 181.3 3.34 4.90 10.68 25.9 4 216.5 1.38 5.99 6.78 38.7 5 243.8 1.81 2.78 12.29 56.3 6 298.8 0.77 1.23 16.03 79.1 7 411.6 0.05 0.14 18.56 85.7 8 516.5 0.02 0.04 19.49 53.0

Supplementary Table S8 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 8. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.00 0.01 19.81 - 2 94.3 0.12 0.41 18.84 78.6 3 159.6 2.12 2.41 13.99 18.7 4 185.9 3.65 5.75 11.39 36.1 5 215.1 2.24 2.80 13.55 63.1 6 281.8 0.51 0.78 16.44 84.8 7 409.1 0.07 0.08 19.17 49.0 8 472.0 0.02 0.15 19.49 89.1

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Supplementary Table S9 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 9. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.01 0.01 19.65 - 2 118.5 0.58 0.96 15.96 12.8 3 164.9 3.85 5.73 10.06 51.8 4 196.4 3.70 5.05 7.46 48.8 5 228.6 4.02 4.93 10.95 53.8 6 273.2 1.31 2.27 16.04 85.9 7 344.8 0.10 0.23 19.11 89.1 8 457.1 0.01 0.02 19.70 51.0

Supplementary Table S10 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 10. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.00 0.00 19.78 - 2 250.5 0.05 0.07 16.40 2.0 3 408.4 1.18 1.90 12.50 1.5 4 508.3 5.35 5.95 6.92 3.5 5 600.3 2.07 2.19 10.39 89.4 6 689.8 0.36 0.56 14.89 88.0 7 774.5 0.01 0.03 19.30 67.9 8 817.8 0.01 0.02 19.48 72.8

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Supplementary Table S11 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 11. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.01 0.02 19.68 - 2 113.6 0.23 0.44 16.56 7.9 3 194.9 3.43 4.14 11.47 25.5 4 249.7 3.57 4.19 10.01 79.9 5 307.3 0.59 0.89 15.47 80.2 6 338.5 0.22 1.77 15.54 83.8 7 394.3 0.77 1.51 17.04 83.6 8 432.4 0.08 1.68 17.65 81.0

Supplementary Table S12 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 12. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.00 0.00 19.98 - 2 270.0 0.46 0.55 18.61 84.8 3 294.6 0.14 0.25 16.58 4.6 4 346.8 1.34 1.84 14.28 84.0 5 410.8 4.57 5.92 8.84 59.0 6 447.3 1.53 2.01 12.25 26.7 7 484.8 0.41 0.50 16.21 49.2 8 526.3 0.02 0.10 17.57 50.3

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Supplementary Table S13 – Energies and g-tensors for the Kramers doublets of the ground multiplet of DyIII for compound 13. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.01 0.02 19.90 - 2 287.8 2.38 6.82 11.77 56.2 3 324.0 3.02 5.50 8.40 42.5 4 406.6 0.37 2.38 14.22 76.5 5 481.0 2.03 7.31 9.69 46.8 6 559.0 2.19 2.44 15.59 57.6 7 697.9 0.43 0.74 16.48 82.7 8 816.8 0.09 0.20 19.09 82.5

Supplementary Table S14 – Energies and g-tensors for the Kramers doublets of the III ground multiplet of Dy for compound 14, H2O rotation A. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.07 0.14 19.67 - 2 94.3 1.10 1.43 17.67 77.3 3 167.2 3.03 3.94 11.18 20.7 4 241.4 0.99 6.74 8.78 85.3 5 311.4 2.85 4.35 10.50 77.1 6 374.0 0.55 1.79 13.55 85.6 7 449.5 0.68 0.82 16.44 85.2 8 621.6 0.01 0.03 19.64 86.9

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Supplementary Table S15 – Energies and g-tensors for the Kramers doublets of the III ground multiplet of Dy for compound 14, H2O rotation B. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.14 0.29 19.51 - 2 72.5 0.96 1.11 17.97 78.1 3 156.7 4.05 4.30 11.18 23.9 4 228.4 0.01 7.23 7.90 84.9 5 299.7 2.28 5.38 10.50 73.8 6 361.5 0.95 2.29 13.32 90.0 7 441.9 0.77 1.01 16.50 82.3 8 611.3 0.02 0.06 19.63 87.5

Supplementary Table S16 – Energies and g-tensors for the Kramers doublets of the III ground multiplet of Dy for compound 14, H2O rotation C. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.30 0.83 18.93 - 2 45.8 0.23 1.14 17.61 80.8 3 144.6 4.27 6.46 9.82 40.2 4 209.8 1.92 5.75 7.81 70.8 5 284.1 3.18 4.93 10.47 74.0 6 346.5 0.34 1.59 13.55 85.5 7 424.3 0.78 0.85 16.55 83.1 8 595.7 0.01 0.04 19.65 87.2

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Supplementary Table S17 – Energies and g-tensors for the Kramers doublets of the III ground multiplet of Dy for compound 14, H2O rotation D. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.05 0.08 19.75 - 2 117.8 1.30 2.39 16.58 67.3 3 178.9 1.26 3.79 10.96 28.4 4 265.0 2.37 6.69 8.99 85.2 5 329.8 1.35 4.65 10.55 74.8 6 391.0 1.31 2.71 13.26 88.0 7 466.7 0.66 0.87 16.47 84.3 8 637.4 0.03 0.05 19.63 87.1

Supplementary Table S18 – Energies and g-tensors for the Kramers doublets of the III ground multiplet of Dy for compound 14, average of the four H2O rotations A – D. The angles between the principal axis of each doublet and the ground state are also given.

-1 Doublet Energy (cm ) gx gy gz Angle (°) 1 0.0 0.1(1) 0.3(3) 19.5(4) - 2 83(31) 0.9(5) 1.5(6) 17.5(6) 74.9 3 162(15) 3(1) 5(1) 10.8(7) 27.3 4 236(23) 1(1) 6.6(6) 8.4(6) 79.8 5 306(19) 2.4(8) 4.8(4) 10.50(4) 74.8 6 368(19) 0.8(4) 2.1(5) 13.4(2) 87.5 7 446(18) 0.72(6) 0.89(8) 16.49(4) 84.0 8 616(18) 0.02(1) 0.05(1) 19.637(7) 86.9

Supplementary Table S19 – Angles between anisotropy axes and terminal iPrO--Dy bond for dysprosium(III) sites in complex 17.

Method Dy1 (°) Dy2 (°) Dy3 (°) Dy4 (°) Dy5 (°) Ab initio 5.8 4.1 2.7 4.7 5.2 Electrostatic 4.6 1.9 5.0 4.0 4.0

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7. Paper four: “The first near-linear bis(amide) f-block complex: a blueprint for a high temperature single molecule magnet”

N. F. Chilton, C. A. P. Goodwin, D. P. Mills and R. E. P. Winpenny, Chem. Commun., 2015, 51, 101.

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The first near-linear bis(amide) f-block complex: a blueprint for a high temperature single molecule Cite this: Chem. Commun., 2015, 51,101 magnet†

Received 21st October 2014, Accepted 5th November 2014 Nicholas F. Chilton, Conrad A. P. Goodwin, David P. Mills* and Richard E. P. Winpenny* DOI: 10.1039/c4cc08312a www.rsc.org/chemcomm

We report the first near-linear bis(amide) 4f-block compound and show (CASSCF) ab initio calculations12 that are often employed to 7,13 that this novel structure, if implemented with dysprosium(III), would examine 4f complexes, pioneered by Chibotaru. Electrostatic have unprecedented single molecule magnet (SMM) properties with an approachessuggestthattheoptimalligandenvironmentto 1 III energy barrier, Ueff, for reorientation of magnetization of 1800 cm . exploit the oblate spheroidal electron density of Dy is axial, where rigorously axial systems have the benefit of maintaining a Since their initial discovery,1 single molecule magnets (SMMs) single, unique quantization axis for the total angular momentum 14 have been lauded as candidates for high density data storage mJ states. A set of unadulterated mJ states implies that the devices.2 A major breakthrough in the field3 occurred in 2003 with probability of quantum tunnelling of the magnetization (QTM) 2 the observation of SMM behavior in a monometallic {TbPc2} is reduced, therefore increasing magnetic relaxation times. 1 4 complex with an energy barrier, Ueff = 230 cm . The ensuing The simplest axial ligand environment is a linear two-coordinate 5 decade saw rapid growth in lanthanide SMMs with the Ueff barrier complex with donor atoms exclusively on a single Cartesian axis; the 1 7 to magnetization reversal increased to 652 cm for another Ueff barrier is so large for the {Dy5}and{Dy4K2} alkoxide complexes 6 1 derivative of {TbPc2}, and 585 cm for a polymetallic Dy@{Y4K2} because of the strongly axially repulsive crystal field potentials along 7 III complex. The highest blocking temperature TB (i.e. the tempera- the local z-direction of each Dy .Othercompoundssuchas 16 ture at which hysteresis is observed) was also increased to 14 K, [(C8H8)2Ln] (ref. 15) or Cloke’s bis(arene) lanthanide complexes 3 3 8 via an N2 radical bridge in a {Tb2N2 }complex. are sometimes described as linear, but lack donor atoms directly on Although three of these milestones employ the TbIII ion, by far the axis. Linear 3d-metal compounds also show remarkable magnetic III 17 the most utilized lanthanide ion in SMMs is Dy by virtue of its behaviour with very high Ueff values. A one coordinate lanthanide 9 + unique electronic structure. Apart from a radical-bridged complex [DyO] has been considered theoretically with a very large Ueff 3 10 III 14 {Dy2N2 }complex, nearly all polymetallic Dy -based SMMs predicted, however such an entity is not chemically feasible. possess negligible interactions between magnetic spin centres, Very low coordination numbers for 4f-ions are difficult to and instead rely on the single ion anisotropy of DyIII (i.e. the local achieve as these are large, electropositive ions, which require a i crystal field environment) to provide the barrier to the reversal of sterically demanding ligand. Such a pro-ligand HN(Si Pr3)2 was i i magnetization. Intra- or intermolecular interactions are often designed, and synthesised from ClSi Pr3 and LiHN(Si Pr3), and this III i detrimental to the performance of Dy SMMs so that doping a was converted to the group 1 transfer agent [KN(Si Pr3)2]withKH. i small amount of the paramagnetic ion into a diamagnetic host Reacting two equivalents of [KN(Si Pr3)2] with samarium(II) diio- III 7 lattice (usually the Y analogue) often results in an increased Ueff. dide yields the mononuclear homoleptic bis(amide) complex, i i An electrostatic model for the design of ideal ligand environ- [( Pr3Si)2N–Sm–N(Si Pr3)2] 1 (Fig.1,seeESI† for details). ments to exploit the maximal anisotropy of DyIII has been Complex 1 is the first near-linear f-element complex, with an postulated,11,12 and shown to be in good agreement with multi- N–Sm–N angle of 175.52(18)1 in the solid state (Fig. 2, see ESI† configurational complete active space Self consistent field for details); this near-linearity contrasts with the bent C–Ln–C II 18–20 angles of [Ln {C(SiMe3)3}2] complexes (Ln = Sm, Yb, Eu). The School of Chemistry, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK. E-mail: [email protected], [email protected] † Electronic supplementary information (ESI) available: Full synthetic details, crystallography, NMR spectroscopy, magnetism, and ab initio and magnetic relaxation methodologies. CCDC 1017031. For ESI and crystallographic data in CIF or other electronic format see DOI: 10.1039/c4cc08312a Fig. 1 Synthetic route to 1.

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Fig. 2 Molecular structure of 1. Hydrogen atoms omitted for clarity. Selected bond lengths (Å) and angles (1): Sm1–N1 2.483(6), Sm1– N2 2.483(6), Sm1C7 3.180(8), Sm1C16 3.169(8), Sm1C19 3.082(8), Sm1C34 3.224(8), N1–Sm1–N2 175.52(18), Sm1–N1–Si1 109.9(3), Sm1– N1–Si2 111.6(3), Si1–N1–Si2 138.5(4), Sm1–N2–Si3 109.8(4), Sm1–N2–Si4 110.8(3), Si3–N2–Si4 138.8(4). Fig. 3 Electronic states and magnetic transition probabilities for the 6 ground H15/2 multiplet of 2 in zero field. The x-axis shows the magnetic bulky iPr groups are vital for the isolation of a homoleptic complex, moment of each state along the main magnetic axis of the molecule. Relaxation commences from the | 15/2 state and only includes pathways as [Sm{N(SiMe ) } (THF) ] exhibits additional O-donors.21 The i 3 2 2 2 which reverse the magnetization. Relaxation probabilities are calculated Sm–N distances in 1 [2.483(6) Å] are longer than those observed based on a magnetic perturbation and are normalized from each departing in [Sm{N(SiMe3)2}2(THF)2] [mean Sm–N 2.433(9) Å] but this is state (see ESI† for details). compensated by 1 exhibiting four short SmCmethine distances [SmC 3.082(7)–3.224(7) Å] that are closer than the analogous III 6 SmCmethyl contacts observed in [Sm{N(SiMe3)2}2(THF)2][SmC is excellent (Fig. S2 and S3, ESI†). Dy has a H15/2 ground multiplet, 21 3.32(1)–3.46(1) Å]. The approximately planar SmNSi2 fragments in which is split by the crystal field into eight Kramer’s doublets with

1 are staggered with respect to each other (twist angle of 44.421), total angular momentum projections mJ = 1/2, 3/2,...15/2. The with the deviation from 901 attributed to agostic SmCmethine ab initio calculations show that the lowest six Kramers doublets are interactions. the almost pure mJ states of mJ = 15/2, 13/2, 11/2, 9/2, 7/2 Formally each nitrogen atom carries a single negative charge and 5/2, sharing a common quantization axis (Fig. 3 and Tables S1 and the SmII ion is divalent, with an [Xe]4f6 configuration. The f 6 and S2, ESI†). The two most energetic doublets are strongly mixed; a 7 configuration leads to a formally diamagnetic F0 ground state, characteristic of low symmetry complexes due to the lack of a 14 with close lying excited states that provide a non-zero magnetic rigorous molecular CN axis. Along the main magnetic axis moment at room temperature. Magnetic measurements on 1 these two states can be expressed as |cabi = 64%|3/2i + give a room temperature magnetic moment of 3.62 mB that falls 26%|81/2i and |ccdi = 68%|1/2i + 31%|83/2i and (Table S2, towards zero at low temperature (Fig. S2 and S3, ESI†). This is ESI†), giving the most energetic Kramers doublet a large gy value of clearly incompatible with interesting low temperature magnetic B17.5 perpendicular to the main magnetic axis. behaviour. However, the structure of 1 is close to the ideal linear Magnetic relaxation in lanthanides follows three possible arrangement to stabilize the large angular momentum states of routes: (1) QTM within the ground doublet (e.g. |15/2i - DyIII and produce monstrous uniaxial magnetic anisotropy. |+15/2i in Fig. 3), (2) thermally assisted QTM (TA-QTM) via Such a DyIII compound is challenging to make; we believe a excited states (e.g. |15/2i - |13/2i - |+13/2i - |+15/2i), or i route via the heteroleptic [Dy{N(Si Pr3)2}2I] treated with the potas- (3) an Orbach process composed of direct and/or Raman sium salt of a large anion might work through precipitation of a mechanisms (e.g. |15/2i - |13/2i - |+15/2i). The most potassium iodide. Other routes canbeimagined,andherewe probable pathway depends on the composition of the states present predictions of the magnetic properties of such a complex, involved and their interactions with phonons. For example, the III intending to inspire synthetic work towards the linear Dy slow magnetic relaxation for {Dy4K2} was shown to occur via the complex, and, more ambitiously, the isoelectronic TbII analogue. first or second excited states (TA-QTM), depending on the i i + III The properties of [( Pr3Si)2N–Dy–N(Si Pr3)2] 2 are predicted by number and location of neighboring Dy ions providing a CASSCF/RASSI/SINGLE_ANISO22 ab initio calculations (see ESI† for source of transverse magnetic field.7 The states with opposing details) employing the structure of 1,whereSmII has been replaced magnetic projections are mixed proportionally to the product of by DyIII. The validity of the method was tested by calculating the the transverse field and the transverse g-factors and therefore variable temperature magnetic behavior of 1, where the agreement TA-QTM will occur via the excited state which has transverse

