Perturbation Expansion of Internally Contracted Coupled-Cluster Theory up to Third Order

The internally contracted multireference coupled-cluster (icMRCC) method is analyzed through third order in perturbation theory. Up to second order, the icMRCC perturbation expansion is equivalent to that of the standard Rayleigh-Schrödinger perturbation theory, which is based on a linear ansatz for the wave function, and the resulting theory is, depending on the employed zeroth-order Hamiltonian, equivalent to either second-order complete active space perturbation theory (CASPT2), N-electron valence perturbation theory (NEVPT2), or Fink's retention of the excitation degree perturbation theory (REPT2). At third order, the icMRCC perturbation expansion features additional terms in comparison to the Rayleigh-Schrödinger perturbation theory, but these are shown to be nearly negligibly small by both analytic arguments and numerical examples. Considering these systematic cancellations, however, may be important in future work on approximations to icMRCC theory. In addition, we provide an extensive set of tests of the second and third-order perturbation theories based on three different zeroth-order Hamiltonians, namely, the projected effective Fock operator as used for CASPT, the Dyall Hamiltonian as used for NEVPT, and the Fink Hamiltonian used for REPT. While the third-order variant of REPT often gives absolute energies that are rather close to values from higher level calculations, the results for relative energies and spectroscopic constants such as harmonic frequencies, give a less clear picture and a general conclusion about any best zeroth-order Hamiltonian does not emerge from our data. For small active spaces, REPT is rather prone to intruder state problems.