This volume provides a treatment of magnetic interactions in terms of the phenomenological spin Hamiltonians that have been such powerful tools in chemistry and physics for the past half century. The first chapter of this volume introduces a number of basic concepts and tools necessary for the development of the theories and methods addressed in the following chapters.
Slater Determinants and Slater–Condon Rules
A serious shortcoming is that neither a Hartree product nor a Slater determinant can be an eigenfunction of the N-electron Hamiltonian operator. Initially, we take a closer look at the one-electron part of the Hamiltonian.
Generation of Many Electron Spin Functions
- Many Electron Spin Functions by Projection
- Spin Functions by Diagonalization
- Genealogical Approach
- Final Remarks
This can be achieved directly with the Serbian variant of the genealogical approach [3,4] illustrated in the branching diagram in Fig.1.5. 1.10 (a) Check that the expressions in Eq. 1.50band1.57 are two different MS components of the same triplet.
Perturbation Theory
Rayleigh – Schr ö dinger Perturbation Theory
To determine the first-order rectified wave function, we need to find the values of the expansion coefficient of Eq.1.65. When excited state energies of the model system (E(0)i) are close to E0(0), the corresponding terms in the summations of ψ(1)andE(2) diverge, unless the matrix elements in these terms are zero.
M ø ller – Plesset Perturbation Theory
This observation also serves to simplify the second-order correction of the energy. 1.83) Again, only the doubly excited determinants need to be considered to calculate the second-order correction for the energy. The expression for the third-order correction of the energy is slightly more complicated, but since the most striking feature introduces the effect of the interaction between excited determinants.
Quasi-Degenerate Perturbation Theory
In the top scheme, the model space is diagonalized and then the effect of the external determinants is included state-by-state. In the lower scheme, all matrix elements in the model space are perturbed, and subsequently the model space is diagonalized.
Effective Hamiltonian Theory
It may be useful to process all matrix elements of the model Hamiltonian. To get an insight into the model parameters, the initial anab Hamiltonian calculation was performed giving the following multideterminant wave functionsΨkand energiesEk.
Atomic Magnetic Moments
Summary This chapter discusses some of the magnetic phenomena that can be observed in systems with a single paramagnetic center. From a classical point of view, this contribution (the spin-magnetic moment) is due to the rotation of the charged electron about its axis.
The Eigenstates of Many-Electron Atoms
Orbital momentum damping: The eigenfunctions of the angular momentum operator ˆl2 are spherical harmonics denoted by the quantum numbers and m. Then the orbital angular momentum of the ground state is defined only from the matrix elementpx|ˆl|pxonly, which is equal to zero, as can be seen from matrix 2.10.
Further Removal of the Degeneracy of the N-electron States
Zero Field Splitting
In the absence of an external field and assuming a quenched orbital angular momentum, the effect of spin–orbit coupling on the ground state levels can be analyzed qualitatively with second-order perturbation theory. 0|V|0 = S,MS|ζSˆ|S,MS0| ˆL|0 (2.12) Regardless of the eSorMS value, this product is strictly zero since we assumed that the ground state has no orbital angular momentum.
Splitting in an External Magnetic Field
This internal field depends on the mean magnetization (M) of the material and is known as the Weiss field. The spin part of the wave function is not written explicitly and can be αorβ.
Combining ZFS and the External Magnetic Field
Sxon|1,0 is most easily obtained by using the expression of Sˆxin terms of the ladder operators Sˆ+andSˆ−. Assuming that the ligands have a closed-shell configuration, the complex can show a splitting of the MS levels in the ground state at zero field. Abstract The description of the magnetic interactions is now extended to more than one magnetic center.
Localized Versus Delocalized Description
The dots in the determinant indicate the other electrons of the system in doubly occupied MOs. Figure 3.3 shows the localized orthogonal orbitals ψa and their ψband product as they appear in the Coulomb integrals of the expression K12. Which state is the ground state, singlet or triplet, depends on the magnitude of the two electronic integrals Jaa, JabandKab.
Model Spin Hamiltonians for Isotropic Interactions
Heisenberg Hamiltonian
Note that the eigenfunctions of the Heisenberg Hamiltonian are multideterminant functions; linear combinations of basic determinants|φ1φ¯2|in| ¯φ1φ2|. The basis of the effective Hamiltonian is the same as for the Heisenberg Hamiltonian. The quartet spin function Q = |ααα is the eigenfunction of the Heisenberg Hamiltonian of equation 3.39 with the eigenvalue −14(J12+J13+J23).
