Spin-orbit interaction model potential

Seijo [120] has performed relativistic ab initio model potential calculations including spin-orbit interaction using the Wood-Boring Hamiltonian. Calculations ere performed for several atoms up to Rn, and several dimer... 

Pacchioni has recently carried out calculations on the low-lying states of Sn2 and Pb2. This author gives the impression that he is the first to carry out a comparative ab initio Cl calculation on these systems. We would like to clarify this further. First, his calculation starts with the Hafner-Schwarz model potentials in comparison to our relativistic ab initio potentials derived from numerical Dirac-Fock solutions of the atoms. Pacchioni s calculations ignore spin-orbit interaction. Our calculations include spin-orbit interaction in a relativistic Cl scheme in comparison to the non-relativistic Cl of Pacchioni. Thus, he obtains a Z), approximately twice the experimental value which he corrects by a semi-empirical scheme to arrive at a value close to our calculated value with a relativistic Cl. Our calculations have clearly demonstrated the need to carry out an intermediate-coupling Cl calculation for Pbj as a result of large spin-orbit contamination. Calculations without spin-orbit, such as Pacchioni s, have little relationship to the real Pb2 molecule. 

In the nuclear shell model, the mutual interaction between the nucleons adds up to a singleparticle average potential consisting of a (spherical) central potential and a spin-orbit interaction. The nucleons are assumed to move independently in this potential. The solutions of the Schrbdinger equation yield single-particle energy levels and wave functions for the individual nucleons (Haxel et al. 1949 Goeppert-Mayer 1949). 

The most obvious criterion would be the use of suitable MO calculations and, in reality, in response to the accumulation of new experimental information, the theoretical investigation of f-element complexes has become a very active area. Highly sophisticated theoretical calculations certainly do not represent a routine exercise. Ab initio calculations [9] require a tremendous computational effort and, therefore approximation to introduce effective relativistic core potentials and spin-orbit interactions [10] would be highly desirable. Sophisticated calculations which treat uniformly all electrons are limited to the Xa model [11] applications of quasi-relativistic corrections [12] have been reported and errors associated with non-relativistic procedures analyzed [13]. Nevertheless, applications are still limited, although with a few exceptions, to highly symmetrical molecules. 

ABSTRACT. Calculation of the rate constant at several temperatures for the reaction +(2p) HCl X are presented. A quantum mechanical dynamical treatment of ion-dipole reactions which combines a rotationally adiabatic capture and centrifugal sudden approximation is used to obtain rotational state-selective cross sections and rate constants. Ah initio SCF (TZ2P) methods are employed to obtain the long- and short-range electronic potential energy surfaces. This study indicates the necessity to incorporate the multi-surface nature of open-shell systems. The spin-orbit interactions are treated within a semiquantitative model. Results fare better than previous calculations which used only classical electrostatic forces, and are in good agreement with CRESU and SIFT measurements at 27, 68, and 300 K. ... 

The spin-orbit operators for the model potential and pseudopotential approximations are one-electron operators. These operators include the effect of the two-electron spin-orbit interaction used in the mean-field approximation to derive the model potential or pseudopotential. Molecular calculations with these potentials therefore include, at least at the atomic level, the two-electron spin-orbit terms. This is just the kind of approximation we are looking for. 

Fig. 5 Schematic representation of the states in the nuclear shell model. The oscillator shells on the left are first split into the individual subshells by deviations of the nuclear potential from the harmonic oscillator, before the spin-orbit interaction creates the groupings of states that produce the correct magic numbers above N = Z = 20. The diagram is schematic and not to scale...

Fig. 6 Schematic representation of the shell model potential and the spin-orbit interaction (top) usually taken as proportional to the derivative of the potential, illustrated via a Woods-Saxon potential (bottom) with a nuclear radius Ro and a surface diffuseness a. A few nuclear levels inside the potential are schematically indicated. It is then obvious that the spin-orbit interaction mainly acts near the surface of the nucleus...

Here another source of a conceptual problem of the seeond order-approach appears. The standard formulation of the J-O theory, even if extended by the dynamic coupling model, is based on the single configuration approximation. This means that in such a description all the eleetron correlation effects are neglected and it is well known that the transition amplitude strongly depends on them. At this point also the spin-orbit interactions should be taken into consideration as possibly important in the description of the spectroscopic patterns of the lanthanides. In the case of all of these possibly important physical mechanisms there is a demand for an extension of the standard Judd-Ofelt formulation. The transition amplitude in equation (10.17) has to be modified by the third-order contributions that originate from various perturbing operators introduced in addition to the crystal field potential that plays a... 

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