Assessing Second-Order Perturbative Corrections to Restricted Active Space CI for Valence Excitations in Organic Molecules

We benchmark second-order perturbative corrections to the Restricted Active Space Configuration Interaction in the hole and particle approximation, RAS(<i>h</i>,<i>p</i>), for valence singlet and triplet excitations in a set of organic molecules. Two partitioning schemes, Epstein-Nesbet (EN) and Davidson-Kapuy (DK), were assessed against NEVPT2 and reference data from the literature. The lack of dynamic correlation in RAS(<i>h</i>,<i>p</i>) leads to a systematic overestimation of singlet excitation energies by ∼0.8 eV, with much smaller errors for triplets. EN perturbation provides only marginal improvement and increases statistical scatter, whereas DK yields a more consistent correction but tends to overcompensate. Introducing an energy level shift in the DK partition effectively removes this bias, producing excitation energies of comparable accuracy to NEVPT2. An optimal shift of ε ≈ 0.55 a.u. for singlets and slightly larger values for triplets was found to be broadly transferable across the data set, with ε in the range 0.4-0.6 a.u. offering a robust compromise. Analysis of second-order contributions shows that the dominant 1<i>h</i>1<i>p</i> terms act synergistically with the variational singles, resembling a state-specific orbital relaxation, while the more expensive 1<i>h</i>2<i>p</i> and 2<i>h</i>2<i>p</i> terms have a minor impact, suggesting a route to cheaper, targeted perturbative schemes. DK-based corrections exhibit weak basis-set dependence, enabling efficient composite strategies in which large-basis RAS(<i>h</i>,<i>p</i>) energies are combined with small-basis DK+shift corrections, achieving significant computational savings with minimal loss of accuracy.