Interaction Hamiltonian spin-orbit

Reference values for the various 2-component relativistic Hamiltonians are provided by the 4-component Dirac-Coulomb Hamiltonian, but we have also included orbital energies obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian. As already mentioned, the Gaunt term brings in spin-other-orbit (SOO) interaction. Since spin-orbit interaction induced by other electrons will oppose the one induced by nuclei we see from Table 3.3 that the spin-orbit splitting of orbital levels is overall reduced. However, one should note that the Gaunt term also modifies /2 levels. 

Both the orbital motion and the intrinsic spin of electron have a magnetic dipole associated with it. The two magnetic dipoles may interact with one another. This feature of atomic and molecular structure is called spin-orbit interaction. The spin-orbit interaction is a coupling of the two different motions of spin of the electron. The Hamiltonian that describes this interaction is a dot product of the angular momentum vectors of the two different types of spin motions of the electron. The proportionality constant is a which can be measured by spectroscopy. 

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

Let us return to the nonadiabatic chemical processes. When a PES has been built, a part of the total Hamiltonian may remain unaccounted for, and this part, acting as a perturbation, induces transitions from the initial to the final state. There are several types of such a perturbation, namely (i) an unaccounted part of the electronic interaction (ii) non-adiabaticity (iii) spin-orbit coupling. 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

The spin-orbit Hamiltonian (HB0) requires some explanation. The energy of interaction between the magnetic moment M and the magnetic field caused by the orbital motion of an electron can be derived as(134)... 

We now will show that spin-orbit coupling can give a spin Hamiltonian term identical to that we obtained from the electron dipolar interaction. Consider the... 

Hamiltonian with the energy from appropriate terms in the true Hamiltonian. The latter terms include the interaction between the external field and the magnetic moment produced by the orbiting electron, the interaction between the external field and the magnetic moment due to electron spin, and the interaction between the orbital magnetic moment and the spin magnetic moment. These interactions may be expressed as a perturbation to the total Hamiltonian for the system where... 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

Electronic Hamiltonian, conical intersections, spin-orbit interaction, 559 Electronic states ... 

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