Spin-orbit interaction numerical calculations

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. 

As described in Sections II.A. 1 and II.A.4, the numerical procedures required to calculate term parameters and C terms induced by the perturbation of the transition moment by spin-orbit coupling are nearly identical. Almost all of the comments concerning the calculation of terms made in Section III.A.1 apply equally well to the spin-orbit-induced C terms. The major difference between the two types of calculation is that the spin-orbit interaction is often significantly larger than the influence of a magnetic field. 

Starting with the method described above, extensive tables of the numerical values of mean energies, integrals of electrostatic and constant of spin-orbit interactions are presented in [137] for the ground and a large number of excited configurations, for atoms of boron up to nobelium and their positive ions. They are obtained by approximation of the corresponding Hartree-Fock values by polynomials (21.20) and (21.22). Such data can be directly utilized for the calculation of spectral characteristics of the above-mentioned elements or they can serve as the initial parameters for semi-empirical calculations [138]. 

However, numerical estimates of the effect of frequency dependence of the coefficients of depolarization based on the calculation of molecular cross sections Gj(na>, Aw) are rather difficult. But it can be shown that in the cases when different Gj magnitudes are nonzero by symmetry selection rules (neglecting the spin-orbital interaction), they are of the same order of magnitude, since they are determined by the same energy denominators and reduced matrix elements of the operator of the dipole moment. 

This potential is more suitable in the numerical calculation because it does not lead to divergence (under r 0) of the spin-orbit interaction —25f(l,j)V lr. 

Early four-component numerical calculations of parity-violating effects in diatomic molecules which contain only one heavy nucleus and which possess a Si/2 ground state have been performed by Kozlov in 1985 [149] within a semi-empirical framework. This approach takes advantage of the similarity between the matrix elements of the parity violating spin-dependent term e-nuci,2) equation (114)) and the matrix elements of the hyperfine interaction operator. Kozlov assumed the molecular orbital occupied by the unpaired electron to be essentially determined by the si/2, P1/2 and P3/2 spinor of the heavy nucleus and he employed the matrix elements of e-nuci,2) nSi/2 and n Pi/2 spinors, for which an analytical expres-... 

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