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successfully used for calculations of the recoil effect in Refs. [14, 34]. The MS is calculated as the expectation value of the recoil operator with the many-electron Dirac wave function. Additionally, we apply an alternative approach which consists in adding the operator H$_{M}$ (2) to the many-electron Hamiltonian H with an arbitrary coefficient λ 

and evaluating the MS by 

Here the derivative is determined numerically and λ is chosen obeying the numerical stability and smallness of the nonlinear terms. We have reformulated the CI-DFS method to adopt the alternative scheme and independently evaluated the normal and specific parts of the MS by both methods. 

## C. Results of the calculations and discussion 

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Here we examine our calculations of the mass shift coefficient K in Li-like ions and compare them with the related results obtained by other authors. In Tables I, II, III, and IV we present numerical results for the coefficient K calculated for the 2 p$_{1}$$_{/}$$_{2}$ − 2 s and 2 p$_{3}$$_{/}$$_{2}$ − 2 s transitions in lithium, Li-like zinc, neodymium, and uranium, respectively. The first line shows the contribution obtained employing the MS operator (2). The entries labeled “NMS”, “SMS”, “RNMS” and “RSMS” represent the corresponding contributions of the mass shift operators. Since the expectation values of the NMS and SMS operators are evaluated with the Dirac wave functions, the values denoted by NMS and SMS in the tables partly contain the relativistic contributions. The values marked by “nr” show the nonrelativistic values of the corresponding contributions, obtained within the same computing procedure but with a 1000-times increased value of the speed of light (in atomic units). We have verified this nonrelativistic limit by comparing our values with the results of the fully nonrelativistic method based on the Schr¨ odinger Hamiltonian and on the same calculation scheme. The values have exactly coincided with each other for all the ions under consideration. To demonstrate the importance of the electron-electron interaction effects we present also the related results obtained with the hydrogenlike wave functions. These values are marked as “hyd” in the tables. Obviously, the CI-DFS approach is not the best for very low- Z Li- 

