Elements of the Energy Operator

In order to find expressions for relativistic matrix elements of the energy operator we have to utilize the following formula for two-particle scalar operators ... 

Let us present in conclusion the expression for the submatrix element of scalar product (7.3), necessary while calculating relativistic matrix elements of the energy operator. It is as follows ... 

Usually, the first way is utilized in practice. This is due to the well developed mathematical technique necessary, by the presence of the expressions for both the matrix elements of the energy operator and of the electronic transitions in various coupling schemes. However, the second method is much more universal and easier to apply, provided that there are known corresponding transformation matrices. Now we shall briefly describe this method. 

Matrix elements of the energy operator in the case of a shell lN... 

The formula for a matrix element of the energy operator of magnetic interactions (its irreducible form is presented by Eq. (19.66)) can be found in a similar way to that for the Coulomb interaction. It is... 

Accounting for the properties of the seniority (quasispin) quantum numbers, we are able to express the matrix elements of the energy operator in terms of the corresponding quantities for the electronic configuration, for which this term has occurred for the first time (/f ajV L/S — ... 

The direct part of the matrix element of the energy operator of the retarding interactions vanishes... 

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... 

Let us also mention that using a number of functional relations between the products of 3n/-coefficients and submatrix elements (/ C(k) / ), the spin-angular parts of matrix elements (26.1) and (26.2) are transformed to a form, whose dependence on orbital quantum numbers (as was also in the case of matrix elements of the energy operator, see Chapters 19 and 20) is contained only in the phase multiplier. In some cases this mathematical procedure is rather complicated. Therefore, the use of the relativistic radial orbitals, expressed in terms of the generalized spherical functions (2.18), is much more efficient. In such a representation this final form of submatrix element of relativistic Ek-radiation operators follows straightforwardly [28]. 

Accounting for the properties of the seniority (quasispin) quantum numbers, we are able to express the matrix elements of the energy operator in terms of the corresponding quantities for the electronic configuration, for which this term has occurred for the first time (/, — /, " a,d,L,S,). This question is considered in detail in [102], The utilization of the relationship between the CFP and the submatrix elements of the operators, composed of unit tensors, which was established in [105], allows one to find a large number of new expressions for the above-mentioned matrix elements. 

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