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ChemComm Communication g-factors above a certain threshold or where its main magnetic suggest that such hypothetical complexes are now chemically axis is non-collinear with that of the ground state. All non-QTM feasible. These metal–organic compounds are becoming of great transitions are induced by the vibrational modes of the lattice importance in molecular magnetism.8,10,27,28 (phonons) which create local oscillating magnetic fields This work was supported by the EPSRC (grant number through modulation of dipolar fields as well as an oscillating EP/K039547/1) (UK). N.F.C. thanks The University of Manchester crystal field potential.23 To a first approximation, we can for a President’s Doctoral Scholarship. R.E.P.W. thanks The associate the probability of a phonon induced transition with Royal Society for a Wolfson research merit award. We would the average magnetic13,14,24 and crystal field perturbation like to thank J. P. S. Walsh for assistance with graphics. matrix elements (see ESI† for details). Compared to all known DyIII complexes the calculated Notes and references properties for 2 areuniquewithverysmalltransverseg-factors and a common principal axis for the lowest six Kramers doublets. 1 R.Sessoli,D.Gatteschi,A.CaneschiandM.A.Novak,Nature, 1993, 365, 141. 2 D. Gatteschi, R. Sessoli and J. Villain, Molecular Nanomagnets, This suggests that both the probability of QTM within the ground Oxford University Press, 2006. doublet and TA-QTM is vanishingly small until the two most 3 L. Sorace, C. Benelli and D. Gatteschi, Chem. Soc. Rev., 2011, 40, 3092. energetic doublets. Orbach relaxation is also strongly disfavoured 4 N. Ishikawa, M. Sugita, T. Ishikawa, S. Koshihara and Y. Kaizu, J. Am. Chem. Soc., 2003, 125, 8694. in the low lying states as magnetic transition probabilities due to 5 D. N. Woodruff, R. E. P. Winpenny and R. A. Layfield, Chem. Rev., phonons are miniscule (Fig. 3). Efficient magnetic relaxation will 2013, 113, 5110. only occur via the highest energy doublets (Fig. 3, Fig. S4 and 6 C. R. Ganivet, B. Ballesteros, G. de la Torre, J. M. Clemente-Juan, E. Coronado and T. Torres, Chem. – Eur. J., 2013, 19, 1457. Tables S4 and S5, ESI†). Therefore the ab initio calculation 7 R. J. Blagg, L. Ungur, F. Tuna, J. Speak, P. Comar, D. Collison, E 1 predicts Ueff 1800 cm for 2 –fargreaterthananycomplex W. Wernsdorfer, E. J. L. McInnes, L. F. Chibotaru and R. E. P. to date. Whilst such calculations may over-estimate the energies Winpenny, Nat. Chem., 2013, 5, 673. 25,26 8 J. D. Rinehart, M. Fang, W. J. Evans and J. R. Long, J. Am. Chem. Soc., of the crystal field states, we can predict a TB in excess of 77 K as 2011, 133, 14236. such temperatures are often around 1/20th of the Ueff value if QTM 9 D. Gatteschi, Nat. Chem., 2011, 3, 830. 10 J. D. Rinehart, M. Fang, W. J. Evans and J. R. Long, Nat. Chem., 2011, within the ground doublet is disfavored, e.g. the TB/Ueff ratios for 3 3, 538–542. {Tb2N2 }, {Mn12}and{Mn6} are approximately 1/16, 1/15 and 11 J. D. Rinehart and J. R. Long, Chem. Sci., 2011, 2, 2078. 1/13 cm1 K1, respectively. Calculations for the TbII analogue 3, 12 N. F. Chilton, D. Collison, E. J. L. McInnes, R. E. P. Winpenny and which is also a 4f9 ion, predict analogous behavior to 2 (Table S6, A. Soncini, Nat. Commun., 2013, 4, 2551. III 13 L. Ungur, J. J. Le Roy, I. Korobkov, M. Murugesu and L. F. Chibotaru, ESI†). The high local symmetry at the Dy site implies that the Angew. Chem., Int. Ed., 2014, 53, 4413. nuclear quadrupole and hyperfine interactions will be axially sym- 14 L. Ungur and L. F. Chibotaru, Phys. Chem. Chem. Phys., 2011, 13, 20086. metric, preventing efficient QTM within the lower energy doublets. 15 (a) F. Mares, K. Hodgson and A. Streitwieser, Jr., J. Organomet. Chem., 1970, 24, C68; (b) K. R. Meihaus and J. R. Long, J. Am. Chem. To examine the stability of 2, we have performed ab initio Soc., 2013, 135, 17952. calculations for modified geometries where the N–Dy–N angle and 16 For example, J. G. Brennan, F. G. N. Cloke, A. A. Sameh and the Dy–N bond lengths have been altered by 0.51 and 0.01 Å, A. Zalkin, Chem. Commun., 1987, 1668. 17 J. M. Zadrozny, D. J. Xiao, M. Atanasov, G. J. Long, F. Grandjean, respectively (Fig. S5, ESI†). The results show that 2 is stabilized F. Neese and J. R. Long, Nat. Chem., 2013, 5, 577. when the Dy–N bond length is shortened and the N–Dy–N angle is 18 C. Eaborn, P. B. Hitchcock, K. Izod and J. D. Smith, J. Am. Chem. closer to 1801 compared to 1, yielding more favorable electronic Soc., 1994, 116, 12071. 19 C. Eaborn, P. B. Hitchcock, K. Izod, Z.-R. Lu and J. D. Smith, properties. These calculations do not take into account the inclu- Organometallics, 1996, 15, 4783. sion of a counter-ion in the structure, which may have conse- 20 G. Qi, Y. Nitto, A. Saiki, T. Tomohiro, Y. Nakayama and H. Yasuda, quences for crystal packing and the local structure of 2. Tetrahedron, 2003, 59, 10409. 21 W. J. Evans, D. K. Drummond, H. Zhang and J. L. Atwood, Inorg. Compound 1 is the first near-linear bis(amide) 4f-block Chem., 1988, 27, 575. complex. It allows us to propose a blueprint for the first 22 (a)G.Karlstro¨m, R. Lindh, P.-Å. Malmqvist, B. O. Roos, U. Ryde, generation of ‘high-temperature’ SMMs, with blocking tem- V. Veryazov, P.-O. Widmark, M. Cossi, B. Schimmelpfennig and P. Neogrady, et al., Comput.Mater.Sci., 2003, 28, 222; (b) V. Veryazov, peratures exceeding that of liquid N2 (77 K). The synthesis of P. Widmark, L. Serrano-Andre´s, R. Lindh and B. O. Roos, Int. J. Quantum the proposed archetypes, viz. the DyIII and TbII analogues of 1, Chem., 2004, 100, 626; (c)F.Aquilante,L.DeVico,N.Ferre´,G.Ghigo, is currently underway in our laboratory, however we believe this P. Malmqvist, P. Neogra´dy,T.B.Pedersen,M.Pitonˇák, M. Reiher and B. O. Roos, et al., J. Comput. Chem., 2010, 31,224;(d)L.F.Chibotaruand is a target many other groups should be pursuing. Calculations L. Ungur, J. Chem. Phys., 2012, 137, 064112. on other fn ions suggest that f9 is ideal; even for the oblate f8 23 A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of TbIII analogue, 4, we find that the pseudo-doublets show strong Transition Ions, Oxford University Press, 1970. 24 L. Ungur, M. Thewissen, J.-P. Costes, W. Wernsdorfer and L. F. Chibotaru, mixing between the |mJi and |+mJi projections, (Tables S7 and S8, Inorg. Chem., 2013, 52, 6328. ESI†), which would lead to strong zero-field QTM. 25R.Marx,F.Moro,M.Do¨rfel,L.Ungur,M.Waters,S.D.Jiang,M.Orlita, J. Taylor, W. Frey, L. F. Chibotaru and J. van Slageren, Chem. Sci., 2014, While 2 would have a huge Ueff, an even higher Ueff barrier 5, 3287. might be possible if dianionic monodentate ligands could be 26 E. Moreno Pineda, N. F. Chilton, R. Marx, M. Do¨rfel, D. O. Sells, i i incorporated, e.g. [( Pr3Si)2C–Dy–C(Si Pr3)2] , containing dianionic P. Neugebauer, S.-D. Jiang, D. Collison, J. van Slageren, E. J. L. McInnes and R. E. P. Winpenny, Nat. Commun., 2014, 5, 6243. methanediides. Our preliminary results suggest this could raise Ueff 27 F. Moro, D. P. Mills, S. T. Liddle and J. van Slageren, Angew. Chem., byafactorof1.2to1.3.Theincredibleadvancesmadeinlow Int. Ed., 2013, 52, 3430. coordination number metal–organic compounds in the last decade 28 R. A. Layfield, Organometallics, 2014, 33, 1084.

This journal is © The Royal Society of Chemistry 2015 Chem. Commun., 2015, 51,101--103| 103

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Supplementary Information for: The First Linear f-block Complex and the

Route to a Liquid N2 Temperature Single Molecule Magnet Nicholas F. Chilton, Conrad A. P. Goodwin, David P. Mills* and Richard E. P. Winpenny*

Contents

1. General Synthetic Methods ...... 2

†† 2. Preparation of [K{NSiiPr3)2}] (KN ) ...... 2

3. Preparation of [(iPr3Si)2N-Sm-N(SiiPr3)2] (1) ...... 3

4. Crystallographic Details and Data...... 3

5. Nuclear Magnetic Resonance Spectroscopy ...... 4

6. Magnetic Measurements ...... 4

7. Ab initio Method ...... 6

8. Magnetic Relaxation...... 12

9. Supplementary References ...... 14

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1. General Synthetic Methods All manipulations were carried out using standard Schlenk techniques, or an Innovative Technology PureLab HE glovebox, under an atmosphere of dry argon. Solvents were dried by refluxing over potassium and degassed before use. All solvents were stored over potassium mirrors except for THF which was stored over activated 4 Å sieves. d6-benzene was distilled from potassium, degassed by three freeze-pump-thaw cycles and stored under

i argon. ClSiPr 3 was dried over Mg turnings and KH was obtained as a suspension in mineral oil and was washed with hexane (3 x 50 mL) and dried in vacuo before use. All other reagents were used as received. 1H, 13C and 29Si NMR spectra were recorded on a Bruker 400 spectrometer operating at 400.2, 100.6 and 79.5 MHz respectively; chemical shifts are quoted in ppm and are relative to TMS (1H, 13C, 29Si). FTIR spectra were recorded as nujol mulls in KBr discs on a Perkin Elmer Spectrum RX1 spectrometer. Elemental microanalyses were carried out by Mr Stephen Boyer at the Microanalysis Service, London Metropolitan University, UK.

†† 2. Preparation of [K{N(SiiPr3)2}] (KN )

Liquid ammonia (28 mL, 1.2 mol) was added to a pre-cooled (–78 °C) solution of ClSiiPr3 (17.618 g, 91.38 mmol), allowed to warm to –60 °C and stirred for 1 hr. The colorless mixture was warmed to room temperature, giving a white precipitate. This was filtered, cooled to –78 °C and BunLi (2.5 M, 36.4 mL, 91.2 mmol) was added drop-wise and the reaction mixture allowed to warm to room temperature and refluxed for 1 hour. Volatiles were removed in vacuo and the yellow oil heated to 140 °C in vacuo for 3 hours to afford

[Li(NHSiiPr3)]n as a yellow oil in essentially quantitative yield. ClSiiPr3 (20.49 mL, 18.46 g,

95.76 mmol) was added to a solution of [Li(NHSiiPr3)]n (16.352 g, 91.2 mmol) in THF (30 mL) and refluxed in a sealed ampoule for 4 days. The pale yellow solution was filtered and volatiles removed in vacuo. Distillation (170 °C oil bath, 10-2 Torr) gave a mixture of

HN(iPr3Si)2 and iPr3SiCl. The mixture was then heated (<100 °C) in vacuo to give

HN(iPr3Si)2 in essentially quantitative yield. A portion of HN(iPr3Si)2 (10.428 g, 31.63 mmol) in toluene (20 mL) was added drop-wise to a pre-cooled (-78 °C) slurry of KH (1.52 g, 38 mmol) in toluene (10 mL) and refluxed for 3 hours. Filtration, followed by removal of volatiles in vacuo, afforded a beige solid. The solid was washed with hexanes (3 x 5 mL) and dried in vacuo to afford the product as an off-white powder, with multiple crops obtained from the washings. Yield: 7.041 g, 61 %. Anal. Calcd. for C18H42Si2NK: C, 58.78; H, 11.51; 1 N, 3.81. Found: C, 58.66 H, 11.46; N, 3.90. H NMR (d6-benzene, 298 K) δ: 1.03 (sept, JHH =

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13 1 7.5 Hz, 6H, CH(CH3)2), 1.43 (d, JHH = 7.5 Hz, 36H, CH(CH3)2). C{ H} NMR (d6-benzene, 29 298 K) δ: 17.48 (s, CH(CH3)2), 20.85 (s, CH(CH3)2). Si NMR (d6-benzene, 298 K) δ: -16.31 (s). IR v/cm-1 (Nujol): 1238 (m), 1217 (m), 1175 (s), 1152 (s), 1099 (m), 1068 (m), 1002 (m), 974 (m), 876 (s), 649 (s), 611 (s).

3. Preparation of [(iPr3Si)2N-Sm-N(SiiPr3)2] (1) A solution of KN†† (2.942 g, 8 mmol) in toluene (15 mL) was added drop-wise to a pre- cooled slurry (–78 °C) of [SmI2(THF)2] (2.194 g, 4 mmol) in toluene (10 mL) and allowed to warm to room temperature with stirring. The mixture was stirred for 4 days, with the precipitation of a pale solid. The supernatant was filtered and the pale solid extracted with toluene (3 x 8 mL). Volatiles were removed in vacuo. The dark green solid was extracted with hexane (3 x 4 mL), concentrated to 2 mL and stored at –25 °C to give the product as dark green blocks (1.786 g, 55 %). Anal. Calcd. for C36H84Si4N2Sm: C, 53.56; H, 10.49; N,

3.47. Found: C, 53.39 H, 10.40; N, 3.38. Magnetic moment (Evans method, d6-benzene, 298 1 K): μeff = 3.73 µB. H NMR (d6-benzene, 298 K) δ: 0.20 (br, 72 H, CH(CH3)2), 6.81 (br, 12 H, 13 1 CH(CH3)2). C{ H} NMR (d6-benzene, 298 K) δ: 29.77 (s, CH(CH3)2), 35.28 (s, CH(CH3)2). 29 -1 Si NMR (d6-benzene, 298 K) δ: 6.06 (s). IR v/cm (Nujol): 1076 (s), 1057 (s), 944 (m), 882 (m), 697 (m), 660 (m).

4. Crystallographic Details and Data CrysAlisProS1 was used for control and integration and SHELXTLS2 and OLEX2S3 were employed for structure solution and refinement and for molecular graphics.

–1 Crystal data for 1: C36H84N2Si4Sm, Mr = 807.77 g mol , space group Pbca, a = 20.509(2), b 3 = 16.0788(19), c = 26.515(2), α = β = γ = 90, V = 8743.5(15) Å , Z = 8, Zʹ = 1 , ρcalcd = 1.227 –3 –1 g cm ; MoKα radiation, λ = 0.71073 Å, μ = 1.477 mm , T = 150 K. 18707 points (7676 unique, Rint = 0.130, 2θ < 50.0°). Data were collected on an Agilent Technologies Supernova diffractometer and were corrected for absorption (transmission 0.908 – 1.000). The structure was solved by direct methods and refined by full-matrix least-squares on all F2 values to give

2 2 2 2 2 1/2 wR2 = {[w(F0 – Fc ) ]/[w(F0 ) ]} = 0.1085, conventional R = Σ(|Fo| – |Fc|)/Σ|Fo| = 2 2 2 2 2 ½ 0.0728 for F values of 7676 with F0 > 2σ(F0 ), S = [Σw(Fo – Fc ) /(n + r – p)] = 0.908 for 412 parameters. Residual electron densities were 0.99 e Å–3 maximum and –1.21 e Å–3 minimum. CCDC 1017031 (1) contains the supplementary crystallographic data for this

118

paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

5. Nuclear Magnetic Resonance Spectroscopy Complex 1 exhibits simple 1H, 13C and 29Si NMR spectra, indicative of a symmetrical species in solution on the NMR timescale. The highly temperature-dependent chemical shift of the methine protons in a variable temperature study is attributed to the paramagnetism of the SmII center (μeff = 3.73 μB at 298 K, Evans method), which is comparable to 18 [Sm{N(SiMe3)2}2(THF)2] (μeff = 3.45 μB at 298 K).

Figure S1 - Variable temperature 1H NMR spectrum of 1.

6. Magnetic Measurements The magnetic properties of 1 were measured with a Quantum Design MPMS XL7 SQUID magnetometer, from 2 – 300 K in a field of 0.1 T. A fresh crystalline sample was ground and fixed with eicosane in an NMR tube under an inert atmosphere. The NMR tube was flame sealed under vacuum and mounted in a straw for attachment to the sample rod. The measurement was corrected for the diamagnetism of the straw, eicosane and the sample, the latter with Pascal’s constants, but not for the NMR tube. The room temperature χMT value of 3 -1 1.64 cm mol K (3.62 μB, very similar to that measured by the Evan’s method) reduces

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3 -1 slowly with cooling until 100 K where it drops rapidly to 0.04 cm mol K (0.56 μB) at 2 K, and is in good agreement with that predicted by ab initio calculations for 1, Figure S2. The χ vs. T plot, Figure S3, shows the characteristic plateau at low temperatures of a temperature independent paramagnetism effect due to mixing of paramagnetic states into the formally

7 diamagnetic F0 ground term. The sharp rise at the lowest of temperatures is due to a small paramagnetic impurity of SmIII. The small differences between the calculated and experimental traces are due to the very subtle nature of the electronic structure of SmII, a feature owed to the strongly mixed close lying excited states.

Figure S2 – Experimental and calculated χMT vs. T for 1.

120

Figure S3 – Experimental and calculated χM vs. T for 1.

7. Ab initio Method Ab initio calculations were performed with MOLCAS 7.8 using the RASSCF, RASSI and SINGLE_ANISO modules. In all cases the 4f ions were treated with the ANO-RCC-VTZP basis set, the N donors and the Si atoms with the ANO-RCC-VDZP basis set, while all C and H atoms were treated with the ANO-RCC-VDZ basis set. For all calculations the 4fn configuration was modelled with a complete active space of n electrons in 7 orbitals. For DyIII and TbII calculations, 21 sextets, 224 quartets and 158 doublets were included in the orbital optimization and 21 sextets, 128 quartets and 130 doublets were mixed by spin-orbit coupling. For the TbIII calculation, 7 septets, 140 quintets and 195 triplets were included in the orbital optimization and 7 septets, 105 quintets and 91 triplets were mixed by spin-orbit coupling. For the SmII calculation, 7 septets and 140 quintets were included in the orbital optimization and mixed by spin-orbit coupling. The SINGLE_ANISO module was employed to calculate the crystal field decomposition for the spin-orbit eigenstates and to yield the crystal field parameters for the ground spin-orbit multiplet of DyIII (Table S3) and TbII. The crystal field parameters were used with PHIS4 to examine the composition of the

121

̂ 푞 wavefunctions using the Hamiltonian (expressions for the 푂푘 operators can be found in the PHI User manual at www.nfchilton.com/phi):

푘 ̂ 푞 ̂ 푞 퐻퐶퐹 = ∑ ∑ 퐵푘푂푘 푘 = 2,4,6푞 =‒ 푘 The small difference in the energies for the ab initio and the crystal field calculations (Tables S1 and S2) is due to the simple nature of the crystal field model, however in this case the departure is very small owing to an extremely strong crystal field potential. The reduction of the principal gz values from those expected for pure mJ states in the ab initio calculation is due to covalent effects14 which are excluded in the crystal field parameterization therefore recovering the expected values.

Table S1 - Ab initio calculated electronic states for 2.

-1 E (cm ) gx gy gz gz angle (°) 0 0.0000 0.0000 19.9044 - 526 0.0002 0.0002 17.0068 0.3 1026 0.0009 0.0011 14.1772 0.5 1426 0.0337 0.0335 11.5169 0.8 1682 0.9868 0.9224 8.9785 2.6 1803 1.6038 3.0759 6.4529 17.6 1836 9.9041 8.9619 1.9276 2.1 1861 2.3210 17.5423 0.5019 3.9

Table S2 - Crystal field calculated electronic states for 2.

