General remarks about amplitude equations

Slater-Condon rules yields the following working expression [5] 

The CC equations are considered as solved when the set of the amplitudes involved in the CC treatment has been determined. One of the methods of solving the CC equations is 

In the case of the full CCSD(F12) model, to obtain the solution of the CC equations the Ti, T2 conventional amplitudes and coefficients have to be determined. Compared to the conventional CCSD scheme, the explicit electron correlation requires an additional equation for the qj coefficients [see Eqs. (41), (47)]. The CCSD(F12) amplitude equations, obtained by projecting Eq. (129) onto the excitation manifolds, have the following form 

(130)-(132), derived with the assumption of the generalized Brillouin condition [31], entirely define the CCSD(F12) model. It is worth to mention that this model is an approximation to the full CCSD-F12 method [41], where more explicitly correlated terms 

The first commutator has been neglected in in both Eqs. (131) and (132), whereas the remaining two commutators were neglected only in the T2 equation. The removal of entire commutators assures the eize-extensivity of the CCSD(F12) energy [5, 41]. The Eqs. (130)-(132) are of a general form that is not yet suitable for the implementation. In the present work very often the expressions vector function and residual are used. They always refer to the many-index quantity, defined by the right hand sites of these equations. The working expressions of the coupled-cluster Ti, T2 and T2/ residuals are discussed in next subsections. 

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