Alternative Transformation for First-Order Energy

In the literature, however, it has most often been suggested to employ perturbation-dependent identity operators (l = U (A)U(A)) for the calculation of within the two-component DKH scheme based on the left-hand 

The alternative approach relies on the insertion of perturhation-dependent identity operators, which will cast the linear response expression given by Eq. (15.7) into the form 

In contrast to the direct DKH transformation in section 15.3.3.1, the alternative transformation with U = U(A) gives rise to additional terms which are not present in Eq. (15.60). In particular, terms containing energy-damped property operators X are present in Eqs. (15.73) and (15.74) and lead to much more complicated expressions compared to the direct transformation with U = U V). For example, already the first-order even term of the transformed property operator is modified as compared to Eq. (15.65) and reads 

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