Nucleus spin-orbit potential

The first term on the right is the operator for the electrons kinetic energy the second term is the operator for the potential energy of attraction between the electrons and the nucleus (r, being the distance between electron i and the nucleus) the third term is potential energy of repulsion between all pairs of electrons ru being the distance between electrons / and j) the last term is the spin-orbit interaction (discussed below). In addition, there are other relativistic terms besides spin-orbit interaction, which we neglect. 

Spin-Orbit Interaction. If an electron of mass m and orbital angular momentum L moves with a velocity v in an electric field (internal to the atom or molecule) Eint, or in an electric potential 0int(r) (due to the nucleus), then it experiences, in addition to Eint, a magnetic field Bint. If the potential is spherically symmetric, then the electric field is simply... 

A are the scalar and vector potentials of an external field, the suffixes, and w refer to the electron and nucleus respectively, and U is the Breit interaction. The spin-orbit interaction of the electron is included in He, that of the nucleus, which is smaller by the factor ( mjM)2, in Hn. In the term U are included the magnetic orbit-orbit and spin-spin interactions, and additional spin-orbit terms, all proportional to (mM) 1. 

The spin-orbit Hamiltonian of Eq. (31) is correct only for a bare nucleus. In the case of many-electron atoms where the nucleus is surrounded by a "core of electrons, the electrostatic potential, U (r), changes more rapidly with r because of the rapid change in shielding by the core as we... 

In addition, the associated spin orbitals (with a maximum occupation number of one electron) acquire different spatial extents, and they are drawn in Figure 2.25. At first sight, it is difficult to recognize that the low-lying a spin orbitals (a) are more spatially contracted than the j6 spin orbitals (b) but an orbital difference plot (c) immediately shows that the oc spins dominate in the region close to the nucleus, where they experience a higher nuclear potential. The analysis may also be carried out numerically, but we will not do so here. 

There is no need to explicitly include terms for direct relativistic effects, such as the dependence of mass on velocity, which are important only in the core region, in the valence-electron Hamiltonian. These terms are included as a consequence of using the Diiac-Fock wave functions. Thus, the Hamiltonian for the valence electrons is composed of the nonrelativistic Hamiltoman for the valence electrons plus the RECPs, which include the effects of the core electrons as well as the relativistic effects on the valence electrons in the core region [37]. The RECPs thns represent, for the valence electrons, the dynamical effects of relativity from the core region, the repulsion of the core electrons, the spin-orbit interaction with the nucleus, the spin-orbit interaction with the core electrons, and an approximation to the spin-orbit interaction between the valence electrons [38], which has usually been found to be quite stnaU, especially for heavier element systems [39-41]. The REP operators can be written as a summation of spin-independent potential and the spin-orbit operator, as written below, and the readers are referred to reference [39] for details. 

Relativistic corrections make significant impact on the electronic properties of heavy atoms and molecules containing heavy atoms. The inner s orbitals are the closest to the nucleus and thus experience the high nuclear charge of the heavy atoms. Thus, the inner s orbitals shrink as a result of mass-velocity correction. This, in turn, shrinks the outer s orbitals as a result of orthogonality. Consequently, the ionization potential is also raised. The p orbitals are iilso shrunk by mass-velocity correction but to a lesser extent since the angular momentum keeps the electrons away from the nucleus. However, the spin-orbit interaction splits the p shells into pi/2 and pj/2 subshells and expands the P3/2 subshells. The net result is that the mass-velocity and spin-orbit interactions tend to cancel for the P3/2 shell but reinforce for the Pi/2. 

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