Quasi-degenerate extension of local N-electron valence state perturbation theory with pair-natural orbital method based on localized virtual molecular orbitals

Chemical phenomena involving near-degenerate electronic states, such as conical intersections or avoided crossing, can be properly described using quasi-degenerate perturbation theory. This study proposed a highly scalable quasi-degenerate second-order N-electron valence state perturbation theory (QD-NEVPT2) using the local pair-natural orbital (PNO) method. Our recent study showed an efficient implementation of the PNO-based state-specific NEVPT2 method using orthonormal localized virtual molecular orbitals (LVMOs) as an intermediate local basis. This study derived the state-coupling (or off-diagonal) terms to implement QD-NEVPT2 in an alternative manner to enhance efficiency based on the internally contracted basis and PNO overlap matrices between different references. To facilitate further acceleration, a local resolution-of-the-identity (RI) three-index integral generation algorithm was developed using LMOs and LVMOs. Although the NEVPT2 theory is considered to be less susceptible to the intruder-state problem (ISP), this study revealed that it can easily suffer from ISP when calculating high-lying excited states. We ameliorated this instability using the imaginary level shift technique. The PNO-QD-NEVPT2 calculations were performed on small organic molecules for the 30 lowest-lying states, as well as photoisomerization involving the conical intersection of 1,1-dimethyldibenzo[b,f] silepin with a cis-stilbene skeleton. These calculations revealed that the PNO-QD-NEVPT2 method yielded negligible errors compared to the canonical QD-NEVPT2 results. Furthermore, we tested its applicability to a large photoisomerization system using the green fluorescent protein model and the ten-state calculation of the large transition metal complex, showcasing that off-diagonal elements can be evaluated at a relatively low cost.