State-specific perturbation theory for configuration interaction singles: Theoretical analysis, benchmarks, and improvement with size-intensive level shifts

We recently proposed a state-specific second-order perturbation theory based on the configuration interaction singles (CIS) state, termed CIS(2). Here, we clarify its theoretical connections, assess the accuracy of its excitation energies in greater detail, and develop size-intensive level shifts to improve both accuracy and convergence of its amplitude equations. We first derive the MP2 structure of its working equations and show that our previously introduced Hybrid partition recovers CIS(D) when the ground-state Fock operator is used and the residual Fock action on the CIS reference is neglected. For this partition, an expanded benchmark shows that using the CIS-state Fock operator improves Rydberg and especially charge-transfer accuracy relative to CIS(D), while slightly worsening valence accuracy. We analyze how the hole and particle densities contribute to these changes in excitation energy, which differ among the classes. We then introduce size-intensive level shifts that preserve the ground-state correlation contribution of noninteracting fragments while leaving the ground-state MP2 energy unchanged. With appropriately calibrated parameters, these shifts improve overall accuracy and substantially accelerate convergence without increasing the formal scaling of the amplitude solve. Different optimal parameters for valence and Rydberg states reveal the limitations of a common parameter. The analysis further shows that shifting the contracted-triples block contributes markedly to the excitation-energy correction in addition to accelerating convergence.