Exact decomposition and approximate power-law scaling of truncation errors in multiconfigurational wave functions

Although truncation errors in multiconfigurational expansions are known empirically to correlate with the discarded weight, a rigorous, scheme-independent theoretical foundation for this relation is lacking. From an orthogonal decomposition of the full configuration interaction (FCI) wave function, we derive an exact expression ΔE = ω(Ed - EFCI)/(1 - ω) and validate it across 20 systems, four orbital representations, and four truncation schemes. In the weak-truncation regime (ω ≪ 1), the mean discarded excitation energy varies slowly, yielding an approximate power law ΔE ∝ ωα, where α depends systematically on the truncation strategy (0.86-0.91) and electronic structure, yet the error remains dominated by ω. Because ω requires the exact FCI wave function, it serves as an a posteriori diagnostic. This framework provides a unified theoretical basis for comparing truncation and orbital schemes and rationalizes the empirical success of coefficient-based selection methods.