Penalty functions for combining coupled-cluster and perturbation amplitudes in local correlation methods with optimized orbitals

Local active space correlation models based on the coupled-cluster doubles (CCD) model like Generalized Valence Bond Perfect Pairing (GVB-PP) and Imperfect Pairing (IP) are attractive methods for treating electron correlation, because they are computationally inexpensive and can describe strong correlations. However, they suffer from symmetry-breaking (SB) in systems with multiple resonance structures, which arises due to neglected correlations. We investigate the extent to which these problems can be removed by using second-order perturbation theory (PT) for weak correlations coupling three different electron pairs, and (infinite-order) coupled-cluster (CC) theory for stronger correlations involving electrons in only one or two pairs. The resulting Three-Pair Corrected Imperfect Pairing (TIP) method is explored here, and it is shown that to robustly combine CC and PT it is necessary to modify several aspects of the basic method. Most importantly, a penalty function term is introduced to ensure the PT amplitudes remain small. Comparison against CC treatment of the three-pair correlations suggests penalty terms will be beneficial for any hybrid CC/PT method that includes orbital optimization. The TIP method greatly reduces SB in aromatic hydrocarbons and recovers a significantly higher fraction of the valence electron correlation energy than IP.