Electrostatic correlated spin-orbit interaction

M k = 0, 2, and 4. They also considered the effect of additional configurations on the spin-orbit interaction to produce the electrostatically correlated spin-orbit interaction. 

Model Hartree-Fock calculations which include only the electrostatic interaction in terms of the Slater integrals F0, F2, F and F6, and the spin-orbit interaction , result in differences between calculated and experimentally observed levels596 which can be more than 500 cm-1 even for the f2 ion Pr3. However, inclusion of configuration interaction terms, either two-particle or three-particle, considerably improves the correlations.597,598 In this way, an ion such as Nd3+ can be described in terms of 18 parameters (including crystal field... 

In a single-configurational non-relativistic approach, the integrals of electrostatic interactions and the constant of spin-orbit interactions compose the minimal set of semi-empirical parameters. Then for pN and dN shells we have two and three parameters, respectively. However, calculations show that such numbers of parameters are insufficient to achieve high accuracy of the theoretical energy levels. Therefore, we have to look for extra parameters, which would be in charge of the relativistic and correlation effects not yet described. 

Most of the variational treatments of spin-orbit interaction utilize one-component MOs as the one-particle basis. The SOC is then introduced at the Cl level. A so-called SOCI can be realized either as a one- or two-step procedure. Evidently, one-step methods determine spin-orbit coupling and electron correlation simultaneously. In two-step procedures, typically different matrix representations of the electrostatic and magnetic Hamiltonians are chosen. 

Corrections to the central-field approximation. Two important terms are omitted in the central-field approximation. The first is the residual electrostatic interactions represented by JC (equation (3.68)). This gives rise to non-central forces which cause the motion of the electrons to be correlated with one another, rather than completely independent as in the central-field approximation. The second terra is the magnetic spin-orbit interaction which was not included in equation (3.65) ... 

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