Orbital Zeeman

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.,... 

The orbital Zeeman interaction is Ms-independent lnvLSMLMs r1MB4 -i) FVL S M M g)... 

The orbital Zeeman interaction involves the matrix element, which is reduced as follows ... 

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 ... 

Here, co represents the Euler angles (orbital Zeeman interaction, we see that it has off-diagonal matrix elements which link electronic states with A A = 0, 1, as well as purely diagonal elements. It is clearly desirable to remove the effect of these matrix elements by a suitable perturbative transformation to achieve an effective Zeeman Hamiltonian which acts only within the spin-rotational levels of a given electronic state rj, A, v), in the same way as the zero-field effective Hamiltonian in equation (7.183). 

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

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 reduction in the orbital Zeeman interaction approaches zero exponentially as Ejy increases and tuo decreases. Note that the quantity Ejy depends solely on the potential energy. 

Quenching of the orbital Zeeman effect and spin-orbit coupling results in an... 

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... 

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

HA5a) 4 0 8 Ge)( ext) rGe) XNe ext)] spin-orbit Zeeman gauge correction... 

Hlb(2a) Jl(2 p BarlGe) orbital Zeeman-kinetic energy correction... 

This can be split into the external field ext( ) = (l/2)(5ext x rGi) and the internal dipolar field AN(rt) = (pL0/4n)r (pLN x rNi) contributions. The term containing the external field represents the electron-electron spin-orbit Zeeman gauge correction... 

The external vector potential Aext(r,) = (l/2)(.Bext x rGi), for a uniform magnetic field, gives the spin-other orbit Zeeman gauge correction term... 

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