-1 E (cm ) gx gy gz gz angle (°) Wavefunctions 0 0.0000 0.0000 20.0000 - 100%| ± 15 2⟩ 517 0.0003 0.0003 17.3327 0.3 100%| ± 13 2⟩ 1032 0.0011 0.0014 14.6629 0.4 100%| ± 11 2⟩ 1427 0.0324 0.0316 11.9909 0.8 100%| ± 9 2⟩ 1675 0.9743 0.9112 9.2952 2.4 96%| ± 7 2⟩ + 4%| ∓ 7 2⟩ 94%| ± 5 2⟩ + 2%| ± 1 2⟩ + 2%| ∓ 5 2⟩ 1798 1.1326 2.7141 6.5704 12.4 + 1%| ± 3 2⟩ + 1%| ∓ 3 2⟩ 64%| ± 3 2⟩ + 26%| ∓ 1 2⟩ + 3%| ± 1 2⟩ 1836 10.0160 8.6757 2.2354 2.1 + 3%| ∓ 3 2⟩ + 2%| ∓ 5 2⟩ + 1%| ± 5 2⟩ 68%| ± 1 2⟩ + 31%| ∓ 3 2⟩ + 1%| ± 5 2⟩ 1861 1.7752 17.6994 0.3270 4.6 + 1%| ∓ 1 2⟩

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Table S3 - Ab initio calculated crystal field parameters for 2. Parameter Value (cm-1) ‒ 2 퐵 2 -1.1917562647038E-01 ‒ 1 퐵 2 -2.1119600305588E-02 0 퐵2 -1.1350199169445E+01 1 퐵2 9.7268699983100E-02 2 퐵2 1.4987459279149E-01 ‒ 4 퐵 4 -7.6752762196053E-04 ‒ 3 퐵 4 -5.4536142276867E-04 ‒ 2 퐵 4 1.0486686143292E-03 ‒ 1 퐵 4 -1.6575458595647E-03 0 퐵4 -7.2335435797888E-03 1 퐵4 -2.0658529932308E-04 2 퐵4 3.6533101842767E-03 3 퐵4 -4.2448546105741E-04 4 퐵4 -9.6344270575107E-04 ‒ 6 퐵 6 -4.2317898861512E-05 ‒ 5 퐵 6 -5.2516010707221E-05 ‒ 4 퐵 6 1.2655146551751E-06 ‒ 3 퐵 6 -1.0642106266008E-05 ‒ 2 퐵 6 -1.7313470374153E-05 ‒ 1 퐵 6 3.8030782269202E-06 0 퐵6 4.6813385093314E-05 1 퐵6 1.3122617258695E-05 2 퐵6 -6.5696276247257E-05 3 퐵6 1.5735760988224E-05 4 퐵6 -2.7776053728306E-07 5 퐵6 -4.7115889505093E-06 6 퐵6 -3.2828109496254E-05

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Table S4 - Magnetic transition probabilities for 2.

| ‒ 15 2⟩ | ‒ 13 2⟩ | ‒ 11 2⟩ | ‒ 9 2⟩ | ‒ 7 2⟩ | ‒ 5 2⟩ | ‒ 푎푏⟩ | ‒ 푐푑⟩ | ‒ 13 2⟩ 100

| ‒ 11 2⟩ 100

| ‒ 9 2⟩ 100

| ‒ 7 2⟩ 96

| ‒ 5 2⟩ 82

| ‒ 푎푏⟩ 67

| ‒ 푐푑⟩ 1 1 29

| + 푐푑⟩ 16 6 85

| + 푎푏⟩ 2 6 59 4

| + 5 2⟩ 6 5 5 10

| + 7 2⟩ 4 7 5 1

| + 9 2⟩ 3

| + 11 2⟩

| + 13 2⟩

| + 15 2⟩

Table S4 cont. - Magnetic transition probabilities for 2.

| + 푐푑⟩ | + 푎푏⟩ | + 5 2⟩ | + 7 2⟩ | + 9 2⟩ | + 11 2⟩ | + 13 2⟩ | + 15 2⟩ | + 푎푏⟩ 97

| + 5 2⟩ 2 99

| + 7 2⟩ 2 1 100

| + 9 2⟩ 100

| + 11 2⟩ 100

| + 13 2⟩ 100

| + 15 2⟩ 100

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Table S5 - Crystal field transition probabilities for 2. | ‒ 15 2⟩ | ‒ 13 2⟩ | ‒ 11 2⟩ | ‒ 9 2⟩ | ‒ 7 2⟩ | ‒ 5 2⟩ | ‒ 푎푏⟩ | ‒ 푐푑⟩

| ‒ 13 2⟩ 17

| ‒ 11 2⟩ 26 10

| ‒ 9 2⟩ 16 27 1

| ‒ 7 2⟩ 14 10 33 9

| ‒ 5 2⟩ 10 14 6 32 12

| ‒ 푎푏⟩ 7 10 17 10 22 12

| ‒ 푐푑⟩ 1 10 11 7 8 21 12

| + 푐푑⟩ 4 7 12 7 15 16 25

| + 푎푏⟩ 2 6 8 24 13 18 31

| + 5 2⟩ 1 3 4 6 19 16 19

| + 7 2⟩ 2 3 7 3 20 14 20

| + 9 2⟩ 2 5 20 8

| + 11 2⟩ 5 3 8 11

| + 13 2⟩ 2 3 4 8

| + 15 2⟩ 2 1 1 3

Table S5 cont. - Crystal field transition probabilities for 2. | + 푐푑⟩ | + 푎푏⟩ | + 5 2⟩ | + 7 2⟩ | + 9 2⟩ | + 11 2⟩ | + 13 2⟩ | + 15 2⟩

| + 푎푏⟩ 19

| + 5 2⟩ 32 17

| + 7 2⟩ 12 34 21

| + 9 2⟩ 10 12 44 16

| + 11 2⟩ 15 20 8 51 3

| + 13 2⟩ 11 11 19 15 62 30

| + 15 2⟩ 1 6 10 17 35 70 100

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Table S6 - Ab initio calculated electronic states for 3.

-1 E (cm ) gx gy gz gz angle (°) 0 0.0000 0.0000 19.9160 - 512 0.0005 0.0005 16.9607 0.4 999 0.0013 0.0016 14.0904 0.6 1399 0.0745 0.0775 11.4118 1.0 1663 1.3571 1.2755 8.8533 3.4 1797 0.7553 3.4376 6.4322 13.8 1846 8.4849 4.0479 2.1986 37.9 1860 5.2299 15.1280 1.0163 5.5

Table S7 - Ab initio calculated electronic states for 4.

-1 E (cm ) gz gz angle (°) 0 17.9410 - 0

392 14.5104 0.2 392

794 11.1393 0.4 794

1197 7.8553 0.7 1207

1597 4.7391 1.2 1606 1935 2.0170 1.9 1953 2098 - -

126

Table S8 – Crystal field calculated electronic states for 4.

-1 E (cm ) gz gz angle (°) Wavefunctions 0 17.9999 - 50%| ± 6⟩ + 50%| ∓ 6⟩ 0 409 14.9993 0.1 50%| ± 5⟩ + 50%| ∓ 5⟩ 409 777 11.9993 0.5 50%| ± 4⟩ + 50%| ∓ 4⟩ 777 1197 9.0001 0.7 50%| ± 3⟩ + 50%| ∓ 3⟩ 1206 1625 5.9992 0.7 50%| ± 2⟩ + 50%| ∓ 2⟩ 1634 1943 3.0001 0.7 50%| ± 1⟩ + 50%| ∓ 1⟩ 1956 2069 - - 100%|0⟩

8. Magnetic Relaxation Provided there are phonons of the correct energy or energy difference, the probability associated with a phonon transition can be determined from magnetic or crystal field origin, irrespective of the relaxation mechanism (Orbach, Raman or Direct). The magnetic transition probability is commonly taken as the average of the x, y and z magnetic perturbations linking 푔 2 퐽 ̂ 2 ̂ 2 ̂ 2 푃푚푎푔 = (|⟨휓푎│퐽푥│휓푏⟩| + |⟨휓푎│퐽푦│휓푏⟩| + |⟨휓푎│퐽푧│휓푏⟩| ) two states,13,14,23 3 and has 2 units of 휇퐵 . The crystal field transition probability on the other hand is commonly overlooked as it is much more difficult to determine. It involves knowledge of the vibrational modes of the molecule and the perturbations that these modes cause to the electronic states. As the magnetic ion resides in a site of near-linear geometry, we estimate the crystal field transition probability by considering only the N-Dy-N bending and symmetrical Dy-N stretching modes. We have performed ab initio calculations based on 2 where we have altered the N-Dy-N angle and the Dy-N bond lengths by ± 0.5° and ± 0.01 Å, respectively. Using the crystal field decomposition provided by SINGLE_ANISO we then performed

127

crystal field calculations with PHIS4 to examine how the small alteration of the molecular geometry would mix the pristine states. These perturbations were evaluated as ̂ 푚표푑푖푓푖푒푑 ̂ 푝푟푖푠푡푖푛푒 푃퐶퐹 = |⟨휓푎│퐻 퐶퐹 ‒ 퐻 퐶퐹 │휓푏⟩| and have been averaged to provide the crystal field transition probability. With initialization in the | ‒ 15 2⟩ state, we only consider transitions that reverse the magnetic moment; that is from any state, only transitions that will increase the magnetic moment towards the | + 15 2⟩ state are included. Then from each state the departing probability is normalized (Tables S5 and S6). In this way we construct the transition probability diagram, showing the barrier to magnetization reversal in zero field (Figure 3 and Figure S2). The crystal field relaxation diagram is in broad agreement with the magnetic one

-1 and suggests a Ueff value of ~ 1700 – 1800 cm .

6 Figure S4 - Electronic states and crystal field transition probabilities for the ground H15/2 multiplet of 2 in zero field. The x-axis shows the magnetic moment of each state along the main magnetic axis of the molecule. Relaxation commences from the | ‒ 15 2⟩ state and only includes pathways which reverse the magnetization.

128

Figure S5 - CASSCF energies for the sextet spin state (relative to the native geometry) for the structural modifications to 2.

9. Supplementary References S1 CrysAlis PRO, Agilent Technologies, Yarnton, England, 2010. S2 G. M. Sheldrick, Acta Crystallogr. A 2008, 64, 112. S3 O. V. Dolomanov, L. J. Bourhis, R. J. Gildea, J. A. K. Howard, H. Puschmann, J. Appl. Crystallogr. 2009, 42, 339. S4 N. F. Chilton, R. P. Anderson, L. D. Turner, A. Soncini, K. S. Murray, J. Comput. Chem. 2013, 34, 1164.

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8. Paper five: “Design Criteria for High-Temperature Single-Molecule

Magnets”

N. F. Chilton, Inorg. Chem., 2015, 54, 2097.

131

132

Communication

pubs.acs.org/IC

Design Criteria for High-Temperature Single-Molecule Magnets Nicholas F. Chilton* School of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

*S Supporting Information

The blossoming of lanthanide SMMs22 was spurred by − ABSTRACT: Design criteria to obtain slow magnetic Ishikawa et al., who, with [Tb(Pc)2] (where Pc is relaxation are theoretically investigated for two-coordinate phthalocyaninate),23 presented the first SMM using just a single III “ ” ± complexes of Dy . It is shown that large energy barriers to lanthanide ion. The sandwich geometry stabilizes the mJ = 6 7 − III magnetic relaxation, Ueff, can be achieved in the absence of states of the F6 ground-state spin orbit multiplet of Tb , and ff fi near-linearity and generally that any two-coordinate the pseudo-D4d symmetry results in small o -axial ligand- eld III 24,25 − complex of Dy is an attractive synthetic target that may terms, therefore disfavoring QTM. However, [Dy(Pc)2] is a −1 fi possess Ueff > 1000 cm . These large Ueff values are poor SMM despite the uniaxial ligand eld because it has an mJ = ±13 ±15 6 immediately diminished if axial ligation is disrupted by /2 ground state and not the maximal mJ = /2 of the H15/2 solvent coordination. multiplet. This seemingly contradictory situation is due to the complex relationship between the proximity of the Pc ligands and the effective ligand-field potential, which determines the ordering rom spin valves1 and transistors2,3 to qubits4,5 and data of the magnetic states. It was also recently shown that “sandwich” F storage bits,6 the proposed applications for single-molecule complexes employing the cyclooctatetrene anion actually magnets (SMMs) are as diverse as they are ground-breaking. generate equatorial ligand fields that stabilize the prolate electron Much of the most interesting SMM physics reported recently density of ErIII,renderingtheDyIII analogues poor − 15,26−30 involves highly air-sensitive lanthanide complexes,7 11 including SMMs. Notwithstanding the progress made with such − 12,13 “ ” the N 3 -bridged DyIII and TbIII dimers by Rinehart et al., the sandwich complexes, there is a much simpler geometry that can 2 fi lanthanide(III) alkoxides by Blagg et al.,14 and the symmetrical provide an axial ligand- eld potential: a linear two-coordinate − 15 complex. In this case, predictable ordering of the m states results [Er(COT)2] species of Meihaus and Long. In contrast to the J original 3d SMMs,6,16 4f SMMs take advantage of the strongly as a simple function of the 4f electron density along the spin−orbit coupled orbital angular momentum and its coordination axis. fi The recent report of a near-linear (N−Ln−N angle θ = interaction with the ligand eld to yield massive magnetic ° II anisotropy. When the ligand field provides a uniaxial potential 175.5 ), pseudo-two-coordinate Sm complex presents exactly this geometry, and the proposed DyIII analogue ([Dy{N- and stabilizes the largest angular momentum projections of the (SiiPr ) } ]+, hereafter 1Dy) was predicted to show a barrier to ground spin−orbit multiplet, there is an effective energy barrier 3 2 2 − magnetization reversal of U ≈ 1800 cm 1 and is so large that to magnetic reorientation, U , resulting in magnetic bistability eff eff 1Dy should display magnetic hysteresis above the temperature of and SMM behavior. The Ueff barrier is therefore directly related 31 fi liquid N2. Indeed, such a strong axial potential reduces the to the ligand- eld splitting, where magnetic relaxation is usually propensity for ground-state QTM as the transverse anisotropies assumed to occur by an Orbach process via the excited ligand- become negligible. The synthesis of a two-coordinate DyIII fi 6,17 eld states, although when the Ueff barrier is small, other 18 complex is challenging enough, let alone the requirement for relaxation processes become competitive because of a higher near-linearity, and therefore it seems worthwhile to examine how degree of mixing between the magnetic states. If uniaxiality rigorous the requirement for an E−Dy−E bond angle near 180° cannot be maintained or the presence of perturbations such as really is. Herein the electronic and magnetic characteristics of transverse magnetic fields is nonnegligable, mixing of opposing some model two-coordinate DyIII complexes of the general angular momentum projections can occur and quantum formula [Dy{ERx}2] are investigated by ab initio calculations, tunneling of magnetization (QTM) results in poor SMM primarily as a function of the bending angle, θ, to yield a set of properties. Therefore, fine control over both the symmetry and structural design guidelines and to assist in the identification of nature of the ligand field is crucial in order to mitigate such target complexes. perturbations and obtain improved SMMs. Rinehart and Long19 In order to make comparisons to the previously proposed DyIII proposed using the intrinsic anisotropic electron density complex 1Dy in a computationally tractable manner, a simplified 20 − i − distributions of the lanthanide ions in an extremely tangible version of the N(Si Pr3)2 ligand in the form of L1 = N(SiH3)2 electrostatic21 manner to design complexes that stabilize these was employed. The simplification of the ligand allows the effect large angular momentum states. The application of this strategy of structural deformations to be examined, and the exact alkyl suggests that an axial ligand field is required for ions whose substituents do not qualitatively change the results. The bending largest angular momentum states have oblate spheroid angle θ was varied along with the torsion angle ϕ in the ranges distributions, such as TbIII and DyIII, while an equatorial field is required for those ions with prolate spheroid states, such as ErIII Received: January 14, 2015 and YbIII. Published: February 10, 2015

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Inorganic Chemistry Communication ° ≤ θ ≤ ° ° ≤ ϕ ≤ ° fi − θ ° 90 180 and 0 90 for a xed Dy N bond length structures for L2 with < 120 were not included because of of r = 2.5 Å (Figure 1a, inset). The average magnetic relaxation unrealistic clashing of hydrogen atoms) were performed. Indeed, these ligands show a similar robustness toward the lowering of θ as L1 does, indicating that this angle generally needs not be a major consideration in the isolation of such two-coordinate dysprosium analogues. Interestingly, all three ligands show an almost linear dependence in Ueff with respect to the bending angle within 90° ≤ θ ≤ 150°, with a gradient of approximately 28 −1 − cm per degree. The dependence of Ueff on the Dy E distance within 2.0 Å ≤ r ≤ 3.0 Å with fixed θ = 180° and ϕ =90° was also studied. Unsurprisingly, a reduction of the bond length leads to a substantial increase in Ueff (Figure S1 in the SI) because of the stronger crystal-field splitting of the ground multiplet. Indeed, 5 minima in the total energies of the S = /2 CASSCF wave functions are observed for each complex (Figures S2 and S3 in the SI), indicating that these cationic complexes are stable. However, these equilibrium bond lengths and angles should not be expected when the complete ligands are employed because of crystal-packing forces and steric effects. To test these conclusions, ab initio calculations were + performed on model [Dy{C(SiMe3)3}2] cations based on two 32 previously reported structures, [Yb{C(SiMe3)3}2](2) and 33 − − [Sm{C(SiMe3)3}2](3), which have C Ln C bending angles of θ = 137.0° and 143.4°, respectively, in a methodology identical with that employed for 1Dy31 (Tables S1−S6 in the SI). The results indicate that, despite the more acute C−Dy−C angles and symmetry-lowering agostic interactions, the main magnetic axis is still found to be parallel to the E−E vector and that efficient relaxation would occur via the fifth or sixth excited state (Figures 2 and S4 in the SI). The calculated barriers for 2Dy and 3Dy are

Figure 1. (a) Relaxation barrier Ueff for model complexes as a function of the bending angle θ, averaged for all torsion angles ϕ. Error bars are 1 standard deviation from the mean of the torsion angles ϕ. Inset: Structure of the model complexes. (b) Zero-field magnetic transition + ϕ ° probabilities for a complex of L1 [Dy{N(SiH3)2}2] with =90 . The x axis shows the magnetic moment of each state (start and end of each arrow) along the main magnetic axis of the molecule. Relaxation |−15 ⟩ commences from the /2 state and only includes pathways that reverse the magnetization. The transparency of each arrow is proportional to the normalized transition probability. barrier for each value of θ, averaged over all torsion angles, was fi calculated with the complete active space self-consistent-field Figure 2. Zero- eld magnetic transition probabilities within the ground- (CASSCF) ab initio method according to the procedure given in state multiplet for 3Dy. The x axis shows the magnetic moment of each state along the main magnetic axis of the molecule. Relaxation the Supporting Information (SI). Figure 1a shows that there does |−15 ⟩ θ commences from the /2 state and only includes pathways that not seem to be a threshold value of where Ueff decreases fi reverse the magnetization. The transparency of each arrow is suddenly and that a strong axial eld can be maintained in the proportional to the normalized transition probability. Inset: main absence of linearity, along with a constant main magnetic axis (x magnetic axis of the ground-state Kramers doublet for 3Dy (blue rod), axis in Figure 1a, inset). Furthermore, despite the low symmetry Color code: Dy, green; Si, pink; C, gray; H, white. in these bent complexes, the strong axial field seems to prevent the mixing of mJ states with opposing projections until the highest energy states, thus not favoring QTM (Figure 1b). This − −1 surprising result indicates that it is not a requirement for an E Ueff = 1247 and 1484 cm , respectively, compared to Ueff = 1837 − θ ° −1 Dy E angle to approach = 180 in order to obtain a large Ueff cm for 1Dy. This is consistent with Figure 1a, where the barrier, but rather just a two-coordinate DyIII complex is needed. deviation of θ from 180° does not completely quench the barrier fi + To con rm that this trend is generally applicable, structural to magnetization reversal, and therefore [Dy{C(SiR3)3}2] fi investigations on model complexes with the simpli ed ligands L2 complexes should be considered highly desirable synthetic − − + = C(SiH3)3 and L3 = CH(SiH3)2 (Figure 1a; note that targets along with [Dy{N(SiR3)2}2] complexes.

