Phenomenological spin Hamiltonians interactions

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

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,... 

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 ... 

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. 

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... 

There is, however, one more interaction which should be (at least phenomenologically) included into the magnetic Hamiltonian the spin-orbit interaction. 

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. 

The term symbols encode the values of the quantum numbers L, S, and /, and an expression for the spin-orbit interaction energy of a many-electron atom follows directly. With the Hamiltonian in Equation 10.21 and with a phenomenological constant, y, specific to the atomic system, the energy associated with spin-orbit interaction in the absence of an exfernal magnetic field is... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

A special focus will be on phenomenological Hamiltonians involving electronic spin interactions. For this it is necessary to define atomic surrogate spin operators—so-called local spin operators—that may be directly related to the effective spins in... 

Ia) is included in the electronic Hamiltonian since, as we shall see, its most important effects arise from interactions involving electronic motions. The interactions which arise from electron spin, 30(5 ), will be derived later from relativistic quantum mechanics for the moment electron spin is introduced in a purely phenomenological manner. The electron-electron and electron-nuclear potential energies are included in equation (2.36) and the purely nuclear electrostatic repulsion is in equation (2.37). The double prime superscripts have been dropped for the sake of simplicity. We remind ourselves that // in equation (2.37) is the reduced nuclear mass, M M2/(M + M2). 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

Phenomenologically, the interaction in the dinuclear acetate and many other compounds with similar temperature-dependences of the magnetic moment can be described as antiferromagnetic couplings of the unpaired species on the adjacent Cu11 atoms. This is expressed mathematically by introducing a term — (where Sj = S2 =i are the spin quantum numbers and J is the coupling constant) into the Hamiltonian operator the values of J are close to 300 cm-1 for the majority of the compounds. 

All other interactions between the particles comprising the molecule are, at least for the present, neglected. In particular there is no mention in the Hamiltonian of any spin-dependent (magnetic) interactions both the phenomenological Pauli (two-component) and the relativistic Dirac (four-component) interaction terms amongst electron and nuclei are replaced by the simple electrostatic model. 

The reduction of (11.3.7) to an approximate form involving the usual spin operators is both involved and uncertain (Appendix 4). Nevertheless, it leads to a Pauli-type Hamiltonian whose terms can be interpreted classically as field-dipole, dipole-dipole interactions and the like. This interpretation provides a basis, insecure though it may be, for a Hamiltonian when one or both particles are nuclei, provided that the nuclear spins and moments are regarded as phenomenological quantities with values to be inferred from experiment. It is then but a short step to the Hamiltonian for a many-particle system, so long as only pairwise interactions are present (as is currently believed to be the case). 

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