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Über dieses Buch

This textbook is the second volume in the Theoretical Chemistry and Computational Modeling series and aims to explain the theoretical basis of magnetic interactions at a level that will be useful for master students in physical, inorganic and organic chemistry.

The book gives a treatment of magnetic interactions in terms of the phenomenological spin Hamiltonians that have been such powerful tools for chemistry and physics in the past half century, starting from the simple Heisenberg and Ising Hamiltonians and ending with Hamiltonians that include biquadratic, cyclic or anisotropic exchange. On the other hand, it also explains how quantum chemical methods, reaching from simple mean field methods to accurate models that include the effects of electron correlation and spin-orbit coupling, can help to understand the magnetic properties. Connecting the two perspectives is an essential aspect of the book, since it leads to a deeper understanding of the relation between physical phenomena and basic properties. It also makes clear that in many cases one can derive magnetic coupling parameters not only from experiment, but also from accurate ab initio calculations.

The book starts with introducing a selection of basic concepts and tools. Throughout the book the text is interlarded with exercises, stimulating the students to not only read but also verify the assertions and perform (parts of) the derivations by themselves. In addition, each chapter ends with a number of problems that can be used to check whether the material has been understood.



Chapter 1. Basic Concepts

In this chapter we examine some basic concepts of quantum chemistry to give a solid foundation for the other chapters. We do not pretend to review all the basics of quantum mechanics but rather focus on some specific topics that are central in the theoretical description of magnetic phenomena in molecules and extended systems. First, we will shortly review the Slater–Condon rules for the matrix elements between Slater determinants, then we will extensively discuss the generation of spin functions. Perturbation theory and effective Hamiltonians are fundamental tools for understanding and to capture the complex physics of open shell systems in simpler concepts. Therefore, the last three sections of this introductory chapter are dedicated to standard Rayleigh–Schrödinger perturbation theory, quasi-degenerate perturbation theory and the construction of effective Hamiltonians.

Coen de Graaf, Ria Broer

Chapter 2. One Magnetic Center

This chapter discusses some of the magnetic phenomena that can be observed in systems with a single paramagnetic center. After shortly reviewing the basics of the magnetic moments of a free atom, we analyze the effect of spin-orbit coupling and an external magnetic field on the


levels of the ground state of larger systems. In a step-by-step procedure, we will first derive the model Hamiltonian to describe the magnetic anisotropy without external field, the so-called

zero-field splitting

. Secondly, the role of the external field is explored and a relation is established with the magnetic susceptibility, a macroscopic quantity. The chapter is closed by discussing the model Hamiltonian that combines the zero-field splitting and the anisotropy of the


-tensor to complete the description of the splitting of the


levels in systems with one, anisotropic, magnetic center.

Coen de Graaf, Ria Broer

Chapter 3. Two (or More) Magnetic Centers

The description of the magnetic interactions is now extended to more than one magnetic center. First it is shown that the two-electron/two-orbital system can be approached from different viewpoints using (de-)localized, (non-)orthogonal orbitals. After this quantum chemical description of the magnetic interaction we discuss the more phenomenological approach based on spin operators. Starting with the standard Heisenberg Hamiltonian for isotropic bilinear interactions, the chapter discusses how biquadratic, anisotropic and four-center interactions can be accounted for within this spin formalism. Furthermore, it is shown how the microscopic electronic interaction parameters can be used to describe macroscopic properties by diagonalization of model Hamiltonians, Monte Carlo simulations and some other techniques.

Coen de Graaf, Ria Broer

Chapter 4. From Orbital Models to Accurate Predictions

Basic understanding and qualitative prediction of the isotropic magnetic coupling between two magnetic centers can be obtained with two well-established valence-only models. This chapter discusses the Kahn–Briat and Hay–Thibeault–Hoffmann models, which have been (and still are) of fundamental importance for understanding the basics of magnetism in polynuclear transition metal complexes. After shortly presenting the basic model for magnetism in organic radicals, we review the most evident magnetostructural relations and then move to the accurate prediction of the magnetic coupling. An overview of the most widely used quantum chemical methods is given, including wave function based methods and approaches within the spin-unrestricted setting such as density functional theory. The last part of the chapter is dedicated to the calculation of the interactions beyond the isotropic magnetic coupling.

Coen de Graaf, Ria Broer

Chapter 5. Towards a Quantitative Understanding

Taking a binuclear copper complex as model system, the isotropic magnetic coupling is decomposed into different contributions. Perturbative expressions of the main contributions are derived and illustrated with numerical examples. An effective Hamiltonian is constructed that incorporates all important electron correlation effects and establishes a connection between the complex


-electron wave functions and the simpler qualitative methods discussed in the previous chapter. Subsequently an outline is given of the analysis of the coupling with a single determinant approach and the biquadratic and four-center interactions are decomposed. The chapter closes with the recently proposed method to extract DFT estimates for these complex interactions.

Coen de Graaf, Ria Broer

Chapter 6. Magnetism and Conduction

After the description of the electron hopping in systems where not all the magnetic centers have the same number of unpaired electrons, a short account is given of the double exchange mechanism in mixed-valence systems. Although this phenomenon can certainly be found in transition metal complexes, it is more common to happen in doped systems in the solid state. Therefore, the second part of this chapter introduces the basics of the quantum chemical approach to magnetic interactions in extended systems. The embedded cluster approach will be contrasted against band structure calculations. Thereafter, some concepts will be introduced that are widely used in the condensed matter physics community. We do not give a full description of all the magnetic phenomena in solid state compounds but rather help the reader with a quantum chemical background to find its way in the rich literature on this topic.

Coen de Graaf, Ria Broer


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