Spin-same orbit interaction

A slightly different situation may be related to the electron spin-same orbit interaction which can be considered as a blend of Fj type operators and gradient derivative elements. 

The spin-orbit term involves the interaction of the spin of the electron with its own orbital angular momentum around the other electrons, and is often called the spin-own-orbit interaction or spin-same-orbit interaction. 

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 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... 

The consequence is that we must treat the spin-orbit and the spin-other-orbit interactions separately we cannot combine them as in the Breit-Pauli Hamiltonian. The reason is that the functions on which the momentum operators operate are derived from the small component, and only in the nonrelativistic limit where the large and small components are related by kinetic balance can we rewrite the spatial part of the spin-other-orbit interaction in the same form as the spin-orbit interaction. The reader... 

The two-electron operator is given in the nuclear frame and not in the reference of either electron. The spin-orbit coupling due to the relative motion of elecrons therefore splits into two parts The total interaction is the coupling of the spin of a selected reference electron with the magnetic field induced by a second electron. The spin-same orbit (SSO) and spin-other orbit (SOO) contributions arise from the motion of the reference electron and the other electron, respectively, relative to the nuclear frame and are carried by the Coulomb and Gaunt terms, respectively. For most molecular application it suffices to include the Coulomb term only, thus defining the Dirac-Coulomb Hamiltonian, but for the accurate calculation of molecular spectra the Gaunt term should be included as well. 

In the nonrelativistic case much has been, and continues to be, learned about the outcome of nonadiabatic processes from the locus and topography of seams of conical intersection. It will now be possible to describe nonadiabatic processes driven by conical intersections, for which the spin-orbit interaction cannot be neglected, on the same footing that has been so useful in the nonrelativistic case. This fully adiabatic approach offers both conceptual and potential computational... 

In other cases, discussed below, the lowest electron-pair-bond structure and the lowest ionic-bond structure do not have the same multiplicity, so that (when the interaction of electron spin and orbital motion is neglected) these two states cannot be combined, and a knowledge of the multiplicity of the normal state of the molecule or complex ion permits a definite statement as to the bond type to be made. 

Orbital phase continuity in triplet state. The orbital phase properties are depicted in Fig. 5c. For the triplet, the radical orbitals, p and q, and bonding n (a) orbital are donating orbitals (labeled by D in Fig. 5c) for a-spin electrons, while the antibonding jt (a ) orbital (marked by A) is electron-accepting. It can be seen from Fig. 5c that the electron-donating (D) radical orbitals, p and q, can be in phase with the accepting (a ) orbital (A), and out of phase with the donating orbital, Jt/a (D) at the same time for the triplet state. So the orbital phase is continuous, and the triplet state of 1,3-diradical (e.g., TMM and TM) is stabilized by the effective cyclic orbital interactions [29, 31]. 

Configuration interaction, which is necessary in treatments of excited states and desirable in calculations of spin densities, is more complex with open-shell systems. This is because more types of configurations are formed by one-electron promotions. These configurations (Figure 5) are designated as A, B, Cq, C(3 G is the symbol for a ground state. Configurations C and Cp have the same orbital part but differ in the spin functions. 

Fig. 2.3. At left, energy levels for a Woods-Saxon potential with Vo — 50 MeV, R = 1.25A1/3 fm and a = 0.524 fm, neglecting spin-orbit interaction. At right, the same with spin-orbit term included. Adapted from Krane (1987).

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