Exploring New Construction Schemes for Extended-Hierarchy Configuration-Interaction Wave Functions

Following the work of Kossoski et al. on hierarchy CI (hCI) we propose an extended way of partitioning the Hilbert space by combining the excitation and the seniority sectors in a more general way. We define the hierarchy parameter, h, involving two "weights" (α values), measuring the importance of excitation (e) and seniority (s) contributions to the wave function according to h = α₁e + α₂s. This formulation generates alternative orderings of the excitation-seniority lattice and enables a systematic examination of how different balances between excitation and seniority influence correlation recovery. Four partitions are considered: positive slope diagonals (PSDs), which is equivalent to the original hCI scheme; negative slope diagonals (NSD); vertical chess horse (VCH); and horizontal chess horse (HCH). These partitions are evaluated for the BeH₂ and for the cubic and linear H₈ dissociation. These results show that the efficiency of a partition depends on the specific excitation-seniority sectors it includes at a given number of determinants. For BeH₂, the PSD/hCI hierarchical ordering (h = 2) incorporates the dominant low-seniority configurations most effectively and reaches near-CISDT quality with compact expansions compared to other partition schemes. For the H₈ systems, NSD and HCH recover static correlation more rapidly in the dissociation regime, whereas VCH remains inefficient across all determinant counts. The present results, obtained in the STO-6G basis for small benchmark systems, are intended as a controlled methodological study of determinant ordering strategies rather than a comprehensive performance assessment. The ehCI framework provides a practical way to analyze how excitation and seniority interact and to design compact CI expansions.