Jacobian coupled cluster

Eq. (9.72), and A state, Eq. (9.96) wavefunctions, respectively, and T is the time-independent, unperturbed cluster operator. The elements of the coupled cluster Jacobian A matrix and of the F matrix are defined 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... 

Exercise 11.9 Derive the coupled cluster Jacobian in Eq. (11.75) as a derivative of the coupled cluster amplitude equations, i.e. prove Eq. (11.77). 

In coupled cluster response theory the poles of the hnear response function and thus the vertical excitation energies are then found as eigenvalues of the coupled cluster Jacobian... 

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 calculation of the zero-order multipliers and Lagrangian requires the solution of one set of linear equations. The resulting expression for the coupled-cluster energy is then variational with respect to the amplitudes as well as their multipliers. Note that the linear equations for the multipliers (13.5.6) are similar in structure to the Newton equations for the amplitudes (13.4.5), both containing the coupled-cluster Jacobian (13.4.4). 

The unsymmetric Jacobian matrix has previously appeared in the optimization of the coupled-cluster wave function (13.4.4) and in the calculation of the coupled-cluster Lagrange multipliers (13.5.6). 

The EOM-CC excitation eneigies therefore correspond to the eigenvalues of the coupled-cluster Jacobian matrix A. Since the Jacobian is unsynimetric, there is no mathematical guarantee that the calculated eigenvalues are real. In practice, however, this is not a problem and the calculated excitation energies are real for any reasonably accurate ground-state wave function. 

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. 

Having developed EOM-CC theory, we now turn to the crucial question of size-extensivity. The ground-state energy corresponds to the itsual coupled-cluster energy and is necessarily size-extensive. We therefore consider only the excited-state energies, for which it is sufficient to analyse the eigenvalues of the Jacobian matrix (13.6.28) [19]. 

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