2098 DOI: 10.1021/acs.inorgchem.5b00089 Inorg. Chem. 2015, 54, 2097−2099

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The proposed synthetic routes to such two-coordinate DyIII (3) Thiele, S.; Balestro, F.; Ballou, R.; Klyatskaya, S.; Ruben, M.; species may involve intermediates with bound solvent molecules Wernsdorfer, W. Science 2014, 344, 1135−1138. (4) Leuenberger, M. N.; Loss, D. Nature 2001, 410, 789−793. that ultimately may or may not be displaced. Therefore, the − impact on the Ueff barrier by additional coordination of (5) Ardavan, A.; Blundell, S. J. J. Mater. Chem. 2008, 19, 1754 1760. tetrahydrofuran (THF) molecule(s) for a number of coordina- (6) Gatteschi, D.; Sessoli, R.; Villain, J. Molecular Nanomagnets; Oxford University Press: Oxford, U.K., 2006. tion numbers and geometries has been examined in a qualitative (7) Mills, D. P.; Moro, F.; McMaster, J.; van Slageren, J.; Lewis, W.; manner (Figure 3). These results show that large decreases in the Blake, A. J.; Liddle, S. T. Nat. Chem. 2011, 3, 454−460. (8) Zadrozny, J. M.; Xiao, D. J.; Atanasov, M.; Long, G. J.; Grandjean, F.; Neese, F.; Long, J. R. Nat. Chem. 2013, 5, 577−581. (9) Meihaus, K. R.; Minasian, S. G.; Lukens, W. W.; Kozimor, S. A.; Shuh, D. K.; Tyliszczak, T.; Long, J. R. J. Am. Chem. Soc. 2014, 136, 6056−6068. (10) Liu, S.-S.; Ziller, J. W.; Zhang, Y.-Q.; Wang, B.-W.; Evans, W. J.; Gao, S. Chem. Commun. 2014, 50, 11418−11420. (11) Demir, S.; Zadrozny, J. M.; Long, J. R. Chem.Eur. J. 2014, 20, 9524−9529. (12) Rinehart, J. D.; Fang, M.; Evans, W. J.; Long, J. R. J. Am. Chem. Soc. 2011, 133, 14236−14239. Figure 3. Ueff values for solvated complexes of L1 (black), L2 (purple), (13) Rinehart, J. D.; Fang, M.; Evans, W. J.; Long, J. R. Nat. Chem. − − θ ° ° and L3 (blue). The E Dy E angles are = 180 (left), 120 (right, top), 2011, 3, 538−542. and 109.5° (right, bottom), and all complexes have ϕ =0°. (14) Blagg, R. J.; Ungur, L.; Tuna, F.; Speak, J.; Comar, P.; Collison, D.; Wernsdorfer, W.; McInnes, E. J. L.; Chibotaru, L. F.; Winpenny, R. E. P. U barrier of 50−70% could be expected upon coordination of Nat. Chem. 2013, 5, 673−678. eff − solvent molecules like THF, and therefore such complexes (15) Meihaus, K. R.; Long, J. R. J. Am. Chem. Soc. 2013, 135, 17952 should ideally be synthesized in the absence of a coordinating 17957. (16) Winpenny, R. E. P. Molecular Cluster Magnets; World Scientific: solvent. However, there does not seem to be any relationship London, 2011. between Ueff and the number of coordinated solvent molecules. (17) Abragam, A.; Bleaney, B. Electron Paramagnetic Resonance of In summary, the design criteria for two-coordinate dysprosium Transition Ions; Oxford University Press: Oxford, U.K., 1970. complexes with favorable SMM properties have been theoret- (18) Zadrozny, J. M.; Atanasov, M.; Bryan, A. M.; Lin, C.-Y.; Rekken, B. ically examined with ab initio calculations, and the task for D.; Power, P. P.; Neese, F.; Long, J. R. Chem. Sci. 2013, 4, 125−138. synthetic chemists pursuing lanthanide-based SMMs is clear; any (19) Rinehart, J. D.; Long, J. R. Chem. Sci. 2011, 2, 2078−2085. DyIII complex with only two anionic donor atoms is desirable, (20) Sievers, J. Z. Phys. B: Condens. Matter 1982, 45, 289−296. where the presence of weak agostic-type interactions should have (21) Chilton, N. F.; Collison, D.; McInnes, E. J. L.; Winpenny, R. E. P.; negligible effects. However, the coordination of solvent Soncini, A. Nat. Commun. 2013, 4, 2551. molecules such as THF has catastrophic consequences on the (22) Woodruff, D. N.; Winpenny, R. E. P.; Layfield, R. A. Chem. Rev. III 2013, 113, 5110−5148. Ueff values. If such pseudo-two-coordinate Dy complexes can be (23) Ishikawa, N.; Sugita, M.; Ishikawa, T.; Koshihara, S.; Kaizu, Y. J. isolated, they should be accompanied by a phenomenal increase Am. Chem. Soc. 2003, 125, 8694−8695. to the current record magnetic relaxation barrier, which should (24) Ishikawa, N.; Sugita, M.; Wernsdorfer, W. Angew. Chem., Int. Ed. result in much higher blocking temperatures, leading the way to 2005, 44, 2931−2935. technologically relevant high-temperature lanthanide SMMs. (25) Ganivet, C. R.; Ballesteros, B.; de la Torre, G.; Clemente-Juan, J. M.; Coronado, E.; Torres, T. Chem.Eur. J. 2013, 19, 1457−1465. ■ ASSOCIATED CONTENT (26) Jiang, S.-D.; Wang, B.-W.; Sun, H.-L.; Wang, Z.-M.; Gao, S. J. Am. − *S Supporting Information Chem. Soc. 2011, 133, 4730 4733. (27) Jiang, S.-D.; Liu, S.-S.; Zhou, L.-N.; Wang, B.-W.; Wang, Z.-M.; Computational details, electronic and magnetic properties of − 2Dy and 3Dy, and structural trends for model complexes of L , Gao, S. Inorg. Chem. 2012, 51, 3079 3087. 1 (28) Le Roy, J. J.; Ungur, L.; Korobkov, I.; Chibotaru, L. F.; Murugesu, L2, and L3. This material is available free of charge via the Internet M. J. Am. Chem. Soc. 2014, 136, 8003−8010. at http://pubs.acs.org. (29) Le Roy, J. J.; Korobkov, I.; Murugesu, M. Chem. Commun. 2014, 50, 1602−1604. ■ AUTHOR INFORMATION (30) Ungur, L.; Le Roy, J. J.; Korobkov, I.; Murugesu, M.; Chibotaru, L. Corresponding Author F. Angew. Chem., Int. Ed. 2014, 53, 4413−4417. *E-mail: [email protected]. (31) Chilton, N. F.; Goodwin, C. A. P.; Mills, D. P.; Winpenny, R. E. P. − Notes Chem. Commun. 2015, 51, 101 103. fi (32) Eaborn, C.; Hitchcock, P. B.; Izod, K.; Smith, J. D. J. Am. Chem. The authors declare no competing nancial interest. Soc. 1994, 116, 12071−12072. (33) Qi, G.; Nitto, Y.; Saiki, A.; Tomohiro, T.; Nakayama, Y.; Yasuda, ■ ACKNOWLEDGMENTS H. Tetrahedron 2003, 59, 10409−10418. The author thanks Prof. R. E. P. Winpenny, Prof. E. J. L. McInnes, and Dr. D. P. Mills for insightful conversations and The University of Manchester for a President’s Doctoral Scholarship. ■ REFERENCES (1) Urdampilleta, M.; Klyatskaya, S.; Cleuziou, J.-P.; Ruben, M.; Wernsdorfer, W. Nat. Mater. 2011, 10, 502−506. (2) Vincent, R.; Klyatskaya, S.; Ruben, M.; Wernsdorfer, W.; Balestro, F. Nature 2012, 488, 357−360.

2099 DOI: 10.1021/acs.inorgchem.5b00089 Inorg. Chem. 2015, 54, 2097−2099

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Supporting Information

Design criteria for high-temperature Single Molecule Magnets

Nicholas F. Chilton

School of Chemistry, The University of Manchester, Oxford Road, Manchester, M19 3PL, United Kingdom [email protected]

Contents

Geometry optimization ...... 1 Ab initio calculations ...... 2

Ueff barrier construction ...... 2 Table S1...... 3 Table S2...... 4 Table S3...... 5 Table S4...... 6 Table S5...... 7 Table S6...... 8 Figure S1...... 9 Figure S2...... 10 Figure S3...... 11 Figure S4...... 12 References ...... 13

Geometry optimization Model ligand geometries were optimized with Density Functional Theory (DFT) as described below using the B3LYP functional1–3 with DFTD3 dispersion corrections4,5 and the def2-TZVP basis6,7 as implemented in ORCA 3.0.2.8 - L1 = N(SiH3)2

The Si-N-Si angle and N-Si bond lengths for L1 were taken from average crystallographic values for similar species (Si-N-Si = 135(5)° and N-Si = 1.71(4) Å)9–11 and the hydrogen positions were optimized. - L2 = C(SiH3)3

The Si-C-Si angles and C-Si bond lengths for L2 were taken from average crystallographic values for similar species (Si-C-Si = 115(1)° and C-Si = 1.84(1) Å)12,13 and the hydrogen positions were optimized. - L3 = CH(SiH3)2

The Si-C-Si angle and C-Si bond lengths for L3 were taken from average crystallographic values for similar species (Si-C-Si = 122(3)° and C-Si = 1.82(2) Å),14–17 the methyl proton fixed with Si-C-H = 90° and C-H = 1.0 Å and the remaining hydrogen positions were optimized.

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THF The entire molecule was optimized.

[Sm{C(SiMe3)3}2] (2) Hydrogen atoms were added and their positions optimized with Ca2+ as a diamagnetic substituent for Sm2+.

Ab initio calculations Ab initio calculations were performed using the CASSCF/RASSI/SINGLE_ANISO approach,18–20 using MOLCAS 7.8.21–23 For all calculations the Dy atom was treated with the ANO-RCC-VTZP basis, the N or C donors and the Si atoms with the ANO-RCC-VDZP basis, while all other atoms were treated with treated with the ANO-RCC-VDZ basis.24–27 The two electron integrals were Cholesky decomposed with the default thresholds. The 4f9 configuration of DyIII was modelled with a complete active space of 9 electrons in 7 orbitals, where 21 sextets, 224 quartets and 158 doublets were included in the orbital optimization and 21 sextets, 128 quartets and 130 doublets were mixed by spin-orbit coupling. The ab initio results, parameterized by a set of crystal field parameters where the z-axis is the main magnetic axis of the ground Kramers doublet,28 were then utilized to estimate the energy barriers to the reversal of magnetization, Ueff.

Ueff barrier construction Firstly it is ensured that the crystal field parameters are such that the main magnetic moment of the ground Kramers doublet is taken as the quantization axis. Then the magnetic transition probability is calculated with PHI29 as the average of the x, y and z magnetic perturbations linking two states, ! ! ! ! ! ! = ! ! ! ! + ! ! ! + ! ! ! , where ! and ! are two !" ! ! ! ! ! ! ! ! ! ! ! ! eigenvectors of the crystal field Hamiltonian. This matrix is symmetric with zeroes on the diagonal. The system is initialized in the ground state with negative magnetization along the quantization axis and any transition elements involving a decrease in magnetic moment are set to zero. The probability of departure from each state is normalized to unity and all possible pathways leading to the positively magnetized component of the ground doublet are examined (keeping in mind that steps which decrease the magnetization are disallowed). The probability for each pathway is taken as the product of the probabilities for each step, normalized such that the sum of the probabilities for all pathways is unity. The energy barrier for each pathway is assigned as the maximum energy encountered in that pathway and the overall Ueff is then the sum of all possible relaxation pathways weighted by their probabilities. It must be stressed that the method used here to estimate the Ueff value for each calculation is not strictly correct as this of course neglects the phonon density of states of these hypothetical compounds as well as the alteration of the crystal field during vibrational modes. However the model is useful in comparison to examine the trends upon complex deformation. Furthermore it is noted that the use of truncated model ligands leads to a small alteration of the -1 calculated Ueff values: Ueff calculated for 1Dy with this approach yields Ueff = 1837 cm , while for a -1 comparable complex of L1 Ueff = 2060 cm (θ = 175° and φ = 40°). Likewise for 2Dy and 3Dy, the -1 -1 barriers are calculated to be Ueff = 1247 cm and Ueff = 1484 cm , respectively, while for comparable -1 -1 complexes of L2 Ueff = 1573 cm (θ = 135° and φ = 50°) and Ueff = 1847 cm (θ = 145° and φ = 60°), respectively.

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Table S1. Ab initio calculated electronic states for 2Dy.

-1 E (cm ) gx gy gz gz angle (°)

0 0.0000 0.0000 19.8736 - 352 0.0013 0.0015 16.9747 0.4 657 0.0332 0.0379 14.0905 0.3 907 0.3792 0.3923 11.2272 0.4 1094 2.5017 2.2152 8.1411 1.1 1219 8.4877 5.4345 4.2645 0.5 1318 15.2567 2.2021 1.1156 0.3 1408 19.5269 0.1921 0.0932 0.4

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Table S2. Crystal-field calculated electronic states for 2Dy, using the ab initio crystal field parameters.

-1 E (cm ) gx gy gz gz angle (°) Wavefunctions

0 0.0000 0.0000 19.9774 - 100% ± 15 2 350 0.0013 0.0016 17.2191 0.4 98% ± 13 2 + 2% ± 9 2 657 0.0334 0.0386 14.3976 0.3 94% ± 11 2 + 5% ± 7 2 88% ± 9 2 + 10% ± 5 2 908 0.3748 0.3916 11.5364 0.5 +2% ± 13 2 76% ± 7 2 + 17% ± 3 2 1094 2.2575 2.5310 8.3776 1.2 +5% ± 11 2 + 2% ∓ 1 2 34% ± 5 2 + 21% ∓ 5 2 +17% ± 1 2 + 12% ∓ 1 2 1218 8.5837 5.5547 4.3417 0.4 +5% ± 9 2 + 3% ∓ 9 2 + 3% ∓ 7 2 +2% ± 7 2 + 2% ± 3 2 + 2% ∓ 3 2 47% ± 3 2 + 23% ∓ 5 2 +13% ∓ 1 2 + 11% ± 7 2 1316 15.3925 2.1481 1.0951 0.3 +3% ∓ 3 2 + 2% ∓ 9 2 +1% ± 1 2 + 2% ± 5 2 53% ± 1 2 + 28% ∓ 3 2

1408 19.6698 0.1701 0.0840 0.4 +10% ± 5 2 + 3% ∓ 7 2 +3% ∓ 1 2 + 2% ± 3 2 + 1% ∓ 5 2

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Table S3. Ab initio calculated crystal field parameters for 2Dy. Parameter Value (cm-1) !! !! 2.01962721927927E-01 !! !! 9.90535471369045E-02 ! !! -7.97415315824689E+00 ! !! 9.45043914084932E-02 ! !! 3.09623722913212E+00 !! !! 2.55865420502294E-03 !! !! 2.94433661425981E-04 !! !! 2.32410301921615E-03 !! !! -5.60993397492546E-04 ! !! -2.34093970354183E-03 ! !! 1.04610326782185E-03 ! !! 1.35613368433382E-02 ! !! 1.24933000434739E-02 ! !! -1.52335628295406E-03 !! !! -2.58151378671642E-05 !! !! -3.74582627683468E-05 !! !! -3.78467505427757E-05 !! !! -1.09824415321907E-05 !! !! -2.61195638356284E-07 !! !! -3.81531217138823E-06 ! !! 1.06249156432690E-05 ! !! -2.25869579446961E-05 ! !! -1.25083861866480E-04 ! !! 2.43263765905571E-06 ! !! 1.46628048009982E-05 ! !! -5.73140016168279E-05 ! !! -7.58183352474061E-07

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Table S4. Ab initio calculated electronic states for 3Dy.

-1 E (cm ) gx gy gz gz angle (°) 0 0.0000 0.0000 19.8900 - 413 0.0007 0.0008 17.0056 0.7 782 0.0167 0.0181 14.1508 0.3 1079 0.2129 0.2202 11.3113 1.3 1290 1.8349 1.8075 8.2396 4.0 1418 7.7901 6.3410 4.1462 1.9 1518 15.1435 2.1177 0.8969 0.4 1645 19.5646 0.0545 0.0278 0.5

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Table S5. Crystal-field calculated electronic states for 3Dy, using the ab initio crystal field parameters.

-1 E (cm ) gx gy gz gz angle (°) Wavefunctions

0 0.0000 0.0000 19.9888 - 100% ± 15 2 410 0.0007 0.0008 17.2774 0.8 99% ± 13 2 + 1% ± 9 2 784 0.0155 0.0171 14.5132 0.3 97% ± 11 2 + 3% ± 7 2 92% ± 9 2 + 7% ± 5 2 1080 0.2095 0.2202 11.6693 1.3 +1% ± 13 2 80% ± 7 2 + 15% ± 3 2 1289 1.8246 1.8611 8.5109 3.8 +3% ± 11 2 + 1% ∓ 1 2 56% ± 5 2 + 30% ± 1 2 1415 7.8311 6.3969 4.2759 1.6 +5% ± 9 2 + 4% ∓ 3 2 + 4% ∓ 7 2 40% ± 3 2 + 20% ∓ 5 2 +10% ∓ 1 2 + 10% ∓ 3 2 1516 15.2198 2.1256 0.9183 0.4 +8% ± 7 2 + 4% ± 5 2 + 4% ± 1 2 +2% ∓ 7 2 + 1% ∓ 9 2 51% ± 1 2 + 28% ∓ 3 2

1645 19.6974 0.0544 0.0279 0.3 +11% ± 5 2 + 3% ± 3 2 +3% ∓ 7 2 + 3% ∓ 1 2 + 1% ∓ 5 2

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Table S6. Ab initio calculated crystal field parameters for 3Dy. Parameter Value (cm-1) !! !! -1.54497001881025E-02 !! !! 1.60949328578478E-02 ! !! -9.33457494690100E+00 ! !! 1.92878051778270E-01 ! !! 3.09394290563682E+00 !! !! 3.07220302128604E-04 !! !! 3.81913879110261E-04 !! !! -4.06823584071651E-04 !! !! 5.97986290483851E-04 ! !! -3.17331013820649E-03 ! !! 2.03510214939157E-03 ! !! 7.92115989320873E-03 ! !! 1.51186002032762E-02 ! !! 6.69929061801084E-04 !! !! 3.85332707821224E-06 !! !! 1.19499993568456E-05 !! !! -6.17502794576279E-06 !! !! -1.54198236037387E-05 !! !! 1.78740571440496E-06 !! !! -1.45391436271930E-05 ! !! 1.81386270585825E-05 ! !! -5.27156054171306E-05 ! !! -8.73350847517333E-05 ! !! -7.77107936900869E-06 ! !! -2.69589025010664E-05 ! !! -5.02559403686099E-05 ! !! 2.60302918806461E-05

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Figure S1. Calculated barrier to magnetization reversal Ueff as a function of the Dy-E bond length r, for θ = 180° and φ = 90°.

