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Evaluation of the two-photon exchange diagrams for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mn /><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mn /></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></…
We present ab initio calculations of the complete gauge-invariant set of two-photon exchange graphs for the ${(1s)}^{2}{2p}_{3/2}$ electron configuration in Li-like ions. These calculations are an important step towards the precise theoretical determination of the ${2p}_{3/2}\ensuremath{-}2s$ transition energy in the framework of QED.