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## III. HIGHER-ORDER ELECTRON-CORRELATION CORRECTIONS TO THE TRANSITION ENERGIES 

Electron-electron interaction within the basic principles of QED is described by exchange of virtual photons. The one-photon exchange leads to the operator 

<img src='content_image/360463.jpg'>

where D$_{µν}$ is the photon propagator, which in the Coulomb gauge is written as 

r$_{12}$ = | r 12 | = | r 1 − r 2 | , r i is the position vector of the i th electron, and α µ = (1 , α ) are the Dirac matrices. 

<img src='content_image/360464.jpg'>

Expanding expression (9) in powers of the photon frequency one can derive a simplified form of the interaction. The low-frequency limit of this interaction consists of two parts, referred to as the Coulomb and the Breit interaction, 

<img src='content_image/360465.jpg'>

The most traditional approach for the treatment of the electron-electron interaction in rel- ativistic many-electron atoms consists in using so-called Breit approximation. In this ap- proximation the total Hamiltonian can be represented as the sum of the one-electron Dirac Hamiltonians and the Coulomb and Breit electron-electron interactions, projected on the positive-energy Dirac’s states. In this way one gets the Dirac-Coulomb-Breit equation. 

Traditional methods for solving the Dirac-Coulomb-Breit equation are the many-body per- turbation theory (MBPT) [48, 49], the multi-configuration Dirac-Fock method [50], and the configuration-interaction (CI) method [3, 34]. All these methods treat the one-photon ex- change exactly and the higher-order electron correlation is accounted for within the Breit approximation only. 

The current level of experimental accuracy demands rigorous QED calculations of two- photon exchange contributions, which for n = 2 states of Li-like ions were performed in Refs. [6, 8–11, 13, 51]. Meanwhile rigorous QED calculations of three- and more photon exchange contributions have not been performed up to now. For high- Z few-electron ions 

