The EOM-CC eigenvalue problem

We now turn our attention to the EOM-CC eigenvalue problems (13.6.14) and (13.6.15). Wth each eigenvalue, there is an associated pair of right and left eigenvectors, which we write in the following way... 

Viewed as a Cl-like eigenvalue problem of H, the EOM-CC equation must have its Hermitian conjugate counterpart ... 

In the PCM-EOM-CC/SACCI approximation the excited electronic states are represented by a linear (Cl-like) expansion build-up on the coupled-cluster wavefunction for the ground state [6-8]. We use the PTE couple-cluster wavefunction, computed in the presence of the frozen Hartree-Fock reaction field, as it leads to a more simpler and physically transparent PCM-EOM theory. The EOM-CC theory leads to a non-Hermitian eigenvalue problem with right and left eigenvalues. 

Overbars are used for the bra coefficients since they are numerically different from the ket coefficients. The EOM-CC states are now determined by applying the variation principle. Differentiation with respect to the ket and bra coefficients yields the unsymmetric eigenvalue problems (assuming real expansion coefficients)... 

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. 

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