Orbital Zeeman operator

We now let the molecule interact with an external magnetic field Bo. The interaction energy, as far as the orbital is concerned, is given by the orbital Zeeman operator... 

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 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 ... 

The q = 1 components of T (L) will be neglected because they involve the mixing of excited states with the ground vibronic state, giving rise to temperature-independent paramagnetism. With this simplification the matrix elements of the orbital Zeeman interaction operating within the ground vibronic state are... 

The point to emphasise concerning this example is the reduced overlap between the ground vibronic wavefunctions associated with the two electronic states, on account of the displacement of the potential energy minima. The expectation value of any electronic operator connecting the two orbital electronic states will be similarly affected. Consider the orbital Zeeman interaction about the z axis, given by... 

The spin-orbit couphng constant / used to be reduced in magnitude by the amount k as compared to the free-ion value The Zeeman operator can be expressed with the help of the relationship... 

The result follows from spin orthogonality. It is perfectly clear from experiments on atoms and molecules (Zeeman effect) that singlet-triplet and other apparently spin-forbidden transitions do occur, so we are led to assume that the spin and orbital motions of an electron are not uncoupled. In the above example, the transition moment will never vanish identically if the state with spin-orbit coupling operator ffjo- Assume that the state with index n and spin function a-j can interact with another state, say tn, with spin perturbation theory, the corrected state xi l is given by... 

The most important relativistic corrections are the one-electron spin-orbit operator, and the relativistic correction to the spin-Zeeman operator. 

Zeeman operator and the spin-orbit coupling (also called paramagnetic spin-orbit term). A ° and A/ are generally small and of opposite sign, so that their effects tend to cancel one another. For transition metal ions, the g-anisotropy and the deviation from the free electron value mainly come from the third contribution, which is usually the most difficult to calculate. Thus, a proper treatment of the spin-orbit coupling (SOC) is crucial for the g-tensor calculation. 

The first term in the brackets is the usual kinetic energy term, while the second term produces the orbital Zeeman effect. The third term describes the second-order interactions corresponding to the atomic contribution to the susceptibility. The second term can easily be converted into the more familiar form of the Zeeman operator as follows ... 

There are electron spin-independent and spin-dependent paramagnetic terms (linear in the vector potential), and a diamagnetic spin-independent term that (quadratic in the vector potential). By substituting (12.12a) or (12.12b) in the spin-free linear terms, one obtains the ZORA form of the Orbital Zeeman (OZ) operator... 

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