Accounting for relativistic and correlation effects as corrections

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. 

One of the easiest ways to improve the results is through the replacement of the matrix element of the energy operator of electrostatic interaction by some effective interaction, in which, together with the usual expression of the type (19.29), there are also terms containing odd k values. This means that we adopt some effective Hamiltonian, whose matrix elements of the 

For pN shells the effective Hamiltonian Heff contains two parameters F2 and 4 i, as well as the constant of spin-orbit interaction. The term with k = 0 causes a general shift of all levels, which is usually taken from experimental data in semi-empirical calculations, and can therefore be neglected. The coefficient at 01 is proportional to L(L + 1). Therefore, to find the matrix elements of the effective Hamiltonian it is enough to add the term aL(L + 1) to the matrix elements of the energy of electrostatic and spin-orbit interactions. Here a stands for the extra semi-empirical parameter. 

For the fN shell we have to take into consideration terms containing expressions of the kind (5.36) and (5.37). As was shown in [127, 129], parameters a and / account for the superposition of all configurations which differ from ground lN by two electrons. If the admixed configurations differ from the principal one by the excitation of one electron, then we have to introduce one extra parameter T, the coefficient of which will be described by the matrix element of the tensorial operator of the type [Uk x Uk x Uk ]°, for which, unfortunately, there is no known simple algebraic expression. This correction is important only for N 3. 

introducing parameters a, / and T we can account for the essential part of the correlation effects. However, it turned out that in the framework of the semi-empirical approach, all relativistic corrections of the second order of the Breit operator improving the relative positions of the terms, are also taken into consideration (operators H2, and H s, described by formulas (1.19), (1.20) and (1.22), respectively). Indeed, as we have seen in Chapter 19, the effect of accounting for corrections Hj and H s in a general case may be taken into consideration by modifications of the integrals of electrostatic interaction, i.e. by representing them in form 

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