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Figure S2. Relative energy of the S = 5/2 CASSCF wavefunction as a function of the Dy-E bond length r, for θ = 180° and φ = 90°.

S10

145

Figure S3. Relative energy of the S = 5/2 CASSCF wavefunction as a function of the E-Dy-E bending angle θ, averaged for all torsion angles φ. Error bars are one standard deviation from the mean of the torsion angles φ.

S11

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Figure S4. Zero-field magnetic transition probabilities within the ground multiplet for 2Dy. The x- axis shows the magnetic moment of each state along the main magnetic axis of the molecule. Relaxation commences from the − 15 2 state and only includes pathways which reverse the magnetization. The transparency of each arrow is proportional to the normalized transition probability. Inset: main magnetic axis of the ground Kramers doublet for 2Dy (blue rod), Dy = green, Si = pink, C = grey and H = white.

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References (1) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785-789. (2) Becke, A. D. Phys. Rev. A 1988, 38, 3098-3100. (3) Becke, A. D. J. Chem. Phys. 1993, 98, 5648-5652. (4) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. J. Chem. Phys. 2010, 132, 154104. (5) Grimme, S.; Ehrlich, S.; Goerigk, L. J. Comput. Chem. 2011, 32, 1456-1465. (6) Schäfer, A.; Horn, H.; Ahlrichs, R. J. Chem. Phys. 1992, 97, 2571-2577. (7) Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2005, 7, 3297-3305. (8) Neese, F. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2012, 2, 73-78. (9) Chilton, N. F.; Goodwin, C. A. P.; Mills, D. P.; Winpenny, R. E. P. Chem. Commun. 2015, 51, 101-103. (10) Goodwin, C. A. P.; Tuna, F.; McInnes, E. J. L.; Liddle, S. T.; McMaster, J.; Vitorica-Yrezabal, I. J.; Mills, D. P. Chem. – Eur. J. 2014, 20, 14579-14583. (11) Goodwin, C. A. P.; Joslin, K. C.; Lockyer, S. J.; Formanuik, A.; Morris, G. A.; Ortu, F.; Vitorica-Yrezabal, I. J.; Mills, D. P. Organometallics 2015, doi: 10.1021/om501123e (12) Eaborn, C.; Hitchcock, P. B.; Izod, K.; Smith, J. D. J. Am. Chem. Soc. 1994, 116, 12071- 12072. (13) Qi, G.; Nitto, Y.; Saiki, A.; Tomohiro, T.; Nakayama, Y.; Yasuda, H. Tetrahedron 2003, 59, 10409-10418. (14) Avent, A. G.; Caro, C. F.; Hitchcock, P. B.; Lappert, M. F.; Li, Z.; Wei, X.-H. Dalton Trans. 2004, 1567-1577. (15) Hitchcock, P. B.; Khvostov, A. V.; Lappert, M. F. J. Organomet. Chem. 2002, 663, 263. (16) Barker, G. K.; Lappert, M. F. J. Organomet. Chem. 1974, 76, C45. (17) Hitchcock, P. B.; Lappert, M. F.; Smith, R. G.; Bartlett, R. A.; Power, P. P. J. Chem. Soc. Chem. Commun. 1988, 1007-1009. (18) Ungur, L.; Chibotaru, L. F. Phys. Chem. Chem. Phys. 2011, 13, 20086-20090. (19) Blagg, R. J.; Ungur, L.; Tuna, F.; Speak, J.; Comar, P.; Collison, D.; Wernsdorfer, W.; McInnes, E. J. L.; Chibotaru, L. F.; Winpenny, R. E. P. Nat. Chem. 2013, 5, 673-678. (20) Ungur, L.; Le Roy, J. J.; Korobkov, I.; Murugesu, M.; Chibotaru, L. F. Angew. Chem. Int. Ed. 2014, 53, 4413-4417. (21) Karlström, G.; Lindh, R.; Malmqvist, P.-Å.; Roos, B. O.; Ryde, U.; Veryazov, V.; Widmark, P.-O.; Cossi, M.; Schimmelpfennig, B.; Neogrady, P.; Seijo, L. Comput. Mater. Sci. 2003, 28, 222-239. (22) Veryazov, V.; Widmark, P.; Serrano-Andrés, L.; Lindh, R.; Roos, B. O. Int. J. Quantum Chem. 2004, 100, 626-635. (23) Aquilante, F.; De Vico, L.; Ferré, N.; Ghigo, G.; Malmqvist, P.; Neogrády, P.; Pedersen, T. B.; Pitoňák, M.; Reiher, M.; Roos, B. O.; Serrano-Andrés, L.; Urban, M.; Veryazov, V.; Lindh, R. J. Comput. Chem. 2010, 31, 224-247. (24) Roos, B. O.; Veryazov, V.; Widmark, P.-O. Theor. Chem. Acc. 2004, 111, 345-351. (25) Roos, B. O.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. J. Phys. Chem. A 2004, 108, 2851-2858. (26) Roos, B. O.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. J. Phys. Chem. A 2005, 109, 6575-6579. (27) Roos, B. O.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. Chem. Phys. Lett. 2005, 409, 295-299. (28) Chibotaru, L. F.; Ungur, L. J. Chem. Phys. 2012, 137, 064112. (29) Chilton, N. F.; Anderson, R. P.; Turner, L. D.; Soncini, A.; Murray, K. S. J. Comput. Chem. 2013, 34, 1164-1175.

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9. Paper six: “Direct measurement of dysprosium(III)···dysprosium(III) interactions in a single-molecule magnet”

E. Moreno Pineda, N. F. Chilton, R. Marx, M. Dörfel, D. O. Sells, P. Neugebauer, S.-D. Jiang, D. Collison, J. van Slageren, E. J. L. McInnes and R. E. P. Winpenny, Nat. Commun., 2014, 5, 5243.

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Received 11 Jul 2014 | Accepted 12 Sep 2014 | Published 13 Oct 2014 DOI: 10.1038/ncomms6243 Direct measurement of dysprosium(III)??? dysprosium(III) interactions in a single-molecule magnet

Eufemio Moreno Pineda1, Nicholas F. Chilton1, Raphael Marx2, Marı´aDo¨rfel2, Daniel O. Sells1, Petr Neugebauer2, Shang-Da Jiang3, David Collison1, Joris van Slageren2, Eric J.L. McInnes1 & Richard E.P. Winpenny1

Lanthanide compounds show much higher energy barriers to magnetic relaxation than 3d-block compounds, and this has led to speculation that they could be used in molecular spintronic devices. Prototype molecular spin valves and molecular transistors have been reported, with remarkable experiments showing the influence of nuclear hyperfine coupling on transport properties. Modelling magnetic data measured on lanthanides is always complicated due to the strong spin–orbit coupling and subtle crystal field effects observed for the 4f-ions; this problem becomes still more challenging when interactions between lanthanide ions are also important. Such interactions have been shown to hinder and enhance magnetic relaxation in different examples, hence understanding their nature is vital. Here we are able to measure directly the interaction between two dysprosium(III) ions through multi-frequency electron paramagnetic resonance spectroscopy and other techniques, and explain how this influences the dynamic magnetic behaviour of the system.

1 School of Chemistry and Photon Science Institute, The University of Manchester, Oxford Road, Manchester M13 9PL, UK. 2 Institut fu¨r Physikalische Chemie, Universita¨t Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany. 3 1. Physikalisches Institut, Universita¨t Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany. Correspondence and requests for materials should be addressed to E.J.L.M. (email: [email protected]) or to R.E.P.W. (email: [email protected]).

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ascinating physics has been observed in lanthanide-based molecules ranging from single nuclear spin detection gz = 20 Fand manipulation1,2, blocking of magnetization at unprecedentedly high temperatures for a molecule3,4, magnetic memory in chiral systems with a non-magnetic ground state5,6 and energy barriers for loss of magnetization, U , an order of eff Dy(1) 44° magnitude higher than observed for polymetallic d-block cages7,8. A diverse range of applications has been envisioned for lanthanide (Ln) containing molecules, including use as qubits J ⊥ for quantum information processing9,10 and prototype devices Dy(1) 11 1 z such as molecular spin valves and transistors have been x y g = 13.9 reported. These advances have been very rapid since the initial z′ Dy(2) report that a terbium phthalocyanine compound could function J || as a single-molecule magnet (SMM)12. The SMM behaviour observed for Ln complexes is due to the Dy(2) large magnetic anisotropy of the individual ions, caused by strong spin–orbit coupling and the crystal field (CF) effects due to the Figure 1 | Structural analysis. (a) Crystal structure of 3. Scheme: Dy, blue; ligand environment13,14. Other interactions, for example N, light blue; O, red; C, grey; H, dark grey. Orientation of the principal intermolecular effects, often favour efficient quantum tunnelling magnetic axes for Dy(1) and Dy(2) in the ground Kramers doublet from ab of magnetization at zero external field, and this factor can mean initio calculations shown as orange arrows and that for Dy(1) from little magnetic hysteresis is seen even in compounds with very electrostatic calculations as green arrow. (b) Schematic of the magnetic 13 large Ueff . When considering polymetallic Ln complexes, the model for the EPR simulation. Relative projection of the principal magnetic situation becomes even more complicated. Radical bridging axes for Dy(1) and Dy(2), along which gz ¼ 20 and 13.9 for Dy(1) and ligands can provide a strong magnetic exchange pathway Dy(2), respectively, and the anisotropic exchange interaction between the between two lanthanide ions, leading to magnetic hysteresis at dysprosium pair. 14 K3,4. In other compounds weak Ln Ln interactions can shift the zero-field quantum tunnelling step to a finite field, ions, three nitrate anions and four 8-quinolinolate ligands, crys- 15 known as exchange biasing , with different effects on cryogenic tallizing with a protonated 8-hydroxyquinolinium counter-cation 16,17 magnetization curves depending on the sign of the exchange and a molecule of MeOH. Dy(1) is bound by three bridging 18 or even mask single-ion slow relaxation modes . More 8-quinolinolate ligands and a single chelating 8-quinolinolate frequently, however, Ln Ln interactions increase quantum ligand, generating an N4O4 coordination environment with a tunnelling rates, leading to apparently lower Ueff values when bicapped trigonal prism geometry, subsequently referred to as the compared with paramagnetic Ln ions doped into a diamagnetic ‘hq pocket’ (Fig. 1a). The Dy(2) site binds to three chelating 8,17,19 lattice . Understanding how Ln Ln interactions can have nitrate anions and the O-atoms from three bridging quinolinate such different and seemingly contradictory influences is ligands, yielding a highly distorted O9 environment subsequently important for improving the performance of Ln SMMs. referred to as the ‘NO3 pocket’. The eight-coordinate hq pocket is Understanding such interactions in polymetallic molecular Ln significantly smaller than the nine-coordinate NO3 pocket: the complexes is far from trivial. Part of this problem is that bulk volumes defined by the donor atoms are 130 and 140 Å3, magnetic measurements provide only indirect evidence of weak respectively. Ln Ln interactions which can be difficult to distinguish from As the two coordination sites are different in compounds 1–7, other contributions to the low energy physics, for example, CF we can take advantage of the lanthanide contraction to dope effects, intermolecular interactions, impurities. In this work we paramagnetic ions into one or the other of the two pockets of study a {Dy2} complex where we can spectroscopically measure a the diamagnetic yttrium or lutetium compound, 1 and 7 Dy(III) Dy(III) interaction that quenches SMM behaviour. respectively20. Crystallographic studies on 1:1 heterobimetallic We combine magnetic measurements, electron paramagnetic Ln-Y compounds (see Methods, Supplementary Tables 6–8 and resonance (EPR) spectroscopy, ab initio calculations and far- Supplementary Data 8–13) show that LuIII occupies the smaller infrared (FIR) spectroscopy to develop a simple model to describe hq pocket in Lu-Y, while for the Dy-Y compound there is a the magnetic interaction in a Dy Dy pair, and provide design distribution between the two pockets as YIII and DyIII have criteria for when this interaction will lead to exchange biasing, similar ionic radii. Therefore for DyIII doped at a low level into 7, III III and when it will lead to collapse of the SMM behaviour. hereafter {Dy@Lu2}, Dy is in the NO3 pocket, while for Dy III doped at a low level into 1, hereafter {Dy@Y2}, Dy is distributed evenly between the two pockets. Results Synthesis and structure. Hydrated lanthanide nitrate Ln(NO3)3 nH2O and 8-hydroxyquinoline (hqH) were combined Magnetic measurements. The room temperature wMT value for 3 in a 1:2.5 mole ratio and heated to reflux in MeOH for 3 h. Slow (26.8 cm3 K mol 1) is close to the expected value for two non- evaporation of this solution gave yellow block-shaped X-ray interacting DyIII ions and declines smoothly on cooling reaching quality single crystals in a yield of 52–75%. Characterization by a minimum at 5 K (Fig. 2a) due to the depopulation of the CF single crystal X-ray diffraction revealed an asymmetric lanthanide states. Below 5 K wMT increases due to a weak ferromagnetic III dimer with formula [hqH2][Ln2(hq)4(NO3)3] MeOH (Ln ¼ Y , interaction between the paramagnetic centres. The low 1;TbIII, 2;DyIII, 3;HoIII, 4;ErIII, 5;YbIII, 6 and LuIII, 7; see temperature magnetization versus applied magnetic field does not Methods, Supplementary Tables 1 and 2 and Supplementary saturate at 1.8 K in a field of 7 T, indicating strong magnetic Data 1–7). Compounds 1–7 are isostructural, crystallizing in the anisotropy (Fig. 2a inset). monoclinic space group P21/c (Fig. 1a; see Methods and The dynamic magnetic behaviour of 3 was investigated Supplementary Tables 3–5). We focus on 3 as it has the most through alternating current (AC) magnetic susceptibility interesting magnetic properties. The dimer consists of two Dy(III) measurements, performed both with a zero and a 1 kG applied

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0.8 1 Hz 28 –1 0.7 10 Hz 70 Hz –3 200 Hz 26 12 0.6 451 Hz –5 K 700 Hz ln( τ ) –1 –1 B U = 41 cm

–1 10 μ 0.5 957 Hz –7 eff –6 A = 1.4 ×10 s 24 1,202 Hz 0 8 mol 0.03 > > 0.55 mol 3 –9 3 0.4 6 0.1 0.2 0.3

22 / cm 1.8 K 4 M 0.3 –1 T / cm Temperature / K χ″ M 3 K χ 2 4 K 0.2

20 Magnetization / N 0 01234567 0.1 18 Field / T 0.0 0 50 100 150 200 250 300 2 4 6 8 101214161820 Temperature / K Temperature / K

0.7 1.10 3.5 K 0T/6T 1T/6T 0.6 1.05 2T/6T 3T/6T 0.5 4T/6T

–1 1.00 5T/6T Experiment

mol 0.4 3 12 K 0.95 0.3 / cm M 0.90 χ″ 0.2

Normalized transmission Calculation 0.1 0.85

0.0 0.80 0.5 1.0 1.5 2.0 2.5 3.0 3.5 10 20 30 40 50 60 70 80 90

χ′ 3 –1 –1 M / cm mol Energy / cm

00 Figure 2 | Magnetic measurements. (a) Experimental wMT(T) and M(H,T) (inset) data (symbols) and best fits (red lines) for 3.(b) wM (T) measured 00 under a 1 kG HDC field (1.55 G AC field) and Arrhenius treatment of wM data for the high temperature process (inset) for {Dy@Y2}. (c) Cole-Cole plots 00 0 (wM versus wM ) showing a single relaxation process above 3.5 K, solid lines are fits of the AC susceptibility data to a modified Debye function for {Dy@Y2}. (d) Normalized FIR transmission spectra at 9 K for 3. direct current (DC) magnetic field. No frequency dependence was but they are not perfect and often require scaling to match observed under zero DC field; whereas at 1 kG a small out-of- absolute experimental values: this has been shown to be a result of phase component is observed, but with no maximum in the neglected correlation energies and possible small structural temperature range 1.8–24 K (Supplementary Fig. 1a–c). By changes at low temperatures22. We have performed such studying doped materials, we can show that the lacklustre calculations on compound 3 with MOLCAS 7.8 (ref. 23; see 6 dynamic behaviour observed in 3 is due to the effect of Methods and Supplementary Table 9). Dy(III) has a H15/2 6 interactions between the Dy(III) ions. ground term, separated from the excited H13/2 term by roughly 1 AC measurements on {Dy@Lu2} revealed only a small out-of- 4,000 cm , and therefore we need only consider the ground phase component below 3 K (Supplementary Fig. 1d–f), similar to multiplet for the magnetic properties. The CF removes the 6 that of the pure compound 3. However the {Dy@Y2} analogue 16-fold degeneracy of the H15/2 multiplet giving eight Kramers 1 15 shows pronounced slow relaxation with frequency-dependent doublets, which are linear combinations of mJ ¼ ± /2 ± /2, 00 peaks in the out-of-phase susceptibility wM to 12 K (Fig. 2b,c and where mJ is the projection of J upon the axis of quantization. Supplementary Figs 2 and 3), when we use an optimal DC field of CASSCF calculations predict markedly different properties for 1 kG (Supplementary Fig. 2a). Treating the highest temperature the two Dy(III) sites in 3 (Supplementary Tables 10 and 11), as 1 AC data with the Arrhenius model gives Ueff ¼ 41 cm and expected given the different coordination environments. For 6 t0 ¼ 1.4 10 s (Fig. 2b inset); however, the lower temperature Dy(III) in the hq pocket, the ground doublet is an almost pure 15 data is markedly nonlinear suggesting the onset of competing mJ ¼ ± /2, as characterized by the calculated gz-value approach- Orbach and Raman processes and this Ueff value should be ing 20 and gx and gy values close to zero. Its main magnetic axis treated with some caution. Fitting the AC susceptibility data to a lies along the Dy-O vector of the terminal 8-hydroxyquinolinolate single Debye function reveals a single dominant relaxation path at ligand, which happens to be co-parallel with the Dy Dy vector 15 higher temperatures (distribution parameter, a ¼ 0.03(1) at 12 K), (Fig. 1a). Electrostatic optimization of the oblate mJ ¼ ± /2 with other pathways becoming competitive at lower temperatures electron density24 provides the same result (8.6° difference to (a ¼ 0.55(2) at 3.5 K; Fig. 2c and Supplementary Fig. 3). From CASSCF) and confirms that this orientation is due to the short these observations we can determine that Dy(III) in the smaller terminal Dy-O bond of 2.248(3) Å (c.f. 2.349(3)–2.384(3) Å for hq pocket is an SMM, whereas Dy(III) in the larger NO3 pocket is the bridging oxygen atoms), identical to the motif in the recently 25 not. The SMM behaviour is quenched in the pure compound 3 described homoleptic [Dy3(hq)9] . The first excited doublet is due to intramolecular Dy Dy interactions, vide infra. well separated from the ground state (B100 cm 1) and has 13 dominant mJ ¼ ± /2 character with its main magnetic axis (gz) almost colinear with that of the ground doublet. This is consistent Ab initio calculations and FIR measurements. Multi- with the observed SMM behaviour of {Dy@Y2}. For Dy(III) in the configurational complete active space self-consistent field NO3 pocket, the ground doublet has g-values of gz ¼ 16.42, (CASSCF) calculations are the most accurate ab initio methods to gy ¼ 1.54, gx ¼ 0.05 (Supplementary Table 11) and hence is not a 6,8,15,21 15 rationalize the magnetic properties of lanthanide systems , pure mJ ¼ ± /2 doublet. The first excited doublet is at only

