Orbital-Zeeman term

The Hamiltonian that describes the interaction of the single magnetic center with the external magnetic field involves the spin-Zeeman term, the orbital Zeeman term, and the operator of the spin-orbit coupling, i.e.,... 

The second-order contribution of the orbital Zeeman term with itself produces... 

The first product is the orbital Zeeman term. The second term is a higher-order correction in which a commutation relation can be used... 

The second-order contribution of the orbital Zeeman term with itself produces a term in the effective Hamiltonian which is quadratic in 5. It therefore has the same form as the diamagnetic susceptibility contribution to the energy it provides the paramagnetic or high-frequency contribution to the susceptibility of the molecular system. The resultant term in the effective Zeeman Hamiltonian is... 

Here we see that the spin-dependent term of the modified vector potential is in fact describing the interaction of the spin with the magnetic field, and the scalar term is describing the interaction of the orbital angular momentum with the magnetic field. These terms are the spin and orbital Zeeman terms respectively. In (15.32) there are also relativistic corrections that arise from the difference between ( ) and rlr. 

Notice that the kinetic relativistic correction applies to both the spin and orbital Zeeman terms, but the relativistic correction from the potential only affects the spin Zeeman term. These corrections are only the first in a truncated expansion, and for a better description of relativistic corrections we turn to the free-particle Foldy-Wouthuysen transformation and the higher transformations that are based on it. 

Hfi includes a nuclear Zeeman term, a nuclear dipole-dipole term, an electron-nuclear dipole term and a term describing the interaction between the nuclear dipole and the electron orbital motion. 

Level-6. The most complete treatment utilizes the basis set of all free-atom terms v,L,Ml,S,Ms) for the given electronic configuration dn, and the calculation of energy levels is performed by involving the operators of the electron repulsion, CF, spin-orbit interaction, orbital-Zeeman and spin-Zeeman terms ... 

We shall use the T-/ -isomorphism that allows us to consider the orbital triplet T2 as a state possessing the fictitious orbital angular momentum L = 1, keeping in mind that the matrix elements of the angular momentum operator L within T2 and P bases are of the opposite signs, L(T2) = —L(P) [2]. As it was shown in our recent paper [10] this approach provides both an efficient computational tool and a clear insight on the magnetic anisotropy of the system that appears due to the orbital contributions. Within T-P formalism the spin-orbital and Zeeman terms can be represented as ... 

The last term in Equation (2.36), Agoz/soc, is a second order contribution arising from the coupling of the orbital Zeeman (OZ) and the spin-orbit coupling (SOC) operators. The OZ contribution in the system Hamiltonian is ... 

It should be appreciated that the Zeeman interactions are usually dominated by the first two terms in (7.232), the electron spin and orbital terms. The other terms are typically between two and four orders of magnitude smaller. For a molecule in a closed shell1 state, only the rotational Zeeman term, the nuclear spin contribution and the susceptibility term survive. 

The Zeeman term is considered first and the hyperfine interaction is discussed in Section 4.2.2. The free electron g-value of = 2.0023 is shifted when the electron is surrounded by material, because of the spin-orbit coupling to the other electron states. The shift is... 

The first two terms are Zeeman terms and the third represents the hyperfine interaction of the electron and nuclear spins, / b and are the Bohr and nuclear magnetons respectively, S is a fictitious effective spin (S = 2 for a simple Kramers doublet), and / is the nuclear spin tensor. The hyperfine tensor is further split into Fermi contact, dipolar, and orbital components according to ... 

The magnetic field splits this state into a triplet with Mj = +1, 0, and — 1, whose energy levels Em depend on A and on the orbital and spin g factors < l and < s. The transition to the Mj = 0 state is a so-called n transition with E//B polarization, which can be observed only in the Voigt configuration while those to the Mj = 1 states are o transitions with ETB polarization, which are seen only in the Faraday configuration. The linear Zeeman term of a 3T2 state with a magnetic field B may be written as ... 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

In the presence of a uniform magnetic field B term (a) gives the orbital Zee-man term, term (b) is the diamagnetic contribution and term (c) is the spin Zeeman term. The latter term is obtained from the Schrodinger equation only... 

For many applications the full molecular Hamiltonian is not necessary it is sufficient to include only relevant energy terms into the model Hamiltonian. One of the model Hamiltonians is the spin Hamiltonian which includes only the angular momentum operators in their mutual interaction (orbit-orbit, spin-orbit, spin-spin interactions) as well as their interaction with a magnetic field (the Zeeman terms orbit-magnetic field and spin-magnetic field). 

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