Spin-other-orbit

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 equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

Foldy-Wouthuysen transformation the spin-same orbit h and spin-other orbit... 

Interaction of the spin magnetic moment of one electron with the orbital motion of another (spin-other orbit term). 

Finally, spin-orbit interaction has often been considered as the cause of states of mixed permutational symmetry. There are, however, a variety of other spin interactions which may accomplish such mixing electron spin-electron spin, electron spin-nuclear spin, spin-other-orbit, and spin rotation interactions. That other such spin interactions may enhance spin-forbidden processes in organic molecules is frequently ignored, though they may be of importance.66,136... 

An interesting peculiarity of the irreducible expressions for the operators of the spin-other-orbit and spin-spin interactions (formulas (19.14) and (19.20), respectively) is that the resulting ranks of the tensors in the orbital and spin spaces are fixed and equal to one for (19.14) and to two for (19.20). 

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

G. Gaigalas, A. Bernotas, Z. Rudzikas, Ch. Froese Fischer, Spin-other-orbit operator in the tensorial form of second quantization, Physica Scripta, 57, 207-212 (1998). 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

The Breit-Pauli spin-orbit Hamiltonian is found in many different forms in the literature. In expressions [101] and [102], we have chosen a form in which the connection to the Coulomb potential and the symmetry in the particle indices is apparent. Mostly s written in a short form where spin-same- and spin-other-orbit parts of the two-electron Hamiltonian have been contracted to a single term, either as... 

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... 

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et a/.112-115 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules. 

Both of these methods essentially use ab initio wavefunctions to deduce the composition of the MOs. An alternative approach is to use semi-empirical functions, obtained from the MINDO or CNDO methods, to calculate the constants directly from the hamiltonian. A major problem in such work is that in many cases only valence shell electrons are included and consequently spin-other orbit effects cannot be calculated directly. Hinkley, Walker, and Richards122 have shown from ab initio calculations that this shielding for a given atom often bears a constant ratio to theZ/r3 terms. Such a ratio could be... 

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). 

Equation (3.170), which is again of order c represents the influence of the external magnetic field on the spin-orbit coupling, and although we have now dealt with all the terms linear in Af we might expect to find similar modifications to the spin-other-orbit and orbit-orbit interactions. In section 3.6 we neglected terms in (A.)2 since they were necessarily of order c 4. This is not true, however, of terms containing (Af)2 or the cross-product A) Af. Inclusion of such terms in the expansion of 7r yields... 

The first term represents the modification to the spin-other-orbit coupling by the external field, the second term is the corresponding modification of the orbit orbit coupling, and the third term represents the electronic contribution to the diamagnetism. As with the original orbit-orbit coupling term, the second term in (3.172) is incorrect since it does not take account of the retardation effects. The correct form of this term is actually... 

The first two terms represent spin-orbit coupling, whilst the second two are normally described as spin other-orbit terms. Following Fontana [39], Chiu pointed out that for matrix elements diagonal in the total electron spin S(S = S + S2), (8.208) is contracted to the sum of two terms,... 

Both terms now contain spin orbit and spin-other-orbit contributions. The constants a and b are defined (in atomic units) as follows ... 

The final result in (8.214) is obtained by putting A2 = 1. The constant A is half the sum of a spin-orbit coupling constant A ] and a spin other-orbit coupling constant A2,... 

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