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B24 cm 1, is strongly mixed and is misaligned with the ground compound 3 (Fig. 3 and Supplementary Fig. 6). We simulate all 32 doublet. This is consistent with {Dy@Lu2} not behaving as an spectra of the doped species with EasySpin , using the effective 1 SMM. The angle between the principal magnetic axes (gz) of the spin formalism (Seff ¼ /2) with gx ¼ gy ¼ 0.1, gz ¼ 13.9. The ground states for the two sites is calculated to be 44° by CASSCF transverse g-values of 0.1 are arbitrarily small numbers and not (Fig. 1a). measurable. The measured gz-value is similar to that calculated As the calculated CF splittings of the ground multiplet of for the ground doublet of the NO3 pocket by ab initio methods Dy(III) in the NO3 pocket lie squarely in the FIR region, we have (Supplementary Table 11) and is indicative of a strongly mixed used magnetic field-dependent FIR absorption spectroscopy22 to ground state. measure them directly. Studies of compound 3 show two field- The Dy Dy exchange significantly perturbs the EPR spectra dependent bands at 39 and 59 cm 1 (Fig. 2d and Supplementary of compound 3 measured at lower frequencies (3.9, 9.7, 24 and Fig. 4) and therefore these are assigned as CF (as opposed to 34 GHz) with multiple resonances observed (Fig. 3 and phonon) transitions. CASSCF predicts the first two CF transitions Supplementary Fig. 9). For example, the 24 GHz spectrum of 3 to be at 24 and 39 cm 1. The agreement is remarkably good has a resonance close to zero field; hence a splitting has been given that CF splittings for lanthanides can be several hundred introduced equal to the microwave energy of B0.8 cm 1. The wavenumbers. The ratio of the energy intervals is correct, and a high-frequency EPR spectra are much broader, and therefore less simple scaling factor22 of 1.6 (see above) gives the experimental sensitive to weak exchange interactions (Supplementary Fig. 10). energies (Supplementary Table 12). As an initial approach to modelling the EPR spectra of the The CASSCF results can be recast in the form of a set of CF coupled system, we can of course use the same approach parameters26. We have applied the FIR scaling factor to these employed for the fitting of the thermodynamic magnetic data, viz. parameters, and then used the scaled CF parameters the Lines model with scaled ab initio CFPs. Simulations of the (Supplementary Table 13) to calculate magnetic observables and low-frequency EPR spectra using the isotropic Lines exchange the magnetic transitions in the FIR spectra (Fig. 2d). This alone parameter (extracted from fitting the magnetometry data) yield gives excellent agreement with experimental wMT(T) from some features of the spectra (Supplementary Fig. 11), however, it 5–300 K (c.f. using unscaled CF parameters, Supplementary is clear that the simulations are far from perfect. It is also clear Fig. 5), but to simulate the low temperature wMT(T) and M(H) from the spectra on doped samples that while the ab initio- data, we need to account for a weak exchange interaction. calculated CFPs provide a decent approximation to the ground One way to simulate such thermodynamic data for orbitally states, they are not a perfect starting point for examining the degenerate ions is the Lines model27, which employs an isotropic exchange interaction (that is, we must get the description of the exchange between the spin component of the angular momenta individual sites correct before tackling the exchange interaction). 5 (S ¼ /2 for Dy(III); see Methods) and has been used previously to Unfortunately due to the low symmetry of both pockets in this model interactions between lanthanides15,28. Employing this molecule, no reduction in the number of allowed CFPs can be 29 1 model with PHI we find JLines ¼þ0.047(1) cm , which made and therefore we are left with 27 parameters to describe gives excellent fits to both wMT(T) and M(H) (Fig. 2a). each pocket. Clearly this problem is over-parameterised and it would not be feasible to find a unique solution. Therefore, a much simpler approach to simulating these EPR studies. Weak exchange can be better defined by EPR spectra has been adopted, invoking a model of two effective spins, 30,31 1 spectroscopy, even between orbitally degenerate ions , but Seff ¼ /2. For the hq pocket we have gx ¼ gy ¼ 0 and gz ¼ 20 from such studies on lanthanide pairs are rare outside minerals, due to its EPR silent nature, SMM behaviour and ab initio calculations. fast spin-lattice relaxation. Compound 3, {Dy@Y2} and {Dy@Lu2} For the NO3 pocket, we have gz ¼ 13.9 from multi-frequency EPR all give rich EPR spectra; the spectra of {Dy@Y2} and {Dy@Lu2} spectroscopy on doped materials, and we take gx ¼ gy ¼ 0.1. As are very similar since in both cases we only see resonances for the both sites are axial in this model, we need only consider the Dy(III) ion in the NO3 pocket. This is because Dy(III) in the hq relative orientation of the local z axes, which we fix at the ab 15 pocket has an essentially pure mJ ¼ ± /2 state, which is EPR initio-calculated angle of 44° (Fig. 1b). We define the molecular silent and hence no g ¼ 20 resonance is observed. coordinate frame as that of the hq pocket (where z is along the gz At low microwave frequencies (9.7, 24 and 34 GHz), the vector of Dy(1) in the hq pocket, coincident with the Dy Dy spectra of {Dy@Y2} and {Dy@Lu2} are dominated by an intense vector, and the two gz vectors define the zx plane). We can then E feature at geff 14, which broadens at higher frequencies use the Hamiltonian [1] to simulate the EPR spectra, where the 4130 GHz (Fig. 3 and Supplementary Figs 6–8). There are other only variables are the components of the anisotropic J-matrix. weaker features that match features from spectra of the pure No agreement could be found for an isotropic J. The next simplest

X-band, 9.7 GHz {Dy@Y2} K-band, 24 GHz {Dy@Y2} Q-band, 34 GHz {Dy@Y2}

{Dy2} {Dy2} {Dy2} EPR intensity / a.u. EPR intensity / a.u. EPR intensity / a.u.

0 200 400 600 800 0 200 400 600 800 1,000 1,200 1,400 0 200 400 600 800 1,000 1,200 1,400 1,600 Field / mT Field / mT Field / mT

Figure 3 | EPR spectroscopy. X- (a), K- (b) and Q-band (c) experimental EPR spectra at 5 K (black traces) and simulated data for {Dy@Y2} (red traces) and 3 (blue traces). See text for simulation parameters.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6243 ARTICLE model has an axially symmetric J-matrix, where we set the magnetic field is increased from zero, the component of the principal component, J||, along the Dy Dy vector (Fig. 1b). doublet that is destabilized by the field begins to strongly mix with the first singlet excited state. This mixing causes an avoided ^ ^ ^ ^ ^ ^ ^ ^ ^ * H ¼2 J? S1xS2x þ S1yS2y þ JkS1zS2z þ mBðS1 g1 þ S2 g2ÞB: crossing between c1 or c2 and c3 between |B| ¼ 100–150 mT and therefore significantly increases the average transition matrix ð1Þ elements between the two components of the doublet to a B 2 Using [1] we can model the EPR spectra of 3 with only two free maximum value of 2.4 mB at |B| ¼ 200 mT. Thus, in an applied 1 variables, and we find excellent fits with J> ¼þ0.525 cm and DC magnetic field further relaxation pathways become available 1 for the magnetization vector, therefore meaning that the applied J|| ¼þ1.52 cm (Fig. 3 and Supplementary Figs 9 and 10). This is a ferromagnetic interaction, consistent with the low tempera- field does not quench fast relaxation modes. For low microwave frequencies the EPR transition energies are on the scale of the ture rise in wMT(T). The spectra are not reproduced by a simple dip 1 dip 1 exchange-induced splitting and even of the avoided crossing, dipolar model (Jzz ¼þ1:48 cm ¼ , Jzx ¼þ1:41 cm , all dip 1 dip hence these EPR experiments are directly measuring the other J ¼ 0cm ; H^ dip ¼2^S1 J ^S2) hence there is a ij probabilities and energies associated with the magnetic relaxation significant exchange contribution, although we note that J|| is very dip processes (see allowed EPR transitions marked on Fig. 4). similar in magnitude to Jzz . The origin of these many efficient relaxation pathways is the III Discussion non-colinearity of the principal axes of the Dy ions. If we consider a scenario where the tilt angle of 44° between the two g With an accurate knowledge of the low energy eigenstates of our z vectors is removed, but the exchange remains unchanged, the system, we are able to explain the lack of SMM behaviour in situation changes considerably. In zero field, while the average compound 3. From the coupled S ¼ 1/ model used to simulate eff 2 transition matrix elements between the ground doublet wave- the EPR spectra, in zero field we have a ground doublet with functions remains zero, the average transition matrix elements wavefunctions c ¼ |mmS and c ¼ |kkS due to the ferromag- 1 2 between these states and the excited singlets are now very small, netic exchange. The two excited states are singlets at B1 and on the order of 10 3 m2, indicating that zero-field relaxation in B2cm 1, which are the symmetric and antisymmetric linear B this case would be hindered. With the application of a magnetic combinations c ¼ p1ffiffi ðÞjþ#""#i jiand c ¼ p1ffiffi ðÞ"#ij #"ij , 3 2 4 2 field along the easy axis (Supplementary Fig. 12) the avoided respectively (Fig. 4). The average transition matrix elements crossings are no longer present, which implies that mixing (average of x, y and z magnetic transition probability) between between the ground doublet and the excited states does not occur the ground doublet wavefunctions vanish in zero field and no and does not induce any transition matrix element between the direct relaxation is possible. Even if we take into account the components of the ground doublet. Therefore this hypothetical experimental perturbations, such as intermolecular dipole and compound would be expected to show slow magnetic relaxation hyperfine fields, direct magnetic relaxation between the two in zero field. components of this doublet are unlikely due to this involving a Compound 3 shows very different physics from the exchange flip of both moments. On the other hand, the average transition bias type of compounds, originally described by Wernsdorfer matrix elements between these doublet states and both the excited et al.33, and also seen in {Dy } complexes16, where a remnant singlets are large, B4 m2, indicating that relaxation processes 2 B magnetization is present in the open hysteresis loops; a feature involving these excited states are readily available. Given the small not observed here because of rapid zero-field relaxation. The energy gaps to these excited states, they present an extremely hypothetical compound where the tilt angle is 0° may possibly efficient method for the reorientation of magnetization and thus show the characteristic QTM steps at the crossing points in explain the lack of any significant out-of-phase component in the Supplementary Fig. 12 once all perturbations are included and AC susceptibility of 3 in zero field. therefore we conclude that it is also the non-colinearity in 3 In a non-zero magnetic field, the situation is more complicated which distinguishes this compound from those of the exchange due to the anisotropic magnetic nature of the system. To illustrate bias type. our point we will discuss the case where the magnetic field is With the nature of the anisotropic magnetic exchange coincident with the principal axis of Dy(1) (as in Fig. 4). As the determined exactly by EPR spectroscopy, we are in a unique position to compare this with the magnetic exchange approximated by the Lines approach. Comparing the lowest four states 2 obtained from EPR spectroscopy (Fig. 4) to those of the Lines model used for fitting the magnetic susceptibility (Supplementary 1 Fig. 13), it is obvious that there are certain similarities. At zero B

–1 ﬁeld, the ground pseudo-doublet of the EPR model has gz 34, B 0 which is identical to that of the Lines model of gz 34, and the gap to the ﬁrst excited state for the EPR model is B1cm 1, also 1 –1 similar to that of the Lines model (B0.7 cm ).

Energy / cm It is also clear that the Lines approach yields two pseudo- –2 XKQ doublets, whereas the excited states of the EPR model are singlets; this is a result of the anisotropic nature of the exchange interaction; such an interaction being excluded by the Lines –3 approach. The high field limits of the two models are very similar, –400 –300 –200 –100 0 100 200 300 400 as would be expected due to the field ‘decoupling’ the individual moments in both cases (Paschen–Back effect). It is in the Field / mT intermediate field regimes where the differences are most Figure 4 | Lowest energy states. All states in the EPR model for 3, apparent, where we observe a number of avoided crossings in calculated as a function of magnetic field coincident with the principal axis the EPR model and none in the Lines model. of Dy(1). Blue line, c1; red line, c2; green line, c3; purple line, c4 (see text). Intramolecular magnetic interactions between lanthanides are Arrows correspond to the EPR resonances observed for this orientation. known to strongly influence magnetic relaxation. Here we can

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6243 quantify this interaction, and demonstrate that it is the angle considered the individual DyIII ions independently, where the ion not in focus was replaced by the diamagnetic LuIII. All calculations were of the complete active between the principal anisotropy axes and the associated 9 III magnetic exchange that has the effect of quenching the SMM space type where the 4f configuration of Dy was modelled with nine electrons in seven orbitals CAS(9,7), using the RASSCF module. The basis sets for these cal- behaviour in this dimetallic compound. Although the ferromag- culations were taken from the ANO-RCC library, where the paramagnetic ion was netic exchange interactions lead to a highly magnetic ground of VTZP quality, the first coordination sphere of VDZP quality and all other atoms state, the local transverse moment due to the asymmetric nature of VDZ quality. Supplementary Table 9 details the states included in the RASSCF and RASSI modules. The RASSI module was used to calculate the g-tensors for the of the dimer makes relaxation very efficient. Many other {Dy2} 6 34 14 H15/2 multiplet and SINGLE_ANISO was used to generate a complete set of CF complexes are reported to be SMMs ; in every {Dy2} SMM parameters from the ab initio-calculated spin–orbit eigenstates35. where the orientation of the anisotropy axes is known, either from symmetry or calculation, they are parallel. In these cases, the CF and lines models. The CF parameters (CFPs) were treated as explained in the Dy Dy interaction is still present, but as it does not induce a main text, and the reference frame rotations were optimized to reproduce the transverse magnetic field at either Dy site, it does not destroy the mutual orientations from the ab initio calculations (Supplementary Table 13). barrier to magnetization reversal. This gives the design criterion These were fixed while the exchange interaction was fitted to the magnetic data. S III The following Hamiltonian was used in the |J, mJ basis for each Dy ion, where that the magnetic moments of individual spins should be aligned 5 III ^ the exchange term acts on the true spins (S ¼ /2) for each Dy and the SDyð1Þ ^ parallel within polymetallic lanthanide SMMs to enhance SDyð2Þ term was evaluated using a Clebsch–Gordan decomposition. yk are the III S measurable magnetization retention at zero field. operator equivalent factors for Dy in the |J, mJ basis, gJ is the Lande´ g-factor for 6 III 4 the H15/2 term of Dy and is equal to /3, I is the identity matrix and JLines is the exchange parameter. Note that the O^ q operators are expressed in the local Methods kDyðnÞ coordinate frame of each DyIII ion, using the given Euler angles. Synthetic methods. Compounds 1–6 were synthesized according to the following general method. To a solution of Ln(NO ) nH O (0.6 mmol), where Ln ¼ YIII, X X 3 3 2 k TbIII,DyIII,HoIII,ErIII,YbIII or LuIII, in MeOH (20 ml) was added 8-hyxdrox- H^ ¼ y Bq O^ q þ Bq O^ q 2J ^S ^S q¼k k kDyðÞ1 kDyð1Þ kDyðÞ2 kDyð2Þ Lines DyðÞ1 DyðÞ2 yquinolinoline (hqH; 220 mg, 1.5 mmol). The solution was then refluxed for 3 h k¼2;4;6 and subsequently filtered. Yellow block crystals were collected after 24 h of slow * m ^ ^ evaporation crystallization for Y (1), Tb (2), Dy (3), Ho (4), Er (5), Yb (6) and Lu þ BgJ JDyðÞ1 I þ JDyðÞ2 I B (7) (Supplementary Table 1). The magnetically dilute samples, {Dy@Y2} and ð2Þ {Dy@Lu2}, were obtained by combining accurately measured amounts of Dy(NO3)3 nH2O and Y(NO3)3 nH2O or Lu(NO3)3 nH2O in a 5:95 molar ratio, following the procedure above. The 1:1 mixed-metal systems were prepared by III III III III III III EPR simulation with the Lines model. Using exactly the same Hamiltonian as reacting Ln(NO3)3 nH2O (where Ln ¼ Tb ,Dy ,Ho ,Er ,Yb and Lu ) above, the EPR was simulated in a non-perturbative manner with PHI. The EPR with Y(NO3)3 nH2O in a 1:1 molar ratio using the procedure described above. absorption spectrum was calculated at discrete field points using the following ^ ^ equation, where HB 0 and HB 0 are the Zeeman Hamiltonians for the components of Crystallography The crystallographic data for 2 was collected on a Bruker x y . the perpendicular microwave magnetic field, Ei and Ej are the energies for the states Prospector CCD diffractometer with CuKa radiation (l ¼ 1.5418 Å). The crystal- |iS and |jS (eigenstates of the above Hamiltonian), respectively, Z is the partition lographic data for 3–5 were collected on an Oxford SMART CCD diffractometer function and Z is the linewidth expressed in energy units. with MoK radiation (l ¼ 0.71073 Å). The data collection for 7 was carried out on a Agilent SUPERNOVA diffractometer with MoKa radiation (l ¼ 0.71073 Å). The E Ei j ffiffiffiffiffiffiffi data for 1 and 6 were collected on a Rigaku Saturn724 þ diffractometer (syn- i;j2dim kBT kB T p 2 X e e ðÞEi Ej EMW * 2 2 8 ln2 chrotron, l ¼ 0.68890 Å) at the I19 beamline at the Diamond Light Source, UK. ^ ^ ffiffiffi Z2ln2 A B ¼ ðjhjjHB 0 jiji þjhjjHB 0 jiji Þ p e 2 x y Z p The structures were solved by direct methods and refined on F using SHELXTL. ioj Z ð3Þ Electron paramagnetic resonance. X-band (B9.7 GHz) and Q-band (B34 GHz) EPR spectra were recorded with a Bruker EMX580 spectrometer, while S-band As the linewidth is evaluated in frequency space, there is no need for the (B3.9 GHz) and K-band (B24 GHz) EPR spectra were recorded with a Bruker frequency-field conversion factor. The absorption spectrum was summed over Elexsys580 spectrometer. The data were collected on polycrystalline powders in the a large number of orientations on a hemisphere using the Zaremba–Conroy– temperature range 5–30 K using liquid helium cooling. High-frequency EPR Wolfsberg spherical integration scheme36. The first derivative of the absorption spectra (260–380 GHz) were recorded in Stuttgart on a home-built spectrometer. spectrum was then obtained to yield the spectral simulations. Its radiation source is a 0–20 GHz signal generator (Anritsu) in combination with an amplifier–multiplier chain (VDI) to obtain the required frequencies. It features a References quasi-optical bridge (Thomas Keating) and induction mode detection. The detector 1. Vincent, R. et al. Electronic read-out of a single nuclear spin using a molecular is a QMC magnetically tuned InSb hot electron bolometer. The sample is located in an Oxford Instruments 15/17T cryomagnet equipped with a variable temperature spin transistor. Nature 488, 357–360 (2012). insert (1.5–300 K). Spectral simulations were performed using the EasySpin 4.5.5 2. Thiele, S. et al. Electrically driven nuclear spin resonance in single-molecule simulation software. magnets. Science 344, 1135–1138 (2014). 3. Rinehart, J. D., Fang, M., Evans, W. J. & Long, J. R. Strong exchange and 3- magnetic blocking in N2 radical-bridged lanthanide complexes. Nat. Chem. 3, Magnetometry. The magnetic properties of polycrystalline samples of 2–6 538–542 (2011). were investigated in the temperature range 1.8–300 K with a Quantum Design 3– 4. Rinehart, J. D., Fang, M., Evans, W. J. & Long, J. R. A N2 radical-bridged MPMS-XL7 SQUID magnetometer equipped with a 7 T magnet. The samples were terbium complex exhibiting magnetic hysteresis at 14 K. J. Am. Chem. Soc. 133, ground, placed in a gel capsule and fixed with a small amount of eicosane to avoid 14236–14239 (2011). movement during the measurement. The data were corrected for the diamagnetism 5. Tang, J. K. et al. Dysprosium triangles showing single-molecule magnet from the gel capsule and the eicosane, with the diamagnetic contribution of the behavior of thermally excited spin states. Angew. Chem. Int. Ed. 45, 1729–1733 complexes calculated from Pascal’s constants. AC susceptibility measurements (2006). were performed with an AC field of 1.55 G oscillating at frequencies between 1 6. Chibotaru, L. F., Ungur, L. & Soncini, A. The origin of nonmagnetic Kramers and 1,400 Hz. doublets in the ground state of dysprosium triangles: evidence for a toroidal magnetic moment. Angew. Chem. Int. Ed. 47, 4126–4129 (2008). Far infrared. FIR spectra were recorded using a Bruker 113v FTIR spectrometer 7. Ganivet, C. R. et al. Influence of peripheral substitution on the magnetic equipped with a mercury light source. For the detection we used an Infrared behaviour of single-ion magnets based on homo- and heteroleptic TbIII Laboratories pumped Si bolometer (operating temperature 1.5 K). 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Urdampilleta, M., Klyatskaya, S., Cleuziou, J-P., Ruben, M. & Wernsdorfer, W. 7.8 using the geometry as elucidated with X-ray crystallography. All calculations Supramolecular spin valves. Nat. Mater. 10, 502–506 (2011).

