Theoretical study on the core-excited states of the allyl using multi-reference methods with core–valence separation (CVS) approximation

The multi-state n-electron valence second-order perturbation theory with core-valance separation (CVS) approximation (CVS-MS-NEVPT2) and static-dynamic-static multi-state multi-reference second-order perturbation theory with CVS (CVS-SDSPT2) were developed based on the internally contracted multi-reference configuration interaction method with single and double excitations with CVS (CVS-icMRCISD) [Song et al., J. Chem. Phys. 160, 094114 (2024)] due to their high similarity and inheritance in theoretical frameworks and computational implementation. Benchmark calculations demonstrate that these perturbation methods significantly improved computational efficiency while maintaining comparable accuracy to the CVS-icMRCISD method. Moreover, the core-excited states of the allyl system were used as a pilot application for three CVS-multi-reference methods. The CVS-icMRCISD method produced four excited states at 282.10, 285.11, 286.18, and 288.07 eV, accurately reproducing four distinctive peaks (A, B, C, and D) in the experimental x-ray absorption spectra (XAS). Theoretical excitation energies of two core-excited states of the allyl cation (282.64 and 286.88 eV) align with the peaks observed at 282.52 and 286.92 eV in the experimental XAS. However, vibrational analysis of the X-A transition suggests that the α band in the experimental XAS might arise from electronic-vibronic coupling between the antisymmetric stretching of C-H bonds in the -CTH2 group and the core-excited state, leaving the assignment of the α band ambiguous. Overall, the CVS-icMRCISD method achieved excellent agreement with experimental results, with an average deviation of 0.25 eV. While the errors of CVS-SDSPT2 and CVS-MS-NEVPT2 methods were slightly larger, both methods maintained acceptable accuracy, making them suitable for medium-sized molecules by significantly reducing computational costs.