III. HIGHER-ORDER ELECTRON-CORRELATION CORRECTIONS TO THE TRANSITION ENERGIES 13 Electron-electron interaction within the basic principles of QED is described by exchange of virtual photons. The one-photon exchange leads to the operator 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. 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, 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 I ( ω ) = e $^{2}$α µ $_{1}$α ν $_{2}$D$_{µν}$ ( ω, r $_{12}$) , (8) D$_{00}$ ( ω, r $_{12}$) = 1 4 πr$_{12}$ , D$_{i}$$_{0}$ = D$_{0}$$_{i}$ = 0 ( i = 1 , 2 , 3) , D$_{il}$ ( ω, r $_{12}$) = ∫ d k (2 π ) 3 exp( i k · r $_{12}$) ω 2 − k 2 + i 0 ( δ$_{il}$ − k$_{i}$k$_{l}$ k 2 ) ( i, l = 1 , 2 , 3) , (9) V ( i, j ) = V$_{C}$ ( i, j ) + V$_{B}$ ( i, j ) = α r$_{ij}$ − α [ α i · α j 2 r$_{ij}$ + ( α i · r$_{ij}$ )( α j · r$_{ij}$ ) 2 r 3 ij ] . (10)