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12. Ishikawa, N., Sugita, M., Ishikawa, T., Koshihara, S-y. & Kaizu, Y. Lanthanide 33. Wernsdorfer, W., Aliaga-Alcalde, N., Hendrickson, D. N. & Christou, G. double-decker complexes functioning as magnets at the single-molecular level. Exchange-biased quantum tunnelling in a supramolecular dimer of single- J. Am. Chem. Soc. 125, 8694–8695 (2003). molecule magnets. Nature 416, 406–409 (2002). 13. Sorace, L., Benelli, C. & Gatteschi, D. Lanthanides in molecular magnetism: old 34. Vancoillie, S., Malmqvist, P.-A. & Pierloot, K. Calculation of EPR g tensors for tools in a new field. Chem. Soc. Rev. 40, 3092–3104 (2011). transition metal complexes based on multiconfigurational perturbation theory 14. Woodruff, D. N., Winpenny, R. E. P. & Layfield, R. A. Lanthanide single- (CASPT2). ChemPhysChem 8, 1803–1815 (2007). molecule magnets. Chem. Rev. 113, 5110–5148 (2013). 35. Chibotaru, L. F. & Ungur, L. Ab initio calculation of anisotropic magnetic 15. Long, J. et al. Single-molecule magnet behavior for an antiferromagnetically properties of complexes. I. Unique definition of pseudospin Hamiltonians and superexchange-coupled dinuclear dysprosium(III) complex. J. Am. Chem. Soc. their derivation. J. Chem. Phys. 137, 064112 (2012). 133, 5319–5328 (2011). 36. Eden, M. & Levitt, M. H. Computation of orientational averages in solid-state 16. Guo, Y.-N. et al. Strong axiality and ising exchange interaction suppress NMR by Gaussian Spherical Quadrature. J. Magn. Reson. 132, 220–239 (1998). zero-field tunneling of magnetization of an asymmetric Dy2 single-molecule magnet. J. Am. Chem. Soc. 133, 11948–11951 (2005). 17. Habib, F. et al. The use of magnetic dilution to elucidate the slow magnetic Acknowledgements relaxation effects of a Dy2 single-molecule magnet. J. Am. Chem. Soc. 133, E.M.P. thanks the Panamanian agency SENACYT-IFARHU for funding and COST 8830–8833 (2011). Action: CM1006 for funding for HF-EPR measurements. N.F.C. thanks The University of 18. Fatila, E. M. et al. Fine-tuning the single-molecule magnet properties of a Manchester for a President’s Doctoral Scholarship. The work at Stuttgart was supported [Dy(III)-radical]2 pair. J. Am. Chem. Soc. 135, 9596–9599 (2013). by the DFG. R.E.P.W. thanks the Royal Society for a Wolfson Merit Award. We also 19. Meihaus, K. R., Rinehart, J. D. & Long, J. R. Dilution-induced slow magnetic thank EPSRC (UK) for funding the National EPR Facility and Service, and for an X-ray relaxation and anomalous hysteresis in trigonal prismatic dysprosium(III) and diffractometer (grant number EP/K039547/1). S.-D.J. thanks the Humboldt foundation uranium(III) complexes. Inorg. Chem. 50, 8484–8489 (2011). for a postdoc fellowship. We thank Professor M. Dressel (Stuttgart) for access to the 20. Aguila`,D.et al. Lanthanide contraction within a series of asymmetric dinuclear far-infrared spectrometer. We thank Diamond Light Source for access to synchrotron [Ln2] complexes. Chem. Eur. J. 19, 5881–5891 (2013). X-ray facilities. 21. Cucinotta, G. et al. Magnetic anisotropy in a dysprosium/DOTA single- molecule magnet: beyond simple magneto-structural correlations. Angew. Chem. Int. Ed. 51, 1606–1610 (2012). Author contributions 22. Marx, R. et al. Spectroscopic determination of crystal field splittings in The synthesis, crystallographic characterization and magnetic data collection and lanthanide double deckers. Chem. Sci. 5, 3287–3293 (2014). processing for all complexes was performed by E.M.P. Low-frequency EPR data was 23. Aquilante, F. et al. MOLCAS 7: the next generation. J. Comput. Chem. 31, collected jointly between D.O.S. and E.M.P. FIR and HF-EPR measurements were 224–247 (2010). performed by R.M., M.D., P.N., S.-D.J. and J.v.S. N.F.C. performed ab initio calculations 24. Chilton, N. F., Collison, D., McInnes, E. J. L., Winpenny, R. E. P. & Soncini, A. and modelled the magnetic data. E.M.P. and N.F.C. jointly performed EPR simulations An electrostatic model for the determination of magnetic anisotropy in with input from R.E.P.W., E.J.L.M. and D.C. The paper was written by E.M.P., N.F.C., dysprosium complexes. Nat. Commun. 4, 2551 (2013). E.J.L.M. and R.E.P.W. with input from the other co-authors. 25. Chilton, N. F. et al. Structure, magnetic behavior, and anisotropy of homoleptic trinuclear lanthanoid 8-quinolinolate complexes. Inorg. Chem. 53, 2528–2534 (2014). Additional information 26. Chibotaru, L. F. & Ungur, L. Ab initio calculation of anisotropic magnetic Accession codes: The X-ray crystallographic coordinates for structures reported in this properties of complexes. I. Unique definition of pseudospin Hamiltonians and study have been deposited at the Cambridge Crystallographic Data Centre (CCDC), their derivation. J. Chem. Phys. 137, 064112 (2012). under deposition numbers CCDC 1011262–1011274. These data can be obtained free of 27. Lines, M. E. Orbital angular momentum in the theory of paramagnetic clusters. charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/ J. Chem. Phys. 55, 2977–2984 (1971). data_request/cif, (or from the Cambridge Crystallographic Data Centre, 12 Union Road, 28. Lukens, W. W. & Walter, M. D. Quantifying exchange coupling in f-ion pairs Cambridge CB21EZ, UK; fax: ( þ 44)1223-336-033; or [email protected]). using the diamagnetic substitution method. Inorg. Chem. 49, 4458–4465 (2010). 29. Chilton, N. F., Anderson, R. P., Turner, L. D., Soncini, A. & Murray, K. S. PHI: Supplementary Information accompanies this paper at http://www.nature.com/ a powerful new program for the analysis of anisotropic monomeric and naturecommunications exchange-coupled polynuclear d- and f-block complexes. J. Comput. Chem. 34, 1164–1175 (2013). Competing financial interests: The authors declare no competing financial interests. 30. Boeer, A. B. et al. A spectroscopic investigation of magnetic exchange between Reprints and permission information is available online at http://npg.nature.com/ highly anisotropic spin centres. Angew. Chem. Int. Ed. 50, 4007–4011 (2011). reprintsandpermissions/ 31. Schweinfurth, D. et al. Redox-induced spin state switching and mixed-valency in quinonoid bridged dicobalt complexes. Chem. Eur. J. 20, 3475–3486 (2014). How to cite this article: Pineda, E. M. et al. Direct measurement of dysprosiu- 32. Stoll, S. & Schweiger, A. EasySpin, a comprehensive software package for m(III) dysprosium(III) interactions in a single-molecule magnet. Nat. Commun. spectral simulation and analysis in EPR. J. Magn. Reson. 178, 42–55 (2006). 5:5243 doi: 10.1038/ncomms6243 (2014).

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Supplementary Figure 1. Out-of-phase χM’’(T) (a), in-phase χM’(T) (b) and χM’T(T) (c) behaviour for 3 measured under a 1 kG DC field with an oscillating magnetic field of 1.55 G. Out-of-phase χM’’(T) (d), in-phase χM’(T) (e) and χM’T(T) (f) behaviour for {Dy@Lu2} measured under a 1 kG dc field with an oscillating magnetic field of 1.55 G.

158

Supplementary Figure 2. a) χM’’(ν) for {Dy@Y2} measured at 7 K and applied DC fields from 0 to 3 kG; b) χM’(T) for {Dy@Y2} under a 1 kG applied DC field; c) experimental data and fits (solid lines) for χM’(ν) under a 1 kG DC field for {Dy@Y2}; d) experimental data and fits (solid lines) for χM’’(ν) under a 1 kG DC field for {Dy@Y2}.

159

Supplementary Figure 3. Frequency dependence of χM’(ν) (a), χM’’(ν) (b) and Cole-Cole plots (χM’’ vs. χM’) (c) showing two relaxation process from 1.8 – 3.0 K for {Dy@Y2}, under a 1 kG applied DC field.

160

Supplementary Figure 4. a) Far-infrared transmission spectra recorded on a pressed powder pellet of 3 at various magnetic fields as indicated; b) Normalized transmission spectra (all spectra divided by the 6T spectrum) of 3. All measurements were performed at 9 K.

161

Supplementary Figure 5. Comparison of CF calculations using the scaled (red) and non- -1 scaled (blue) CFPs, using the exchange parameter JLines = +0.047(1) cm . The poor agreement of the calculation with the non-scaled CFPs to the experimental χMT values prevents any meaningful determination of JLines. The magnetization data is insensitive to JLines as the magnetic field quickly overcomes this small interaction.

162

Supplementary Figure 6. (a) X-, (b) K-band and (c) Q-band experimental (black trace) and simulated (blue and red trace) EPR spectra for {Dy@Lu2} at 5 K. Simulation of a single Dy(III) in NO3 pocket with gx = gy = 0.1, gz = 13.9 (red trace). Simulations of Dy in NO3 pocket and small fraction of 3 (blue trace) (see main text for simulation parameters). Statistically a small amount of 3 should be present in the doped material (i.e. 0.25% of the total is {Dy2}). Due to the low transition probability of Dy(III) in the NO3 pocket a small intensity would be expected compared with the strong transition probabilities for 3.

163

Supplementary Figure 7. Experimental (black trace) and simulated (blue trace) spectra for {Dy@Y2} at 5 K and (a) 260 GHz; (b) 320 GHz; (c) 350 GHz and (d) 380 GHz. Simulation obtained using Seff = ½ with gx = gy = 0.1, gz = 13.9.

164

Supplementary Figure 8. Experimental (black trace) and simulated (blue trace) spectra for {Dy@Lu2} at 5 K and (a) 260 GHz; (b) 320 GHz; (c) 350 GHz and (d) 380 GHz. Simulation obtained using Seff = ½ with gx = gy = 0.1, gz = 13.9.

165

Supplementary Figure 9. Experimental (black trace) and simulated (blue trace) spectra for 3 at S-band and 5 K. Simulated as the exchange-coupled system, using the parameters and model described in the main text.

166

Supplementary Figure 10. (a) 260, (b) 320, (c) 350 and (d) 380 GHz experimental HF-EPR spectra of compound 3 at 5 K (black trace) and simulated EPR spectra (blue trace). Simulated as the exchange coupled system, using the parameters and model described in the main text.

167

Supplementary Figure 11. (a) S-, (b) X-, (c) K- and (d) Q-band experimental EPR spectra (black trace) and simulated (blue trace) EPR spectra for compound 3 at 5 K. Simulated using the Lines model with scaled ab-initio CFPs and the exchange parameter extracted from magnetometry measurements. The poor agreement indicates that the Lines model is inadequate in this case.

168

Supplementary Figure 12. Four lowest lying states of 3 where the 44° tilt has been removed, calculated as a function of magnetic field coincident with the easy axis. This model uses the EPR exchange parameters determined for 3.

169

Supplementary Figure 13. Four lowest lying states of 3, calculated as a function of magnetic field coincident with the principal axis of Dy(1). This model uses the scaled ab-initio CFPs, -1 ab-initio rotations and the Lines exchange parameter of JLines = +0.047(1) cm .

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Supplementary Table 1. Elemental analysis and yield (%) for compounds 1 – 7 Formula Yielda % Elemental analysis: Found (calculated) C H N Ln 1 75.4 49.39 3.24 10.02 15.9 [hqH2][Y2(hq)4(NO3)3]·MeOH (49.18) (2.99) (10.00) (15.77) 2 51.9 43.89 2.88 8.90 25.25 [hqH2][Tb2(hq)4(NO3)3]·MeOH (43.74) (2.69) (8.81) (25.20) 3 74 43.65 2.87 8.85 25.67 [hqH2][Dy2(hq)4(NO3)3]·MeOH (43.49) (2.76) (8.92) (25.43) 4 58.7 43.48 2.86 8.82 25.96 [hqH2][Ho2(hq)4(NO3)3]·MeOH (43.23) (2.67) (8.78) (25.78) 5 72.2 43.32 2.84 8.79 26.23 [hqH2][Er2(hq)4(NO3)3]·MeOH (43.16) (2.62) (8.69) (26.05) 6 72.8 42.93 2.82 8.71 26.89 [hqH2][Yb2(hq)4(NO3)3]·MeOH (42.95) (2.87) (8.88) (27.03) 7 67.7 42.8 2.81 8.68 27.11 [hqH2][Lu2(hq)4(NO3)3]·MeOH (42.93) (2.78) (8.56) (27.13) a. Calculated based on 8-hydroxyquinoline

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Supplementary Table 2. Elemental analysis and yield (%) for doped compounds 8 – 13 Formula Elemental analysis: Found (calculateda) C H N Y Ln Lu 8 ca. 5% Tb@1 49.1 3.22 9.95 15.01 1.41 - (48.92) (3.13) (9.84) (15.07) (1.06) 9 ca. 5% Dy@1 49.07 3.22 9.95 15.00 1.44 - (48.97) (3.18) (10.10) (14.70) (1.87) 10 ca. 5% Ho@1 49.06 3.22 9.95 15.00 1.46 - (49.02) (3.04) (9.79) (14.85) (1.64) 11 ca. 5% Er@1 49.05 3.22 9.95 15.00 1.48 - (48.93) (3.19) (9.73) (14.98) (1.53) 12 ca. 5% Yb@1 49.02 3.22 9.94 14.99 1.53 - (48.89) (3.06) (9.74) (14.92) (1.76) 13 ca. 5% Dy@7 42.85 2.81 8.69 - 1.26 25.78 (42.54) (2.64) (8.56) (1.17) (25.64) a) Calculated based on 5% doping by molar ratio

172

Supplementary Table 3. Crystallographic information for dimers 1 – 4

1 2 3 4

Data Collection Space group P21/c P21/c P21/c P21/c T (K ) 100.15 100.15 106.5(1) 102.4(8) Cell dimensions a / (Å) 20.9895(8) 21.0148(16) 20.9630(6) 20.8835(7) b / (Å) 11.6630(5) 11.6603(7) 11.5969(3) 11.5805(3) c / ( Å) 20.4338(8) 20.3588(17) 20.3323(6) 20.2541(7) β / (°) 114.458(5) 114.416(10) 114.349(3) 114.287(4) V / Å 3 4553.3(4) 4542.5(7) 4503.2(2) 4464.8(3) Crystal System monoclinic monoclinic monoclinic monoclinic Rint 0.0942 0.0843 0.0320 0.0613 Rσ 0.0766 0.0745 0.0484 0.0821 Wavelength (Å) 0.6889 1.54184 0.71073 0.71073 θ range (°) 1.938–25.503 4.440–68.250 2.933–26.372 2.943–26.372 Completeness (%) 98.68 100 100 100 Refinement Formula Y2N8O15H36C46 Tb2N8O15H36C46 Dy2N8O15H36C46 Ho2N8O15H36C46 Molecular Mass (g mol-1) 1118.65 1258.67 1265.83 1270.69 Z 4 4 4 4 No of reflections 9184 8290 9191 9113 ρ calc. / g cm-3 1.632 1.840 1.867 1.890 a R1(I > 2σ)(I)) 0.0452 0.0408 0.0348 0.0351 a wR2 0.1017 0.0840 0.0797 0.0646 a R1(all data) 0.0605 0.0666 0.0474 0.0595 a wR2 (all data) 0.1134 0.0935 0.0878 0.0728 a 2 2 1/2 R1 = ||Fo| - |Fc||/|Fo|, wR2=[w(|Fo| - |Fc|) /w|Fo| ]

173

Supplementary Table 4. Crystallographic information for dimers 5 – 7

Data Collection 5 6 7 T (K) 101.6(5) 100.15 164(2) Cell dimensions a / (Å) 20.8408(18) 20.9279(6) 20.8413(12) b / (Å) 11.5872(6) 11.6145(3) 11.5713(5) c / (Å) 20.2631(18) 20.4114(7) 20.2950(12) β / (°) 114.303(11) 114.324(4) 114.204(7) V / Å3 4459.6(7) 4520.9(3) 4464.1(5) Crystal System monoclinic monoclinic monoclinic Rint 0.1429 0.0407 0.0536 Rσ 0.1941 0.0442 0.1066 No. of reflections 9092 9148 9046 Wavelength (Å) 0.71073 0.6889 0.71073 θ range (°) 2.821–26.373 2.548–25.503 3.062–26.371 Completeness (%) 100 99.1 99.1 Refinement Formula Er2N8O15H36C46 Yb2N8O15H36C46 Lu2N8O15H35C46 Space group P21/c P21/c P21/c Molecular Mass (g mol-1) 1275.35 1286.91 1290.77 Z 4 4 4 ρ calc. / g cm-3 1.899 1.891 1.921 a R1(I > 2σ)(I)) 0.0678 0.0344 0.0557 a wR2 0.1530 0.0965 0.0989 a R1(all data) 0.1298 0.0367 0.0852 a wR2 (all data) 0.1956 0.0999 0.1128 a 2 2 1/2 R1 = ||Fo| - |Fc||/|Fo|, wR2=[w(|Fo| - |Fc|) /w|Fo| ]

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Supplementary Table 5. Selected bond lengths for dimers 1 – 7.