Ising Hamiltonian
4J|φ1φ¯2| (3.47) Assuming that the spatial part is the same in both functions (currently only rotational degrees of freedom are considered), the energy difference between the two determinants gives an estimate of the magnetic coupling through EL S−EH S=. Taking into account that the eigenfunctions of the Ising Hamiltonian are not necessarily the spin eigenfunctions, we use a notation to characterize the eigenfunctions, which consists of the MS-value of both magnetic centers. Figure 3.6 summarizes the energy levels of the Heisenberg and Ising Hamiltonians for both systems.
Comparing the Heisenberg and Ising Hamiltonians
76 3 Two (or more) magnetic centers In the general case of S1 = S2, the distance between the lowest state and the group of degenerate first excited states of the Ising Hamiltonian is given by the value of the smallest spin. These eigenmodes share the common feature that at least one of the local MS values is not equal to ±MSmax. Therefore, in any practical application focused on magnetic interactions, one should only consider the eigenstates of the Ising Hamiltonian with MS=0 or MSmax.
From Micro to Macro: The Bottom-Up Approach
Monte Carlo Simulations, Renormalization
Therefore, we have shown that the partition function of the entire system can be written in terms of properties that only depend on half the number of centers. After calculating the energy of this spin distribution, a trial step in conformational space is taken by inverting the spin at one of the lattices. When J>0, that is for ferromagnetic interactions, the step is accepted because the energy of the system is lowered by the spin flip.
Complex Interactions
Biquadratic Exchange
Calculating the eigenvalues of the triplet and singlet functions is slightly more involved, but follows exactly the same mechanics and can be derived as a useful exercise. Use the results to verify the Heisenberg Hamiltonian eigenvalues of the singlet and triplet spin functions. 4λS (3.74c) and the eigenvalues of the Heisenberg Hamiltonian expanded with a term for the biquadratic exchange.
Four-Center Interactions
The Hamiltonian of this system is a sum of the standard two-body interactions plus Pˆ1234, a four-body operator that cyclically permutes the four spin functions. To determine the result of the sum of four-spin operators, we will develop step by step the action of (SˆA ˆSD)(SˆB ˆSC). In principle, the diagonalization of this matrix should yield the necessary relations to extract the bilinear exchange parameters and the strength of the four-centre interaction.
Anisotropic Exchange
A rigorous description of the antisymmetric interaction is obtained by including the matrix elements in the four determinants spanning the model space. The triplet block and the diagonal elements are exactly the same as in the Hamiltonian considering only the symmetric part of the anisotropic interaction. Construct the 3×3 effective Hamiltonian and extract the different J values by comparing them to the matrix elements of the Heisenberg Hamiltonian given in Eq. 3.39.
Qualitative Valence-Only Models
The Kahn – Briat Model
Abstract Basic understanding and qualitative prediction of the isotropic magnetic coupling between two magnetic centers can be achieved with two well-established valence models. The last part of the chapter is dedicated to the calculation of the interactions beyond the isotropic magnetic coupling. This results in the following expression for the energy of the singlet and triplet states.
The Hay – Thibeault – Hoffmann Model
Furthermore, by expressing the integrals using local orbitals ψa and ψbin instead of molecular orbitals φ1 and φ2, the expression can be written even more compactly. This brings us to the final expression of the HTH model for singlet-triplet splitting. The direct Kab exchange favors the triplet and thus the parallel alignment of the torques.
McConnell’s Model
The third simplification consists in limiting the sum to only the shortest contacts. The closest contacts in folded dimers are formed by the aligned carbon atoms of the benzene ring. The conclusions about the nature of the ground state in the benzylic dimer obtained from the McConnell model are in agreement with those from exact ab initio calculations.
Magnetostructural Correlations
Plus and minus combinations of px and bridge piorbitals in the right column of the MO diagram interact with 3dxy. In the reasoning of the HTH model, the difference in orbital energy ε of the two magnetic orbitals is directly related to the magnetic coupling strength, cf. Left and right Magnetic orbitals for Cu site and V site, respectively. Middle superposition of two magnetic orbitals.
