Spin-other-orbit interactions

The spin of one electron can interact with (a) the spins of the other electrons, (b) its own orbital motion and (c) the orbital motions of the other electrons. This last is called spin-other-orbit interaction and is normally too small to be taken into account. Interactions (a) and (b) are more important and the methods of treating them involve two types of approximation representing two extremes of coupling. 

The submatrix element of the energy of the spin-other-orbit interaction... 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

Equation (3.144) represents the interaction of the spin of electron j with the orbital motion of electron i relative to electron j it therefore makes a contribution to the spin-other-orbit interaction. 

We are now in a position to present the total electronic Hamiltonian by summing over all possible electrons i. We must be very careful, however, not to count the various interactions twice. Thus on summing over i, we modify all terms which are symmetric in i and j by a factor of 1 /2. These terms are the electron-electron Coulomb interaction (3.141), the orbit-orbit interaction (3.145), the spin-spin interaction (3.151) and the spin-other-orbit interaction from (3.144) and (3.153). 

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... 

Judd, Crosswhite, and Crosswhite (10) added relativistic effects to the scheme by considering the Breit operator and thereby produced effective spin-spin and spin-other-orbit interaction Hamiltonians. The reduced matrix elements may be expressed as a linear combination of the Marvin integrals,... 

Although this interaction has properties very similar to the spin-other-orbit interaction, it is distinct enough to require the additional paramters k = 2, 4, and 6. 

All the terms of the operator (164) are the order Z (ctZy in r.u., i.e. Z (aZ) o where So is the characteristic binding energy. The terms in the first line of Eq(164) describe the relativistic orbit - orbit interaction, the terms in the second line describe the spin - other orbit interaction (unlike the spin-orbit interaction that is included in the one-electron Dirac equation) and the terms in the third line describe the spin-spin interaction. 

The second part, the spin-other-orbit interaction, is due to interelectronic interactions and has the effect of partially counterbalancing the field of the bare nuclei. Its sign is opposite to that of the first part. This operator is a two-electron operator because of the r term. 

It can be shown (Veseth, 1970) that all electron-nuclear distances, r ) can be referred to a common origin, and, neglecting only the contribution of spin-other-orbit interactions between unpaired electrons, the two-electron part of the spin-orbit Hamiltonian can be incorporated into the first one-electron part as a screening effect. The spin-orbit Hamiltonian of Eq. (3.4.2) can then be written as... 

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