Linear Scaling Calculations of Excitation Energies with Active-Space Particle-Particle Random Phase Approximation

We developed an efficient active-space particle-particle random phase approximation (ppRPA) approach to calculate accurate charge-neutral excitation energies of molecular systems. The active-space ppRPA approach constrains both indexes in particle and hole pairs in the ppRPA matrix, which only selects frontier orbitals with dominant contributions to low-lying excitation energies. It employs the truncation in both orbital indexes in the particle-particle and the hole-hole spaces. The resulting matrix, the eigenvalues of which are excitation energies, has a dimension that is independent of the size of the systems. The computational effort for the excitation energy calculation, therefore, scales linearly with system size and is negligible compared with the ground-state calculation of the (N-2)-electron system, where N is the electron number of the molecule. With the active space consisting of 30 occupied and 30 virtual orbitals, the active-space ppRPA approach predicts excitation energies of valence, charge-transfer, Rydberg, double and diradical excitations with the mean absolute errors (MAEs) smaller than 0.03 eV compared with the full-space ppRPA results. As a side product, we also applied the active-space ppRPA approach in the renormalized singles (RS) T-matrix approach. Combining the non-interacting pair approximation that approximates the contribution to the self-energy outside the active space, the active-space $G_{\text{RS}}T_{\text{RS}}$@PBE approach predicts accurate absolute and relative core-level binding energies with the MAE around 1.58 eV and 0.3 eV, respectively. The developed linear scaling calculation of excitation energies is promising for applications to large and complex systems.