Accurate Computational Models
- The Reference Wave Function and Excited
- Difference Dedicated Configuration Interaction
- Multireference Perturbation Theory
- Spin Unrestricted Methods
- Alternatives to the Broken Symmetry Approach
An important factor in the accurate prediction of parameters of magnetic coupling (and other electronic structures) is the correct choice of the reference wave function. This eliminates any second-order perturbation contribution from the 2h-2p determinants to the off-diagonal elements in the model space. In the limit of zero overlap of the magnetic orbitals, ˆS2BS becomes equal to Smax and the following expression appears.
Decomposition of the Magnetic Coupling
Valence Mechanisms
Therefore, we rewrite the CAS in terms of the orthogonal localized Cu orbitalsandb, shown in Fig.5.2. However, there is also an indirect interaction between the two determinants via the ionic determinants |aa|and|bb| as shown in the lower part of the figure. Instead, one must use fourth-order perturbation theory. representation of the interaction between the neutral determinants ΦI and ΦJ through the bridging ligand.
Beyond the Valence Space
148 5 Towards a quantitative understanding Table 5.1 Breakdown of the magnetic CAS(2,2) coupling in the binuclear Cu2+ complex with a double azido bridge. Analogously, the 1h-2pexcitations can be considered to introduce the relaxation of the metal-to-ligand charge transfer excitations. The addition of these determinants is ferromagnetic in most cases, but usually less than the effect of the 2h-1p determinants.
Decomposition with MRPT2
152 5 Towards a quantitative understanding Table 5.3 Decomposition of the CASPT2 contribution to magnetic coupling in the binuclear Cu2+ complex with a double azido bridge. Table 5.3 shows the contribution of different excitation classes to the composition of the example studied above with DDCI. First, the 1h-1 pexations, which cause spin polarization and relaxation of the ionic determinants.
Mapping Back on a Valence-Only Model
A rigorous way to construct a valence-only model with ab initio methods is to make use of the effective Hamiltonian theory presented in Chap.1. The magnetic coupling extracted from the DDCI energies of the lowest singlet and triplet states is -158 meV. The estimate of J based on Eq.5.8 is in excellent agreement with the result of the full DDCI calculation.
Analysis with Single Determinant Methods
2−1 =2Kab (5.23) Step 2 consists of the optimization of the magnetic orbitals of the BS determinant in the fixed field of the doubly occupied orbitals, a so-called frozen core (FC). The optimization of an enb means that the Sˆ2expectation value of the BS determinant is not exactly one as for the BS-ROKS determinant (see Problems). The lifting of the restrictions on the spin symmetry in the nuclear orbitals introduces different α- and β-spin orbitals, and is therefore responsible for the spin polarization of the core electrons in response to the parallel (HS) or antiparallel (BS) unpaired electrons.
Analysis of Complex Interactions
Decomposition of the Biquadratic Exchange
Hint: Compare the matrix elements of the ionic determinants with the non-Hund states and consider the relative energies of the non-Hund states involved in the coupling. The first column, labeled K, is the result of the diagonalization of the model space with only neutral determinants. This illustrates the role of indirect coupling of non-Hund states to neutral determinants via ionic ones; without significant contribution of ionic determinants, i.e.
Decomposition of the Four-Center Interactions
Setting the energy of the diionic state - third column in the middle of the figure - to 2U, is the total contribution from the six pathways. However, the very large prefactor in the perturbative estimate means that the ring exchange is not necessarily negligible in all cases. As long as U is not too large and large, one can expect significant four-center interactions when the geometry of the system is square.
Complex Interactions with Single
Write the four symmetry-matched CSFs that arise from linear combinations of the four determinants MS = 0. The expressions of states after the configuration interaction given in eq 5.5 can give a hint about the CSFs. Calculate the expected energy values of the four CSFs and place them on the diagonal of the matrix.
Electron Hopping
The interaction matrix elements of the initial and final states are easily determined with the Slater–Condon rules. The contribution of inactive doubly occupied orbitals is the same in all three cases as is the one-electron termhab. Numerical estimates of the jump parameter are relatively easy to obtain with the various computational schemes discussed in Chapter 4.
Double Exchange
The sum of the three terms can be considered as the jump parameter, similar to the expressions given in Ex.6.1band6.1c. The Anderson–Hasegawa model describes the transfer of electrons from site A to B in the field of spin moments SA and SB, which are described as classical vectors. The second simplification arises from the fact that τ ≪Kand justifies the neglect of the quadratic term in the square root.
A Quantum Chemical Approach to Magnetic Interactions
Embedded Cluster Approach
Periodic Calculations
Goodenough–Kanamori Rules
Spin Waves for Ferromagnets