(Å) Y Tb Dy Ho Er Yb Lu Ln1—Ln2 3.4606(5) 3.5056(6) 3.4854(4) 3.4659(4) 3.4456(8) 3.4222(4) 3.4033(5) Ln1—O1A 2.243(2) 2.260(4) 2.248(3) 2.241(3) 2.236(9) 2.212(3) 2.208(6) Ln1—O1B 2.340(2) 2.361(4) 2.349(3) 2.337(3) 2.304(7) 2.313(2) 2.304(5) Ln1—O1C 2.365(2) 2.372(5) 2.358(4) 2.341(4) 2.323(9) 2.344(2) 2.303(7) Ln1—O1D 2.347(3) 2.390(4) 2.384(4) 2.372(3) 2.373(7) 2.318(3) 2.343(6) Ln1—N1A 2.469(2) 2.496(3) 2.476(3) 2.462(3) 2.448(8) 2.438(3) 2.427(6) Ln1—N1B 2.516(3) 2.539(4) 2.535(3) 2.525(3) 2.496(8) 2.497(3) 2.483(6) Ln1—N1C 2.499(4) 2.573(5) 2.561(4) 2.537(4) 2.540(1) 2.520(4) 2.496(7) Ln1—N1D 2.545(3) 2.530(5) 2.522(5) 2.500(5) 2.490(1) 2.474(4) 2.440(1) Ln2—O1N 2.402(2) 2.428(5) 2.416(3) 2.399(3) 2.388(7) 2.369(3) 2.364(6) Ln2—O1B 2.312(2) 2.328(5) 2.318(4) 2.298(3) 2.292(7) 2.280(3) 2.276(6) Ln2—O1C 2.287(2) 2.308(3) 2.298(3) 2.286(3) 2.278(7) 2.253(3) 2.248(5) Ln2—O1D 2.285(2) 2.312(4) 2.301(4) 2.280(4) 2.280(9) 2.247(2) 2.231(7) Ln2—O2N 2.478(2) 2.495(4) 2.478(3) 2.471(3) 2.464(7) 2.460(3) 2.454(5) Ln2—O4N 2.455(3) 2.498(5) 2.475(4) 2.458(4) 2.445(9) 2.444(4) 2.424(7) Ln2—O6N 2.427(3) 2.511(5) 2.499(4) 2.487(4) 2.480(1) 2.401(3) 2.439(7) Ln2—O7N 2.468(4) 2.468(4) 2.463(4) 2.443(4) 2.430(1) 2.452(4) 2.426(6) Ln2—O8N 2.481(4) 2.463(3) 2.451(3) 2.436(3) 2.400(8) 2.459(4) 2.396(5)

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Supplementary Table 6. Crystallographic information for mixed-metal systems (1:1)

Data Collection Y:Tb Y:Dy Y:Ho T (K) 100.15 139(1) 100.15 Cell dimensions a / (Å) 20.9511(12) 20.9166(12) 20.9304(13) b / (Å) 11.6457(3) 11.6153(5) 11.6242(5) c / (Å) 20.3330(9) 20.330(2) 20.3448(10) β / (°) 114.319(6) 114.331(9) 114.266(7) V / Å3 4520.9(4) 4500.6(6) 4512.5(4) Crystal System monoclinic monoclinic monoclinic Rint 0.0470 0.1001 0.0553 Rσ 0.0709 0.1664 0.0872 No. of reflections 9243 9188 9078 Wavelength (Å) 0.6889 0.71073 0.6889 θ range (°) 1.985–25.502 2.934–26.373 2.004–25.502 Completeness (%) 100 100 99 Refinement Formula YTbN8O15H36C46 YDyN8O15H36C46 YHoN8O15H36C46 Space group P21/c P21/c P21/c Molecular Mass (g mol-1) 1188.66 1192.24 1194.67 Z 4 4 4 ρ calc. / g cm-3 1.746 1.760 1.758 a R1(I > 2σ)(I)) 0.0457 0.0625 0.0420 a wR2 0.1142 0.1092 0.0802 a R1(all data) 0.0635 0.1270 0.0609 a wR2 (all data) 0.1260 0.1373 0.0916 a 2 2 1/2 R1 = ||Fo| - |Fc||/|Fo|, wR2=[w(|Fo| - |Fc|) /w|Fo| ]

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Supplementary Table 7. Crystallographic information for mixed-metal systems (1:1)

Data Collection Y:Er Y:Yb Y:Lu T (K) 150.0(2) 100.15 150.0(2) Cell dimensions a / (Å) 20.8897(8) 20.8637(18) 20.926(3) b / (Å) 11.5976(3) 11.6041(7) 11.5849(9) c / (Å) 20.3126(8) 20.3299(16) 20.340(2) β / (°) 114.304(5) 114.274(10) 114.245(13) V / Å3 4485.0(3) 4486.8(7) 4496.0(9) Crystal System monoclinic monoclinic monoclinic Rint 0.0443 0.0833 0.0927 Rσ 0.0696 0.1673 0.1994 No. of reflections 9163 9075 9096 Wavelength (Å) 0.71073 0.71073 0.71073 θ range (°) 3.055–26.371 3.289–26.37 3.203–26.372 Completeness (%) 100 99 99 Refinement Formula YErN8O15H36C46 YYbN8O15H36C46 YLuN8O15H36C46 Space group P21/c P21/c P21/c Molecular Mass (g mol-1) 1197.00 1202.78 1204.71 Z 4 4 4 ρ calc. / g cm-3 1.773 1.781 1.780 a R1(I > 2σ)(I)) 0.0426 0.0646 0.0701 a wR2 0.0877 0.1158 0.0777 a R1(all data) 0.0667 0.1385 0.1641 a wR2 (all data) 0.0973 0.1438 0.1032 a 2 2 1/2 R1 = ||Fo| - |Fc||/|Fo|, wR2=[w(|Fo| - |Fc|) /w|Fo| ]

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Supplementary Table 8. Occupancy of Ln in 50% Ln@[Y2(hq)4(NO3)3][hqH2]·MeOH.

Y-Ln % hq pocket % NO3 pocket Tb 45 55 Dy 56 44 Ho 64 36 Er 69 31 Yb 71 29 Lu 84 16

178

Supplementary Table 9. States included for Dy complex Spin RASSCF roots RASSI states 5 /2 21 21 3 /2 224 128 1 /2 158 130

179

Supplementary Table 10. ab-initio results for Dy(1) (hq pocket) -1 o a b E (cm ) gx gy gz Angle/ Dominant mJ states 0.0 0.01 0.01 19.75 - 106.0 0.15 0.16 16.91 1.9 221.2 0.21 0.37 13.79 3.6 311.1 3.83 4.92 9.54 4.9 383.3 1.30 5.68 8.64 82.3 439.7 2.27 2.90 12.63 84.4 515.1 0.50 1.15 17.90 84.8 598.1 0.14 0.36 19.20 75.5 a) Angle of gz with respect to gz of ground Kramers doublet b) Calculated using the CFPs in Supplementary Table S13 with PHI

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Supplementary Table 11. ab-initio results for Dy(2) (NO3 pocket) -1 o a b E (cm ) gx gy gz Angle/ Dominant mJ states 0.0 0.05 1.54 16.42 - 23.9 1.29 7.72 10.71 72.7 38.8 2.45 4.17 7.15 80.0 86.8 2.42 3.77 14.06 22.8 127.4 0.86 1.32 15.98 68.3 158.5 2.34 5.27 10.24 76.1 222.2 0.61 3.62 6.01 61.3 265.5 1.33 7.79 12.68 49.9 a) Angle of gz with respect to gz of ground Kramers doublet b) Calculated using the CFPs in Supplementary Table S13 with PHI

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Supplementary Table 12. Comparison of the ab-initio calculation and FIR spectroscopy for 3. A scaling factor of 1.6 was used. -1 FIR (cm ) ab-initio, NO3 scaled ab-initio, ab-initio, hq scaled ab-initio, pocket NO3 pocket pocket hq pocket 39 23.9 38.2 106.0 169.6 59 38.8 62.1 221.2 353.9 86.8 139.9 311.1 497.8

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Supplementary Table 13. Ab initio Crystal Field Parameters (CFPs) for 3, scaled by factor of 1.6 derived from FIR results (Supplementary Table 12). -1 -1 CFP Dy(1) hq pocket (cm ) Dy(2) NO3 pocket (cm ) 82 14 81 -16 757 179 -287 -55 -162 -36 671 -63 1726 876 20 64 -89 77 21 -80 155 649 -223 -35 -1175 1016 82 -217 43 176 -90 460 -242 -130 298 443 61 174 19 135 21 -15 -8 254 -59 -153 -664 -178 126 157 771 560 -121 518 Euler rotations (α, β, γ), in degrees, (using the PHI convention) to transform the above CFPs in the local reference frames defined by the magnetic axes of ground Kramers doublets of Dy(1) and Dy(2), to the molecular frame (see main text): Dy(1) = (-126, 33, 86) Dy(2) = (-44, 78, 146)

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10. Conclusion

The use of theoretical methods for understanding and predicting the magnetic properties of transition metal complexes has been expounded. This work shows how the judicious application of such methods in conjunction with complementary experimental techniques can be leveraged to gain significant new insight into the anisotropic magnetic properties of molecules.

It has been shown, through detailed EPR spectroscopy, magnetic measurements and theoretical calculations, how the complex interplay of single-ion magnetic anisotropy and magnetic exchange interactions in a dimetallic NiII complex can be resolved. Such work highlights the necessity for in-depth studies of the magnetic properties so that the origin of certain magnetic effects can be correctly identified.

II The anomalous magnetic anisotropy of Mn in a Ru2Mn triangle has been shown to originate from strong antisymmetric exchange interactions with RuIII ions. Despite the antiferromagnetic coupling between the two RuIII ions removing their angular momentum contribution to the ground state, the antisymmetric exchange component is successful in transmitting some of the strong anisotropy to the = 5 2 ground state, highlighting another mechanism whereby the inclusion of 4d metal ions can provide a 푆 ⁄ source of interesting physics.

The use of a simple electrostatic model for the determination of the magnetic anisotropy of DyIII complexes has been shown to be a very useful approach, not only for the understanding of strongly anisotropic species, but for the design of new candidate complexes.

The prediction of highly magnetically anisotropic two-coordinate DyIII compounds has shown that there is large scope for improvement upon the current generation of SMMs. This class of compounds is expected to bring major advances to prototype devices as well as providing a platform to understand the nature of the slow magnetic relaxation in lanthanide complexes in great detail.

The direct measurement of the DyIII···DyIII interaction in an asymmetric dimetallic complex highlights the subtlety of such complexes. The strong magnetic anisotropy coupled with weak exchange interactions renders the absolute determination of both a very difficult task. It was shown how this can be achieved with a complementary suite of experimental techniques along with computational approaches, and suggests that the commonplace analysis utilizing the Lines approach to magnetic exchange is limited.

185

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11. Outlook

The research studies presented herein have served to outline some experimental and theoretical methods by which the magnetic anisotropy of transition metal complexes may be determined. While many advances in the understanding of complex magnetic phenomena have been made in recent years, there is still much to be done. The ultimate goal would be the ability to arbitrarily tailor the magnetic properties of target materials to specific applications; this is clearly a long way off, however is perhaps a realistic goal.

Two experimental techniques that will provide further information are single crystal spectroscopic and magnetic measurements, and optical spectroscopy. Orientation- dependent single crystal experiments can be mapped from the crystal frame onto the molecular frame, providing a direct measurement of the anisotropic properties. While these experiments are more challenging, the information available far surpasses that contained in polycrystalline or powder data. Separately, while EPR spectroscopy provides vital information on the low energy magnetic states, investigation into the high energy spectrum is also required to fully understand the electronic structure. The use of optical spectroscopies, both absorption and emission on polycrystalline, powder and single crystal samples, should be performed to gather this information.

Although a generally applicable understanding of the origin of the magnetic anisotropy remains hidden, a design principle for the next generation of lanthanide-based SMMs based on a simple electrostatic model has been presented. The theoretical proposal for a two-coordinate complex of DyIII is an extremely challenging synthetic target which may or may not be achievable. It is clear, however, that the predicted properties of the target are so desirable that it is worth pursuing every available avenue to isolate such a species.

Even if these complexes are not themselves accessible, species possessing the same motif of transoid anionic donors should prove to be a step in right direction, further enhancing the barrier to slow magnetic relaxation.

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While much effort has been expended, some detailed herein, on how to enhance the barrier to magnetization reversal in lanthanide complexes, the true mechanism of the relaxation process itself remains elusive. It is well understood that the interaction of the magnetic system with quantized lattice vibrations (phonons), often referred to as spin- lattice relaxation, allows individual molecules to traverse their magnetic states thereby returning to equilibrium population. However, the exact nature of these interactions is not known and this must be thoroughly investigated in order to fully understand how the dynamic magnetic properties can be engineered.

Parallel to the work outlined herein, the understanding of the magnetic anisotropy of complexes of the 5f elements is only in its infancy. This is largely because only a limited number of facilities worldwide are able to handle these radioactive compounds, therefore limiting the experimental data available. Furthermore, theoretical approaches are far less adept for complexes of the 5f elements as all the electronic interactions are of a similar order of magnitude, rendering the techniques outlined in the introduction far from perfect. These ab initio approaches make a number of assumptions and while it has been shown that these assumptions provide a reasonable approximation to reality for complexes of the 3d and 4f elements, there is scope for improving these calculations to make them amenable to 5f-based complexes. In general, actinide complexes require larger active spaces than the minimal fn spaces employed for lanthanide complexes, which can be tackled directly or through the use of Restricted Active Space Self-

Consistent Field (RASSCF) calculations. Secondly, the SOC is larger for actinide complexes and therefore should be included in the orbital optimization procedure.

Given the rapid development of computational infrastructures, simplified instrumental procedures and an ever growing toolkit for highly air-sensitive inorganic chemistry, the magneto-chemical community are well equipped to tackle the challenges posed in the field of magnetic anisotropy. The future is bright; the pathway to technologically relevant compounds has never looked so clear.

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12. Appendix: Further Ph.D. publications

1. S. K. Langley, N. F. Chilton, B. Moubaraki and K. S. Murray, Chem. Commun., 2013,

49, 6965.

2. S. K. Langley, D. P. Wielechowski, V. Vieru, N. F. Chilton, B. Moubaraki, B. F.

Abrahams, L. F. Chibotaru and K. S. Murray, Angew. Chem. Int. Ed., 2013, 52, 12014.

3. S. König, N. F. Chilton, C. Maichle-Mössmer, E. Moreno Pineda, T. Pugh, R.

Anwander and R. A. Layfield, Dalton Trans., 2013, 43, 3035.

4. C.-Y. Lin, J.-D. Guo, J. C. Fettinger, S. Nagase, F. Grandjean, G. J. Long, N. F. Chilton and P. P. Power, Inorg. Chem., 2013, 52, 13584.

5. G. F. S. Whitehead, J. Ferrando-Soria, L. G. Christie, N. F. Chilton, G. A. Timco, F.

Moro and R. E. P. Winpenny, Chem. Sci., 2014, 5, 235.

6. N. F. Chilton, G. B. Deacon, O. Gazukin, P. C. Junk, B. Kersting, S. K. Langley, B.

Moubaraki, K. S. Murray, F. Schleife, M. Shome, D. R. Turner and J. A. Walker, Inorg.

Chem., 2014, 53, 2528.

7. S. K. Langley, L. Ungur, N. F. Chilton, B. Moubaraki, L. F. Chibotaru and K. S.

Murray, Inorg. Chem., 2014, 53, 4303.

8. S. K. Langley, D. P. Wielechowski, V. Vieru, N. F. Chilton, B. Moubaraki, L. F.

Chibotaru and K. S. Murray, Chem. Sci., 2014, 5, 3246.

9. S. K. Langley, D. P. Wielechowski, V. Vieru, N. F. Chilton, B. Moubaraki, L.F.

Chibotaru and K. S. Murray, Chem. Commun., 2015, 51, 2044.

10. B. M. Day, N. F. Chilton and R. A. Layfield, Dalton. Trans., 2015, doi:

10.1039/c5dt00346f.

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