Spin-Zeeman term

Under the SH formalism it is understood that the energy levels are reconstructed by considering only the formal spin kets S,Ms) that are under the action of the SH involving the spin-spin interaction and the spin-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.,... 

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 ... 

The restricted summation in equation (7.221) means that, for this contribution, the Zeeman energy depends on the x and y components of S only (i.e. it has cylindrical symmetry). It is therefore referred to as the anisotropic correction to the electron spin Zeeman term, which has the isotropic form gsH sB S. 

We shall return to the results for para-H2 in due course, but first repeat the Zeeman calculations for ortho-H2, including the nuclear spin Zeeman term in equation (8.179). The required matrix elements are as follows ... 

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... 

The terms contributing to Hm = f(Sh B) are compiled in Table 4.6(g). The leading contribution is the (electron spin) Zeeman term... 

The (spin) Zeeman term entering the total magnetic Hamiltonian is expressed... 

Let us investigate the situation of having an orbitally non-degenerate atomic term 2S+ S or a molecular state 2S+ A and 25+1B, respectively. In the absence of thermally accessible excited states the only contribution to the spin Hamiltonian is an isotropic (spin) Zeeman term... 

The exchange Hamiltonian in the presence of the magnetic field is enriched by the (spin) Zeeman term, hence... 

Note that the spin-Zeeman term involves the total electron spin S. 

The diagonal term gel arises from the spin-Zeeman term S Bext, while the anisotropic part has two contributions. The direct term arises from the P operator (eq. (10.79)) and the relativistic correction from the spin-Zeeman term (eq. (10.83)), with additional contributions coming from the combination of V2LG and the spin-orbit operators P and Pe . 

We consider the familiar case of spin space first. In this case, the vector h in equation (2) is given by an external magnetic field B, hQ vanishes, and Hi represents the spin Zeeman term in the one-electron Hamiltonian. Using for the electron spin magnetic dipole moment... 

The fourth term is the spin Zeeman term and represents the interaction of the electron spin with the external magnetic field, and under the substitution fur = 2s it takes the form e/m)s B. 

Here we have expanded the scalar quadruple product. The dominant term is the first term, which is a relativistic correction to the spin Zeeman term. The corrected spin Zeeman operator is... 

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. 

As for the Pauli Hamiltonian, there is a kinematic correction to the entire Zeeman term, but only the spin Zeeman term has a relativistic correction from the potential. Expanding the kinematic factors to order we have ep — I — p /lm c and... 

The first term is simply the spin-Zeeman term, the seeond term is the reduced mass correction, and the fliird term is a gauge correction to the SOC, which has been written in somewhat simplified form here (explained in detail elsewhere [60]). 

Finally, the terms in O Eq. 11.43 arise from the spin-Zeeman term, the last term in O Eq. 11.40. 

Here, is the kinetic energy correction to the spin-Zeeman term given in the Breit-Pauli... 

Considering the expansion of the spin-Zeeman term in O Eq. 11.40, when we insert the magnetic vector potential arising due to the nuclear magnetic moments, we find two contributions that are bilinear in the electron spin and nuclear magnetic moments, namely ... 

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