Phenomenological spin Hamiltonians

These fields become important in magnetic resonance spectroscopies, where the interaction between electronic and nuclear spins is considered in phenomenological Spin Hamiltonians. Note... 

The phenomenological Spin Hamiltonian widely applied in quantum chemistry—often also referred to as the Dirac-Heisenberg-Van... 

Though the true electron spin operators were employed here as well as in the Breit-Pauli Hamiltonian, the phenomenological Spin Hamiltonian, in which the spin coupling is an exchange effect, is in sharp contrast to the Breit-Pauli Hamiltonian, that is including the (magnetic) spin-spin interactions. Since the exchange effect is an effect introduced by the Pauli principle imposed on the wave function, we may write the electron-electron interaction as an expectation value,... 

The magnetic properties of a-02, which is the most extensively studied phase, have always been interpreted on the basis of the following phenomenological spin Hamiltonian ... 

In a recent paper (Jansen and van der Avoird, 1985), two of us have proposed replacing the phenomenological spin Hamiltonian (139) by a spin Hamiltonian from first principles. By this qualification we mean that our Hamiltonian can be derived directly from the known properties of the 02 molecules and their interactions. Such a Hamiltonian, which applies not only to a-02, but also to any of the condensed phases, looks as follows ... 

The energy states of the dimer (Fig. 26.3) can be described by a phenomenological spin Hamiltonian, e.g. the Heisenberg Hamiltonian... 

From the examples considered above, it is clear that much information about the nature of the electron distribution, in particular about the distribution of spin and of induced currents, can be obtained from a careful study of spin-Hamiltonian parameters. At this point, then, it must be asked whether the parameters in the phenomenological spin Hamiltonian used by the experimentalist can always be accounted for in this way or whether other types of coupling might appear. 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

The Heisenberg-Dirac-van Vleck (HDVV) spin Hamiltonian [1] is a well-known phenomenological model that became very popular and routinely used in the last... 

As illustrated above, the microscopic explanation of observed magnetic properties hinges on the construction of an appropriate model. In most instances, simplifications have to be weighed and phenomenological models can be employed, such as the Heisenberg spin Hamiltonian. 

The operator [157] is a phenomenological spin-orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined fine-structure splittings via the so-called spin-orbit parameter Aso (see the later section on first-order spin-orbit splitting). In terms of its tensor components, the phenomenological spin-orbit Hamiltonian reads... 

The phenomenological spin-orbit Hamiltonian ought not to be used for computing spin-orbit matrix elements, though. An example for a failure of such a procedure will be discussed in detail in the later subsection on a word of caution. 

A phenomenological spin-orbit Hamiltonian, formulated in terms of tensor operators, was presented already in the subsection on tensor operators. Few experimentalists utilize an effective Hamiltonian of this form (see Eq. [159]). Instead, shift operators are used to represent space and spin angular... 

Let us remark that in crystals consisting of aromatic molecules, to which the theory of Sternlicht and McConnell (26) was applied, the excited triplet states are not three-fold degenerate even when an external magnetic field is absent. Due to the dipole spin-spin interaction between electrons the degeneracy is totally or partially removed, depending on the symmetry of the excited state wavefunction. By a phenomenological description of this splitting the so-called Spin-Hamiltonian is usually applied... 

The evolution of the density matrix is governed by Eq. (2.10) in which the Hamiltonian for the spin system must be specified. It is noted here that the relaxation effects arising from dissipative interactions between the spin system and the lattice have not been included in the equation. The nuclear spin Hamiltonian contains only nuclear spin operators and a few phenomenological parameters that originate from averaging the full Hamiltonian for a molecular system over the lattice coordinates. These magnetic resonance parameters can, at least in principle, be deduced by quantum chemical calculations [2.3]. The terms that will be needed for discussion in this monograph will be summarized here. 

We now turn to the spin-dependent terms listed in Section 11.3, and discuss some examples of their observable effects. Such effects are most commonly interpreted in terms of a phenomenological Hamiltonian, which usually contains only spin operators (for the various nuclei and for the total electron spin) and applied fields, together with numerical parameters that serve as coupling constants . This spin Hamiltonian H describes a model spin system whose behaviour may be determined by solving ... 

In order to account phenomenologically for spectra within the doublets, the spin Hamiltonian for each doublet is written... 

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