Coupled cluster vector function

The coupled cluster amplitude equations are often collectively called the coupled cluster vector function e with elements. 

In the case of the coupled cluster wavefunction the equations for the wavefunction parameters, i.e. for the coupled cluster amplitudes are simply the equations for the coupled cluster vector function in Eq. (9.81). The constraints are then = 0 and the coupled cluster Langrangian (Christiansen et ai, 1995a, 19986) is given as... 

The Jacobian matrix A can be shown [see Exercise 11.9) to be the first derivative of the time-independent coupled cluster amphtude equations, i.e. the coupled cluster vector function e, Eq. (9.81), with respect to the time-independent amplitudes... 

We shall now set up a coupled-cluster optimization scheme that avoids the solution of the linear equations (13.4.5) in each iteration. In the canonical representation, we may invoke (13.2.44) to write the coupled-cluster vector function (13.4.3) and its Jacobian (13.4.4) of the nth iteration in the form... 

The key assumption underlying the calculation of excitation energies as eigenvalues of the Jacobian (13.6.41) and (13.6.42) is that the first column vector of the EOM-CC Hamiltonian (13.6.33) vanishes. This assumption is satisfied for the standard coupled-cluster models such as CCSD and CCSDT, for which the first column contains the coupled-cluster vector function (13.2.23). By contrast, in other models, where certain terms are omitted as in quadratic Cl (discussed in Section 13.8.2), the first column does not vanish and EOM-CC theory can no longer be applied. 

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. 

Consider the left-hand side of the coupled-cluster amplitude equations (13.2.23). Its elements constitute a vector function... 

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