## H

1 Nelec

4mc2

Darwin .

8m 2c2

Here F, and B, indicate the (electric and magnetic) fields at the position of particle i. HZeeman has the s • B term from eq. (8.28) and a relativistic correction, and Hmv is the mass-velocity correction, as is also present in eq. (8.18). H|jO and H;?arwm are spin-orbit and Darwin type correction with respect to an external electric field. It should be noted that the generalized momentum operator contains magnetic fields via the vector potential p = p + A, and eq. (8.35) therefore implicitly includes higher order effects.

Two electron operators:

2mC2 ;=1 j#j r3 geMB Nie Vc Si • ("";/■ X p j )

ee 2

p Nelec Nelec

The sums run over all values of i and j, excluding the i = j term, and there is consequently a factor of 1/2 included to avoid overcounting. H is a spin-orbit operator, describing the interaction of the electron spin with the magnetic field generated by its own movement, as given by the angular momentum operator r/j x p,. H is a spin-other-orbit operator, describing the interaction of an electron spin with the magnetic field generated by the movement of the other electrons, as given by the angular momentum operator r/j x pj. H ee and H O° are spin-spin and orbit-orbit terms, accounting for additional magnetic interactions, where the orbit-orbit term comes from the Breit correction to Vee (eq. (8.34)). The (two-electron) Darwin interaction HDearwin contains a 5 function, which arise from the divergence of the field (V F) from the (electron-electron) potential energy operator, i.e. V • (V(1/r)) = _4n5(r). The spin-spin interaction H If also has a 5 function, which comes from taking the curl of the vector potential associated with the magnetic dipole corresponding to the electron spin. A mathematical reformulation leads to a term involving the divergence of the r/r3

operator, giving V • (r/r3) = (4rc/3)8(r). Such terms are often called contact interactions, since they depend on the two particles being at the same position (r = 0). In the spin-spin case, it is normally called the Fermi Contact (FC) term. Operators involving one nucleus and one electron:

HSS = "Z'gA f^ - 3 ^'^A'1') - ^ •Ia )d(r„ ) (8'37)

p Nelec Nnuclei

The H operator is the one-electron part of the spin-orbit interaction, while the HSe°and HS°° operators in eq. (8.36) define the two-electron part. The one-electron term dominates and the two-electron contribution is often neglected or accounted for approximately by introducing an effective nuclear charge in H(corresponding to a screening of the nucleus by the electrons). The effect of the spin-orbit operators is to mix states having different total spin, as for example singlet and triplet states.

The equivalent of the spin-other-orbit operator in eq. (8.36) 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 within the Born-Oppenheimer approximation, and it is normally denoted the Paramagnetic Spin-Orbit (PSO) operator.The spin-spin term is analogous to that in eq. (8.36), while the term describing the orbit-orbit interaction disappears owing to the Born-Oppenheimer approximation. The spin-orbit and (one-electron) Darwin terms are the same as given in eq. (8.18), except for the quantum field correction factor of gejuB.

All of the terms in eqs (8.35)-(8.37) may be used as perturbation operators in connection with non-relativistic theory,5 as discussed in more detail in Chapter 10. It should be noted, however, that some of the operators are inherently divergent and should not be used beyond a first-order perturbation correction.

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